Dependencies

Let \(\pi :(\alpha ,T,\mu )\to (\beta ,S,\nu )\) be a factor map of measure-preserving systems on probability spaces (\(\alpha \) standard Borel), and let \(P, R\) be finite partitions of the total and factor systems, supplied with the generator reductions \(h(T)=h(T,P)\), \(h(S)=h(S,R)\), the relative-generator reduction \(h(T\mid \pi ^{-1}\mathcal B_\beta )=h(T,P\mid \pi ^{-1}\mathcal B_\beta )\), and the partition-level identity of Theorem 10.20 for \(P\) over \(\pi ^{-1}R\). Then the entropy of the total system splits as

the factor entropy plus the fibrewise entropy relative to the pulled-back factor \(\sigma \)-algebra \(\pi ^{-1}\mathcal B_\beta =\operatorname {comap}\pi \, m_\beta \).

Let \(T\) preserve \(\mu \) on a standard Borel probability space, let \(\mathcal A\le m_\alpha \) be forward invariant, and let \(P, Q\) be finite partitions such that (i) each cell of the \(n\)-fold \(P\)-join lies \(\mu \)-a.e. inside a single cell of the \(n\)-fold \(Q\)-join (a refinement witness \(g\)), and (ii) the join filtration of \(Q\) saturates \(\mathcal A\): \(\bigvee _n \sigma \bigl(\bigvee _{k{\lt}n}T^{-k}Q\bigr)=\mathcal A\). Then

The saturation hypothesis is essential: the identity is false without it.

If \(g\) is an ultrametric growth function and \(g(v)\neq g(w)\) (with \(v,w,v+w\) nonzero), then \(g(v+w)=\max (g(v),g(w))\).

For a permutation \(\sigma \in \mathfrak {S}_d\), the permutation dynamics \(\mathrm{adPerm}\, \sigma \) is conjugation \(x\mapsto P_\sigma \, x\, P_\sigma ^{\dagger }\) by the permutation matrix, a unital \(*\)-endomorphism; on diagonal matrices it acts by \((\mathrm{adPerm}\, \sigma )(\operatorname {diag}v)=\operatorname {diag}(v\circ \sigma )\). Paired with a \(\sigma \)-invariant diagonal state \(\rho _\mu \) (\(\mu \circ \sigma =\mu \), Definition 17.27) and a projection partition (Definition 17.28), this is the abelian corner: it mirrors the classical system \((\operatorname {Fin}d,\mu ,\sigma )\) with the measurable partition \(c^{-1}(\cdot )\).

For a finite measure, a measure-preserving \(T\) and integrable \(g\), the Birkhoff averages \(n \mapsto \texttt{birkhoffAverage}\, \mathbb {R}\, T\, g\, (n+1)\, x\) are a.e. bounded above (and, applied to \(-g\), a.e. bounded below).

For all \(a, b \in \mathbb {R}\) with \(a + b {\lt} 0\),

For a.e. \(x\), for every forward level \(i\) and backward level \(s\) with \(\mathrm{expEnum}\, \mathrm{lam0}\, d\, i + \mathrm{expEnum}\, \mathrm{mu0}\, d\, s {\lt} 0\),

For a finite measure, measure-preserving \(T\) and an a.e. measurable \(F\) with \(F\, x \le F\, (Tx)\) for a.e. \(x\), one has \(F \circ T =^{\text{a.e.}} F\).

Under the same hypotheses, for a.e. \(x\) the EReal limsup of the normalized cocycle is dominated by its liminf.

For a.e. \(x\) and all \(t \in \mathbb {R}\),

Let \( V \) be the Oseledets filtration of the time-one data, and fix \( t_0 \in \mathbb {R}\). Then for almost every \( x \) and every level \( i \),

For \(\mu \)-a.e. \(x\) and every threshold index \(k\in [1,d)\), the block term \((e^{\lambda ^0_{k-1}}-e^{\lambda ^0_k})\cdot P^{c_k}_n(x)\) converges, where \(P^{c}_n\) is the band projector \(\mathbf1_{(c,\infty )}(q_n(x))\).

For \(\mu \)-a.e. \(x\) and every \(t\), \(\mathrm{lambdaSublevel}(x,t) \le \mathrm{vslow}(x,e^t)\).

For a measure-preserving \(T\) and a finite partition indexed by a nonempty \(\iota \),

Neither ergodicity nor any ergodic theorem enters: this is the Algoet–Cover engine fed the uniform competing measure, whose partition-function bound \(\int \exp \bigl(i_n - n\log \# \iota \bigr)\, d\mu \le 1\) is the lemma .

For a.e. \(x\),

Under the same hypotheses, \(\bar d_\mu (x) = \operatorname {finrank}_{\mathbb {R}} E\) for \(\mu \)-almost every \(x\): the measure is exact-dimensional with dimension equal to the ambient dimension.

If \(\mathrm{IsOseledetsFiltration}\ \mu \ T\ A\ k\ \lambda \ V\) holds, then \(\mu \)-a.e. each interior flag level is exactly a growth-sublevel set: for every \(i\) and vector \(v\),

For \(\mu \)-a.e. \(x\), every \(c{\gt}0\) that is not one of the limiting eigenvalues \(e^{\lambda _{\mathrm{sing}}(x,i)}\) of \(\Lambda (x)\) satisfies

For a finite measure, measure-preserving measurable \(T\), an integrable subadditive cocycle with normalized integrals bounded below, there is an integrable \(G\) with \(\texttt{cdiv}\, g\, n\, x \to G\, x\) for \(\mu \)-a.e. \(x\).

Let \(T\) be ergodic and let \((i_n)\) be any sequence of functions satisfying, for \(\mu \)-a.e. \(x\) and all \(n\), the Breiman telescoping identity

where \(g_k\) is the conditional information function of \(P\) given \(\mathcal C_k\). Then

Let \(T\) be an ergodic measure-preserving transformation of a standard Borel probability space and \(P\) a finite measurable partition (nonempty index). Then the concrete information functions of Definition 11.1 satisfy

the measure of the atom of \(x\) in the \(n\)-fold join decays like \(e^{-n\, h(P,T)}\) (entropy equipartition, the AEP of ergodic theory).

For integrable \( g, g' \) and a measure-preserving flow \( \varphi \), for almost every \( x \) the combined fluctuation along the integer orbit vanishes:

Let \(E\) be a finite-dimensional real inner-product space (Borel-measurable) and let \(\mu \) be a probability measure on \(E\) absolutely continuous with respect to an additive Haar measure \(\nu \) (e.g. Lebesgue). Then for \(\mu \)-almost every \(x\) the local-dimension quotient \(\log \mu (\bar B(x,r)) / \log r\) converges, as \(r \to 0^+\), to the ambient dimension \(\dim _{\mathbb {R}} E = \operatorname {finrank}_{\mathbb {R}} E\).

Fix a real time \( t_0 \). For almost every \( x \), both \( n^{-1}\log \left\lVert A(t_0, \varphi (n)x) \right\rVert \) and \( n^{-1}\log \left\lVert (A(t_0, \varphi (n)x))^{-1} \right\rVert \) tend to \( 0 \) as \( n \to \infty \).

For a measure-preserving \(T\) and integrable \(g\), the orbital tail \(n^{-1}\, g(T^{[n]}x)\) tends to \(0\) for \(\mu \)-a.e. \(x\).

The inclusion \(A_n\hookrightarrow A_{n+1}\) appends a fresh qubit at the far end, \(x\mapsto x\otimes 1\), reindexed along \(\mathrm{Equiv.prodComm}\) because \(\mathrm{Qbits}\, (n+1)\) places the fresh factor on the left. It is unital, multiplicative and \(*\)-preserving — a unital \(*\)-homomorphism realising the tower inclusion.

The far-end inclusion is injective: \(x\otimes 1\) (reindexed) determines \(x\) through the entry identity \((\mathrm{appendQubit}\, x)_{(0,a),(0,b)} = x_{a,b}\).

The far-end site adjoined by \(\mathrm{appendQubit}\) is a fresh, maximally mixed degree of freedom, so pairing the \((n+1)\)-level maximally mixed state against \(\mathrm{appendQubit}\, x\) reproduces the \(n\)-level pairing against \(x\):

This is the compatible-marginals condition for the tracial state along the inclusion tower.

The junk cell of Definition 13.38 is literally empty, not merely null: a four-case skew-lattice reduction shows every torus point, lifted to the unit square, lands in \(R_1 \cup R_2\) after subtracting one of the four lattice vectors \((0,0),(0,1),(-1,0),(-1,1)\). Hence the two golden rectangles tile the torus exactly.

\(\operatorname {awSymbFull} : \mathbb {T}^2\to (\operatorname {Fin} 5)^{\mathbb {Z}}\), \(\operatorname {awSymbFull} x\, k = \operatorname {awSymb} x\, k\), records the branch symbol of the forward orbit of \(x\) at every integer time \(k\), landing in the two-sided full shift on the five golden branches.

If \(\det A(x) \neq 0\) for all \(x\), \(A\) is measurable, \(T\) is measure-preserving, and \(\log ^+\left\lVert A \right\rVert , \log ^+\left\lVert A^{-1} \right\rVert \in L^1(\mu )\), then the backward system \((T^{-1}, B)\) satisfies the same four hypotheses: \(\det B \neq 0\) everywhere, \(B\) measurable, and \(\log ^+\left\lVert B \right\rVert , \log ^+\left\lVert B^{-1} \right\rVert \in L^1(\mu )\).

For \(A : X \to \mathrm{Matrix}\, (\mathrm{Fin}\, d)\, (\mathrm{Fin}\, d)\, \mathbb {R}\) and \(T : X \simeq _m X\), the backward generator is

Its cocycle is taken over \(T^{-1}\).

The Bernoulli suspension flow is the time-translation flow on the suspension (mapping-torus) of the two-sided Bernoulli shift \(T\) over \(\operatorname {bernZ}\nu \) with constant roof \(\tau \equiv 1\): points are orbits \([x, s]\) of the identification \((x, s) \sim (Tx, s - 1)\), the flow is \(\zeta _t[x,s] = [x, s+t]\), and it preserves the normalized suspension measure \(\hat\mu \) (the product of \(\operatorname {bernZ}\nu \) with Lebesgue on the fibres; the constant roof makes the normalizing constant \(1\)). It is a MeasurePreservingFlow of \(\hat\mu \) (Definition 8.1).

The witness partition of the suspension measure \(\hat\mu \) is the base time-\(0\) coordinate partition of \(\operatorname {bernZ}\nu \) pulled back along the base projection (factor map) \(\pi : [x,s] \mapsto T^{\lfloor s \rfloor } x\), which is measure-preserving onto \(\operatorname {bernZ}\nu \). Its cells are indexed by \(\operatorname {Fin}(\operatorname {card} \alpha _0)\); the crux mass identity is that pulling back does not change cell masses, so the \(j\)-th cell carries the single-symbol mass \(\nu \{ a_j\} \) of the corresponding symbol.

The \(n\)-th binary digit \(d_n(y)=\mathbf1_{T^{-n}(\operatorname {binCell}1)}(y)\in \{ 0,1\} \) records whether \(T^n y\) lands in the right half; the \(N\)-th dyadic partial sum

() is the truncated binary expansion of the canonical representative \(\operatorname {rep}(y)\in [0,1)\) ().

The binary partition is a one-sided generator for the doubling map: \(\bigvee _{n}\operatorname {comap}(T^n)\, \sigma (\alpha )=m_{\mathbb T}\).

A symbol map \(\varphi : A \to B\) lifts coordinatewise to the one-block code \(\mathrm{blockCode}\, \varphi : \mathrm{Shift}\, A \to \mathrm{Shift}\, B\), \((\mathrm{blockCode}\, \varphi \, x)(k) = \varphi (x_k)\), with induced word map \(\mathrm{wordMap}\, \varphi \, n : (\operatorname {Fin}n \to A) \to (\operatorname {Fin}n \to B)\).

The block entropy of the tower at level \(n\) is the von Neumann entropy of the product state, \(\mathrm{blockEntropy}\, \rho \, n = S(\rho ^{\otimes n})\).

The block entropy grows exactly linearly in the level:

At the maximally mixed single-qubit state \(I/2\) the linear law reads \(\mathrm{blockEntropy}\, (I/2)\, n = n\cdot \log 2\), recovering issue #70’s \(n\cdot \log 2\) claim as an equality — the maximal case, where \(S(\rho ) = \log 2\) is largest; for a general \(\rho \) the honest law is \(n\cdot S(\rho )\le n\cdot \log 2\).

For the concrete faithful family \(\rho _r = \tfrac 12\! \left(\begin{smallmatrix} 1 & r \\ r & 1 \end{smallmatrix}\right)\) (\(0 {\lt} r {\lt} 1\)) every nonempty block carries strictly positive entropy, \(\mathrm{blockEntropy}\, \rho _r\, n {\gt} 0\) for \(n\ge 1\); its single-qubit entropy is the binary entropy \(h_2\! \big((1+r)/2\big) {\gt} 0\), so the spatial rate is a positive constant — the tower is genuinely alive.

The Borel \(\sigma \)-algebra of the circle lies below the one generated by the dyadic arcs, \(m_{\mathbb T}\le \operatorname {generateFrom}(\operatorname {dyadicArcSet})\).

The bottom exponent \(\lambda _d = \Gamma _d - \Gamma _{d-1}\) is lower semicontinuous in the generator: \(\lambda _d(A) \le \liminf _i \lambda _d(B_i)\).

For \(f\in C(\mathbb {T}^2,\mathbb {R})\) with \(\mathcal C_s\)-complexification (\(s {\gt} 2\)), \(\left\lvert \rho (k) \right\rvert \le C\theta ^{k}\) with the same rate \(\theta = \lambda _+^{-(s-2)/4}\); this is the bridge identifying the probabilistic autocovariance with the centred correlation of Theorem 13.50.

\(h(\operatorname {catTorus}, \operatorname {catAWPartition}) \le \log \! \big((3+\sqrt5)/2\big)\). The positive-measure atoms of the join inject into admissible golden itineraries, whose weighted count satisfies the exact recurrence \(W(n+1) = \lambda \cdot W(n)\) via the golden weight identity \(\sum _{\operatorname {src} e' = b} w(e') = \lambda _+\, w(b)\); the atom-count backbone converts this growth rate into the entropy bound.

The golden two-box partition of \(\mathbb {T}^2\) into the projected branches of the two golden rectangles \(R_1 = [0,\varphi )\times [0,\varphi )\) and \(R_2 = [\varphi ,\varphi ^2)\times [0,1)\) (in the eigen-coordinates), a six-cell \(\operatorname {MeasurePartition}\) (five branch cells plus a junk cell).

For \(f\in \mathcal C_s\) there is a constant \(B\ge 0\) with, for every sample size \(n\ge 1\) and threshold \(\varepsilon {\gt} 0\),

the tier-3 headline law: the empirical average concentrates around the space mean at the Chebyshev rate.

For \(f, g\in \mathcal C_s\) (\(s {\gt} 2\)) there is a constant \(C\ge 0\) with, for every \(k\),

A real-observable corollary records \(\left\lvert \int f\, (f\circ \operatorname {catTorus}^{[k]}) - (\int f)^2 \right\rvert \le C\theta ^{k}\) (), the Green–Kubo input downstream. This is the repository’s first quantitative-rate mixing statement.

Because the derivative cocycle of the cat map is the constant hyperbolic matrix \(\texttt{cat}_{\mathbb {R}} = \bigl(\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\bigr)\), the length-\(n\) top-exponent estimator \((1/n)\log \left\lVert \texttt{cat}_{\mathbb {R}}^{\, n} \right\rVert \) is deterministic and converges to \(\log \lambda _+\) at the explicit rate

with \(C_0 = \left\lvert \log C \right\rvert \), \(C = (\left\lVert \texttt{cat}_{\mathbb {R}} \right\rVert + \lambda _+ + 1)/\sqrt5\).

For \(f\in \mathcal C_s\) the rescaled Birkhoff variance converges to the Green–Kubo variance,

together with a linear variance bound \(\operatorname {Var}(S_n)\le B\, n\) ().

Let \(\operatorname {catLift}:\mathbb {R}^2\to \mathbb {R}^2\) () be the linear lift of the cat map to the universal cover — the continuous linear map with matrix \(M\), which genuinely lifts \(\operatorname {catTorus}\): the covering projection \(\mathbb {R}^2\to \mathbb {T}^2\) satisfies \(\pi \circ \operatorname {catLift} = \operatorname {catTorus}\circ \pi \) (). Its derivative cocycle in the sense of Definition 13.1 is the constant matrix \(M\) at every point (), and the top Lyapunov exponent of this genuine derivative cocycle, over the genuine ergodic base \(\operatorname {catTorus}\), is

the formalized hyperbolicity of the Arnold cat map. (Reading the derivative on the torus manifold itself via \(\texttt{mfderiv}\) is a documented gap: Mathlib has no manifold-derivative API for \(\texttt{AddCircle}\) endomorphisms; the lift and the map share the same derivative everywhere because the covering projection is a local diffeomorphism with identity derivative.)

Realized as a constant cocycle with generator the cat-map matrix \(M\) over an ergodic base (here the doubling map — the spectrum depends only on \(M\)), the two Lyapunov exponents are

Honesty caveat (as in the Lean docstring): the cocycle here is the constant matrix, not the derivative cocycle of the genuine toral automorphism; the genuine dynamics is treated below.

\(\lambda _1 + \lambda _2 = 0\): the cocycle is conservative (\(\det M = 1\)).

Instantiating Definition 9.53 at the Arnold cat map Definition 13.21 under the constant unit roof, with base generator the cat map’s own derivative cocycle (the constant hyperbolic matrix \(\operatorname {cat}_{\mathbb {R}}\)), yields a genuine Definition 8.3 on the cat suspension.

For \(\hat\mu \)-a.e. orbit class \(q\), the growth rate of Definition 9.56 converges to the flow Lyapunov exponent \(\log \! \big((3+\sqrt5)/2\big)\) — the log of the cat matrix’s top eigenvalue.

For base observables \(f, g\in \mathcal C_s\) with \(g\) centred (\(\int g = 0\)) and bounded measurable fibre profiles \(\psi , \chi \), the centred flow correlation of the fibre-product observables \(F[x,s] = f(x)\psi (s)\), \(G[x,s] = g(x)\chi (s)\) on the constant-unit-roof cat suspension decays as

Instantiating the suspension functor twice at time \(1\) yields two suspension-flow factor maps: the first () is injective — the conjugacy stage, cat suspension \(\cong \) SFT\(_5\) suspension — and the second () is a genuine non-injective flow factor onto the two-symbol suspension with a strict flow-entropy drop \(h(\zeta ^{(2)}_1)\le \log 2 {\lt} \log \! \big((3+\sqrt5)/2\big) = h(\zeta ^{\mathrm{cat}}_1)\). All three levels are alive, with the merged level bracketed in \([\log \lambda _+ - \log 2, \log 2]\) by Theorem 13.61.

The map \(\operatorname {catTorus}:\mathbb {T}^2\to \mathbb {T}^2\), \((\operatorname {catTorus} y)_i = \sum _j M_{ij}\cdot y_j\), with the integer cat-map matrix \(M\) acting by integer scalar multiplication on each circle coordinate. Since \(\det M = 1\), the integer matrix \(M^{-1}\) (with rows \((1,-1)\) and \((-1,2)\)) induces a two-sided inverse, so \(\operatorname {catTorus}\) is a continuous additive automorphism of the compact group \(\mathbb {T}^2\). Throughout this section \(\mathbb {T}^2\) carries the product Haar probability measure (the normalization for which Mathlib’s multivariate Fourier basis is stated).

Realized as a constant cocycle with generator \(M\) over the genuine ergodic Arnold cat map \(\operatorname {catTorus}\), the two Lyapunov exponents are \(\log \bigl((3+\sqrt5)/2\bigr)\) and \(\log \bigl((3-\sqrt5)/2\bigr)\) — the same spectrum as Theorem 13.19, now over the hyperbolic toral automorphism itself rather than a surrogate base.

A measurable \(g:\mathbb {T}^2\to \mathbb C\) with \(g(\operatorname {catTorus} x) = l\cdot g(x)\), \(\left\lvert l \right\rvert = 1\), \(l\ne 1\), vanishes a.e. — re-deriving Theorem 13.31 through the mixing interface rather than the direct Fourier argument.

Every atom of the \(n\)-fold forward join of the \(5 \times 5\) grid partition under \(\operatorname {catTorus}\) has \(\operatorname {volume}\) at most \((9\sqrt5/25)\cdot \lambda \cdot \mu ^n\). Two points sharing every grid cell for \(n\) steps stay \(1/5\)-close along the orbit, so a nearest-integer lift of their difference telescopes into an eigencoordinate slab \(\left\lvert \operatorname {eigCoordU} \right\rvert \le (3/10)\mu ^{n-1}\), \(\left\lvert \operatorname {eigCoordS} \right\rvert \le 3/10\); translation invariance and the projection contraction give the measure bound.

\(h(\operatorname {catTorus}) = \log \! \big((3+\sqrt5)/2\big) = \log \lambda _+\). This is the exact Kolmogorov–Sinai entropy of a hyperbolic system, formalized end-to-end; we have not located a prior formalization.

\(\log \! \big((3+\sqrt5)/2\big) \le h(\operatorname {catTorus})\).

\(h(\operatorname {catTorus}) \le \log \! \big((3+\sqrt5)/2\big)\).

\(0 {\lt} h(\operatorname {catTorus})\).

For any finite measurable partition \(P\) of \(\mathbb {T}^2\) whose \(n\)-fold refinement under \(\operatorname {catTorus}\) eventually has at most \(C\cdot e^{\, n\log \lambda _+}\) non-empty atoms (\(C\ge 1\), \(\lambda _+ = (3+\sqrt5)/2\)),

As in the doubling-map case this is the per-partition bound, not the system inequality; the rate \(\log \lambda _+\) is the genuine top Lyapunov exponent of the cat map (Theorem 13.25), and the atom-count growth is the honest named geometric input. The sharp system-level equality \(h_\mu = \log \lambda _+\) is a documented wall (it needs an Adler–Weiss Markov generating partition for the upper bound and Pesin/Ledrappier–Young machinery for the lower bound).

For arbitrary measurable sets \(A,B\subseteq \mathbb {T}^2\),

i.e. \(\operatorname {catTorus}\) is strongly mixing for the Haar probability measure.

The normalized cocycle is \(\texttt{cdiv}\, g\, n\, x = g\, (n+1)\, x / (n+1)\), the sequence whose a.e. limit is the content of Kingman’s theorem; its EReal coercion is \(\texttt{ecdiv}\, g\, n\, x\), used where \(-\infty \) cannot be excluded a priori.

For every abelian (diagonal) operational partition the correlation density matrix is a rank-one idempotent at every resolution \(n\) — a Gram collapse onto the column-\(0\) diagonal entry — so its von Neumann entropy is \(0\) (); while for the non-commuting partition \(\operatorname {cexPartition}\) the length-\(2\) correlation matrix is not idempotent, so its entropy is strictly positive. This is issue #59’s “entropy strictly above every abelian restriction”, stated honestly at the per-resolution level: the system-level CNT/ALF rate is the disclosed \(0 = 0\) of Theorem 17.41, so the non-commutativity lives in the per-resolution correlation matrices.

If \(M\) is self-adjoint and \(M v = c\, v\) for \(v \neq 0\), then \((\mathrm{cfc}\, f\, M)\, v = f(c)\cdot v\).

For positive-definite \(A,B\), \(\ \log (A\otimes B) = \log A\otimes 1 + 1\otimes \log B\).

For all faithful parameters \(0{\lt}r,s{\lt}1\), the chain-seal predicate holds: \(\mathrm{ChainSealed}\, r\, s\). Dephasing site \(0\) of the chain at the tracial block admits no faithful-ancilla Stinespring recovery, at every level \(n\).

For a differentiable \(T\), the \(n\)-th cocycle iterate of the derivative generator represents the derivative of the \(n\)-th iterate of the map:

For the faithful local state \(\rho _r\) and diagonal reference \(\operatorname {diagState} s\) (\(0{\lt}r,s{\lt}1\)), the predicate \(\mathrm{ChainSealed}\) asserts that at every level \(n\) the partial-dephasing channel on \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{2^{n}}(\mathbb {C})\) admits no faithful-ancilla Stinespring recovery that simultaneously inverts it on the coherent state \(\rho _r\otimes \mathrm{mm}_n\) and on the reference \(\operatorname {diagState} s\otimes \mathrm{mm}_n\), where the block is the tracial (maximally mixed) block state \(\mathrm{mm}_n\). This is the chain-level, per-stage (channel-level) seal: single-step per stage, not a flow seal, with the recovery quantified over all faithful-ancilla Stinespring dilations.

For any Kraus channel with all four states faithful, saturation \(S(\rho \| \sigma )=S(\Lambda \rho \| \Lambda \sigma )\) implies IntertwinesIt.

For a Kraus channel \(\Lambda \) with all four states \(\rho ,\sigma ,\Lambda \rho ,\Lambda \sigma \) faithful, if data processing is saturated (\(S(\rho \| \sigma )=S(\Lambda \rho \| \Lambda \sigma )\)), then the two \(-\log \) modular quadratic forms agree at the output cyclic vector \(\xi =\operatorname {vec}((\Lambda \rho )^{1/2})\):

with \(W=\texttt{petzWChanVec}\), \(\Delta =\sigma \otimes (\rho ^{-1})^{\top }\), \(\Delta _{\mathrm{out}}=(\Lambda \sigma )\otimes ((\Lambda \rho )^{-1})^{\top }\).

Under entropy saturation, the channel contraction intertwines the modular unitary power on the output cyclic vector, \(W\bigl(\Delta _{\mathrm{out}}^{it}\, \xi \bigr)=\Delta ^{it}\, (W\xi )\), with \(W\xi =\operatorname {vec}(\rho ^{1/2})\).

For a countable family of cells \(s : \iota \to \mathrm{Set}\, \alpha \), the countable Shannon entropy is \(\; c H_\mu (s) = \sum _i \operatorname {negMulLog}\, \mu (s_i) = -\sum _i \mu (s_i)\log \mu (s_i)\), an unconditionally convergent tsum. Downarowicz’s Fact 1.1.4 supplies the finiteness criterion (): if the index-weighted masses satisfy \(\sum _i i\cdot \mu (s_i) {\lt} \infty \), the entropy terms are summable, so \(cH_\mu (s)\) is a genuine finite sum. This is the form in which the Keane–Serafin construction certifies finite entropy of its limit partition.

For Kraus operators \(K\) with \(\sum _i K_i^{\dagger }K_i=1\) and \(X\) self-adjoint with spectrum in \((0,\infty )\),

Let \(g_k\) be the conditional information function of \(P\) given the \(k\)-th Breiman conditioning \(\sigma \)-algebra \(\mathcal C_k = \sigma (T^{-1}\bigvee _{j=0}^{k-1}T^{-j}P)\), and let \(g^{*} = \sup _k g_k\) be the Chung maximal information function (, valued in \([0,\infty ]\)). Then for every cell \(P_{i_0}\) and every \(\lambda {\gt} 0\),

Consequently \(g^{*} \in L^1(\mu )\), with \(\int g^{*}\, d\mu \le H(P) + 1\) () — Chung’s \(L^1\) domination, the one genuinely analytic leaf of the SMB proof.

For every unital \(*\)-endomorphism \(\Phi \), state \(\rho \), operational partition \(X\), and resolution \(n\), the correlation entropy obeys the uniform-in-\(n\) reservoir ceiling

The reservoir is \(\log (d^{2}) = 2\log d\), not the naive single-copy ceiling \(\log d\): the correlation-matrix construction lives on a \(d^{2}\)-dimensional Gram factorization, so the honest cap is \(\log (d^{2})\).

The entropy of a partition is the infimum von Neumann entropy rate

and the CNT dynamical entropy of \(\Phi \) in the state \(\rho \) is the supremum over all finite operational partitions, \(h_\Phi (\rho )=\sup _{k,X}h_\Phi (\rho ,X)\) (valued in \(\overline{\mathbb {R}}\)). The \(\inf _n\) form is the honest analogue of the subadditive limit \(\lim _n S(\cdot )/n\) and sidesteps an operator-Fekete argument.

The full CNT/ALF dynamical entropy of any finite-dimensional dynamics vanishes, \(h_\Phi (\rho ) = 0\).

The abelian-corner CNT dynamical entropy of \(\mathrm{adPerm}\, \sigma \) in the state \(\rho _\mu \) is the supremum of the partition rate over all projection operational partitions, \(h^{\mathrm{ab}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) =\sup _{k,c}h_{\mathrm{adPerm}\, \sigma }(\rho _\mu ,\mathrm{projPartition}\, c)\).

Suppose every state carries positive mass (\(\mu _i{\gt}0\) for all \(i\)). Then the abelian-corner CNT dynamical entropy equals the classical Kolmogorov–Sinai entropy of the permutation system:

The sharp abelian-corner entropy is dominated by the full diagonal version, \(h^{\mathrm{ab}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) \le h^{\mathrm{ab,full}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu )\), since every projection partition \(\mathrm{projPartition}\, c\) is diagonal.

The full abelian-corner CNT dynamical entropy is the supremum of the partition rate over all operational partitions whose operators are diagonal (soft / POVM partitions of the diagonal subalgebra), a wider family than the sharp projection partitions of \(\mathrm{cntDynamicalEntropyAbelian}\) (Definition 17.32).

For a \(\sigma \)-invariant state with every mass positive, the full diagonal CNT dynamical entropy equals the classical Kolmogorov–Sinai entropy of the permutation system, \(h^{\mathrm{ab,full}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) = h_\mu (\sigma )\). Both sides are \(0\).

In finite dimension the full diagonal supremum vanishes, \(h^{\mathrm{ab,full}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) = 0\), as does the sharp projection supremum \(h^{\mathrm{ab}}_{\mathrm{adPerm}\, \sigma }(\rho _\mu ) = 0\) .

For each projection partition \(\mathrm{projPartition}\, c\), the CNT partition entropy in the abelian corner equals the classical Kolmogorov–Sinai partition entropy of the corresponding measurable partition \(c^{-1}(\cdot )\):

For every operational partition \(X\), the CNT partition entropy is \(0\): \(h_\Phi (\rho ,X) = \inf _{n\ge 1}S(\mathrm{corrMatrix}\, n)/n = 0\).

For every partition the sequence \(n\mapsto S\bigl(\mathrm{corrMatrix}\, \Phi \, \rho \, X\, n\bigr)\) is bounded above by the fixed reservoir \(\log (d^{2})\), uniformly in the step count \(n\). Monotonicity in \(n\) is deliberately not claimed — it is unproven in general and forced flat in the saturated regime; only the uniform reservoir cap is asserted.

\(0 {\lt} h(\operatorname {catTorus}, \operatorname {coarseAWPartition})\); indeed \(\log \lambda _+ - \log 2\le h(\operatorname {catTorus}, \operatorname {coarseAWPartition})\) (), strictly positive because \(\lambda _+ = (3+\sqrt5)/2 {\gt} 2\). The merged symbolic factor is thus a two-symbol system whose entropy lies in the nontrivial band \([\log \lambda _+ - \log 2, \log 2]\).

The two-cell \(\operatorname {MeasurePartition}\) of \(\mathbb {T}^2\) into the two projected golden rectangles \(R_1, R_2\); its symbolic factor is the merged itinerary \(k\mapsto \operatorname {src}(\operatorname {awSymb} x\, k)\).

\(h(\operatorname {catTorus}, \operatorname {coarseAWPartition})\le \log 2\), strictly below the system entropy \(\log \lambda _+\) (): the coarse partition, unlike the fine generator, is far from generating. This is the abstract \(\log (\operatorname {card})\) ceiling for a partition into two cells.

Given \(A : X \to \operatorname {Matrix}(\operatorname {Fin}d)(\operatorname {Fin}d)\, \mathbb {R}\) and \(T : X \to X\), define \(A^{(n)} = \texttt{cocycle}\, A\, T\, n\) by recursion on \(n\):

Unfolding, \(A^{(n)}(x) = A(T^{n-1}x)\cdots A(Tx)\, A(x)\), with the newest factor on the left. The matrix norm throughout is the scoped L2 operator norm, which is submultiplicative; vectors live in \(\mathrm{EuclideanSpace}\; \mathbb {R}\, (\operatorname {Fin}d)\) and the action is via \(\texttt{toEuclideanCLM}\).

For all \(m, n \in \mathbb {N}\) and \(x \in X\),

For \(c\ge 0\) and \(n\ge 1\),

Writing \(B = \mathrm{backwardGen}\, A\, T\), the cocycle of \(B\) over \(T^{-1}\) satisfies

If \(T\) is differentiable and uniformly expanding with constant \(K {\gt} 1\), then the derivative of the \(n\)-th iterate stretches every vector by at least \(K^n\):

For any generator \(A\) and map \(T\), \(A^{(n+1)}(x) = A(T^{[n]}x)\cdot A^{(n)}(x)\).

Over an invertible measure-preserving base \(T : X \simeq X\), the one-sided iterated cocycle (indexed by \(\mathbb {N}\)) extends to a two-sided \(\mathbb {Z}\)-indexed cocycle \(\operatorname {cocycleZ} A\, T\, n\): for \(n \ge 0\) the forward cocycle, and for \(n {\lt} 0\) the backward cocycle of the inverse dynamics with generator \(x \mapsto (A(T^{-1}x))^{-1}\).

\(\operatorname {cocycleZ}(m + n)\, x = \operatorname {cocycleZ} m\, (\operatorname {baseIter} n\, x) \cdot \operatorname {cocycleZ} n\, x\), where \(\operatorname {baseIter}\) is the \(\mathbb {Z}\)-iterate of the base. This is the keystone that lets the discrete object assemble into a continuous-time cocycle.

Let \(\mu \) be an invariant probability measure with \(\mu \ll \operatorname {volume}\), let \(T\) be differentiable with everywhere nonvanishing Jacobian, let \(\xi \) be an injectivity partition, and assume \(\log \rho \in L^1(\mu )\) for the density \(\rho = d\mu /d\operatorname {vol}\) and \(\log \left\lvert \det D T \right\rvert \in L^1(\mu )\). Then the conditional Shannon entropy of \(\xi \) given the pulled-back \(\sigma \)-algebra \(T^{-1}\mathcal{A}\) is

independently of the partition (no generating hypothesis).

For any finite measurable partition \(P\) of a standard Borel probability space and any sub-\(\sigma \)-algebra \(\mathcal A\le m_\alpha \),

On a standard Borel probability space, the conditional Shannon entropy of a finite family of cells \(s\) given a sub-\(\sigma \)-algebra \(\mathcal A\) is

the \(\mu \)-average of the pointwise entropy computed against the regular conditional probability kernel \(\kappa =\operatorname {condExpKernel}\mu \, \mathcal A\). It measures the information about the partition remaining after conditioning on \(\mathcal A\); conditioning on the trivial \(\sigma \)-algebra recovers the ordinary entropy.

For a finite measure, a measure-preserving measurable \(T\) and integrable \(g\), the conditional expectation onto the \(\sigma \)-algebra \(\texttt{invariants}\, T\) of \(T\)-invariant sets is a.e. \(T\)-invariant:

For a sub-\(\sigma \)-algebra \(\mathcal A\) of a standard Borel space, the conditional information function of \(P\) given \(\mathcal A\) is

the conditional probability realized by the regular conditional probability kernel; exactly one indicator survives at each point. Its integral is the conditional entropy: for \(\mathcal A \le m_\alpha \), \(\int g_{\mathcal A}\, d\mu = H(P \mid \mathcal A)\) (); in particular \(g_{\mathcal A}\) is integrable.

The relative entropy of the system given \(\mathcal A\) is the \(\overline\mathbb {R}\)-valued supremum \(h(T\mid \mathcal A) = \sup _\alpha h(T,\alpha \mid \mathcal A)\) over all finite measurable partitions, mirroring Definition 10.13.

For a sub-\(\sigma \)-algebra \(\mathcal A\le m_\alpha \) with \(T^{-1}\mathcal A\le \mathcal A\) (forward invariance), the relative entropy of \(T\) on a partition \(\alpha \) given \(\mathcal A\) is the Fekete limit of the conditional iterated-join entropies,

the conditional sequence being subadditive by the conditional analogues of Theorem 10.4 and Lemma 10.5.

Under the hypotheses of Theorem 17.12, for every continuous \(g\),

For a contraction \(W\) with defect \((1-W^{\dagger }W)\xi =0\), positive-definite shifts \(X,\mathrm{Out}\), writing \(\eta =\mathrm{Out}^{-1}\xi \), \(b=X^{-1}(W\xi )-W\eta \), \(Y=W^{\dagger }XW\),

In the setting of Lemma 17.10, with the compression bound \(W^{\dagger }XW\le \mathrm{Out}\), if the total resolvent gap \(\operatorname {Re}\langle \xi ,(W^{\dagger }X^{-1}W-\mathrm{Out}^{-1})\xi \rangle \) vanishes, then \(X^{-1}(W\xi )=W(\mathrm{Out}^{-1}\xi )\).

Let \(W\) be a contraction (\(W^{\dagger }W\le 1\)), \(\Delta ,\Delta _{\mathrm{out}}\) positive definite with compression bound \(W^{\dagger }\Delta W\le \Delta _{\mathrm{out}}\), cyclic-norm condition \(\left\lVert W\xi \right\rVert =\left\lVert \xi \right\rVert \), and the scalar \(-\log \) saturation \(\operatorname {Re}\langle \xi ,W^{\dagger }(-\log \Delta )W\xi \rangle =\operatorname {Re}\langle \xi ,(-\log \Delta _{\mathrm{out}})\xi \rangle \). Then for every \(t{\gt}0\),

For a dynamics \(\Phi \), a state \(\rho \) (a density matrix on \(\mathbb {C}^d\)) and an operational partition \(X\), the depth-\(n\) correlation density matrix \(\mathrm{corrMatrix}\, \Phi \, \rho \, X\, n\) is the matrix on the classical index set \((\operatorname {Fin}k)^{\operatorname {Fin}n}\) with entries

It is a genuine density matrix: its trace is \(1\) by the telescoping identity (Lemma 17.24) together with \(\operatorname{tr}\rho =1\), and it is positive semidefinite because the quadratic form \(x^{\dagger }Mx\) equals \(\operatorname{tr}(T\rho T^{\dagger })\ge 0\) for \(T=\sum _f x_f\, \mathrm{refine}(n,f)\).

For every unital \(*\)-endomorphism \(\Phi \) of \(\operatorname {Matrix}_2(\mathbb {C})\), the depth-\(1\) correlation matrix of the Pauli partition in the maximally mixed state \(I/2\) is itself the maximally mixed state on the \(4\)-element classical index set \((\operatorname {Fin}4)^{\operatorname {Fin}1}\):

The CNT correlation matrix is the Gram matrix of its Hilbert–Schmidt vectors:

A measure-preserving flow \(\varphi \) is measure-continuous if for every measurable set \(A\) the real measure of the symmetric difference \(\varphi _t^{-1} A \, \triangle \, A\) tends to \(0\) as \(t \to 0\).

\(\varphi \) is an a.e. coboundary of \(u\) over \(\mu \) if \(\varphi = u\circ T - u\) holds \(\mu \)-almost-everywhere. This is the measurable analogue of Definition 15.1.

On \(\texttt{BiShift}\ \alpha _0 = (\mathbb {Z}\to \alpha _0)\), set \(\operatorname {dist}(x,y) = (1/2)^{N}\) where \(N\) is the least \(\left\lvert j \right\rvert \) at which \(x_j\ne y_j\) (); the symmetric cylinders \(\{ y : \forall \, \left\lvert j \right\rvert {\lt}N,\ y_j = x_j\} \) are the balls. The metric is built against the pre-existing product topology, so \(\texttt{CompactSpace}\) and \(\texttt{BorelSpace}\) are inherited with no diamond.

\(\texttt{catProj}:\mathbb {R}^2\to \mathbb {T}^2\) is the coordinatewise quotient projection; it intertwines the linear map \(\texttt{cat}_{\mathbb {R}} = \bigl(\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\bigr)\) upstairs with \(\texttt{catTorus}\) downstairs and is distance-nonincreasing.

For a lifted return defect \(e\), \(\texttt{catShadowSol}\ n\ e\) solves the cohomological equation \((\texttt{cat}_{\mathbb {R}}^n - 1)\, w = e\) in closed form in the hyperbolic eigenbasis — \(1\) is not an eigenvalue of any power of the hyperbolic matrix, so \(\texttt{cat}_{\mathbb {R}}^n - 1\) is invertible on \(\mathbb {R}^2\).

An observable \(\varphi \) is a coboundary for a bare map \(T:X\to X\) if \(\varphi = u\circ T - u\) for some transfer function \(u:X\to \mathbb {R}\), with no regularity assumed on \(u\).

\(T\) has the exponential closing property \(\texttt{ExpClosing}\, T\, \alpha \, \delta \, K\) if every point \(x\) that almost \(n\)-returns (\(\operatorname {dist}(x,T^n x)\le \delta \)) is shadowed by a genuine \(n\)-periodic point \(p\) (\(T^n p = p\)) whose orbit \(\alpha \)-Hölder-shadows that of \(x\) with total cost controlled by the return gap:

Summing the per-step cost (rather than hardcoding a per-term \(\min \)-exponent bound) is deliberate: the individual terms decay like \(\theta ^{\min (i,\, n-i)}\) (two-sided Anosov) or \(\theta ^{\, n-i}\) (one-sided/expanding), and both sum to a constant multiple of the endpoint gap. Abstracting the sum keeps the crux estimate regime-agnostic.

\(F\) is a flow coboundary for \(\Phi \) if it is the flow-time derivative (read integrally) of some transfer function \(u\):

No integrability is assumed — the interval integral is read as an opaque real, so the predicate is a bare cohomological one mirroring Definition 15.1.

On a metric space, \(\varphi \) is a Hölder coboundary for \(T\) if \(\varphi = u\circ T - u\) with \(u\) Hölder continuous (some exponent \(0 {\lt} r\), constant \(C\)). This is the target regularity of the Livšic theorem; forgetting the modulus of \(u\) shows a Hölder coboundary is a coboundary.

The induced base observable of a flow observable \(F\) is the one-lap roof integral \(\texttt{inducedBaseCocycle}\ F\ x = \int _0^{\tau x} F([x,s])\, ds\).

The rich point is the sequence obtained by concatenating all finite words (enumerated through the alphabet’s countable encoding). Every finite word appears as a block, so every target is approximated by some forward iterate.

A safe symbol for \(M\) is a symbol \(s\) allowed adjacent to every symbol: \(M(a,s)\) and \(M(s,a)\) hold for all \(a\). This is strictly stronger than irreducibility of \(M\); a general irreducible SFT need not have one, and dense orbits for such SFTs are not delivered here. The golden-mean shift has \(s = 0\).

For \(M:\operatorname {Fin}k\to \operatorname {Fin}k\to \texttt{Bool}\), the carrier \(\texttt{SFTCarrier}\ M = \{ x : \forall i,\ M(x_i, x_{i+1})\} \) is the closed shift-invariant set of admissible sequences; as a subtype () it inherits the ultrametric and, being closed in the compact full shift, is compact. The SFT shift is the restriction of \(\sigma \), still \(2\)-Lipschitz.

\(\varphi \) has vanishing periodic sums for \(T\) if every Birkhoff sum around a periodic orbit vanishes: whenever \(T^n p = p\), one has \(S_n\varphi \, p = \sum _{i{\lt}n}\varphi (T^i p) = 0\). This is the obstruction class the Livšic theorem certifies as complete.

For a finite index type \(n\), a density matrix is a structure bundling a matrix \(\rho \in \operatorname {Matrix}_n(\mathbb {C})\) together with proofs that \(\rho \) is positive semidefinite and has unit trace, \(\operatorname{Tr}\rho = 1\). It is the finite-dimensional state of a quantum system on \(\mathbb {C}^n\).

The rank of a density matrix is strictly positive, \(0 {\lt} \operatorname {rank}\rho \).

The real eigenvalues \(\lambda _i\) of a density matrix \(\rho \) are nonnegative, each at most \(1\), and sum to \(1\): \(\sum _i \lambda _i = \operatorname{Tr}\rho = 1\).

For a probability vector \(\mu :\operatorname {Fin}d\to \mathbb {R}_{\ge 0}\) (\(\sum _i\mu _i=1\)), the associated diagonal state is \(\rho _\mu =\operatorname {diag}\mu \), a density matrix on \(\mathbb {C}^d\).

For a block index \(\mathrm{blk}\), the partial-dephasing channel \(\Delta \otimes \mathrm{id}_{\mathrm{blk}}\) on \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{\mathrm{blk}}(\mathbb {C})\) dephases the first qubit (killing its coherences) and leaves the block untouched; its Kraus operators are \(K_i\otimes 1\) for the qubit-dephasing Kraus operators \(K_i\) (the diagonal projections). It factorises on product states, \((\Delta \otimes \mathrm{id})(x\otimes \beta ) = (\Delta x)\otimes \beta \).

For a self-map \(T\) of \(E = \operatorname {EuclideanSpace}\, \mathbb {R}\, (\operatorname {Fin}d)\), the derivative cocycle generator is the matrix-valued map

the matrix representing the Fréchet derivative \(D_xT = \operatorname {fderiv}\, \mathbb {R}\, T\, x\), transported along the star-algebra equivalence between \(d\times d\) real matrices and continuous linear endomorphisms of \(E\). Since that equivalence is an \(L^2\)-operator-norm isometry, the generator has the same norm as \(D_xT\). Feeding it to the iterated cocycle construction of Definition 2.2 yields the tangent cocycle of the smooth system.

If \(\det A(x) \neq 0\) for every \(x\), then \(\det A^{(n)}(x) \neq 0\) for all \(n, x\).

Let \(\operatorname {bern}\nu \) be the Bernoulli (i.i.d. product) measure on the full shift with single-symbol law \(\nu \), and suppose \(\nu \) charges two distinct symbols \(i \ne j\) with positive mass. Then there is a \(\operatorname {bern}\nu \)-conull set \(s\) with

the single-symbol Shannon entropy (Hnu in the sources). No ergodicity or positive-entropy hypothesis remains: both standing conditionals of Theorem 12.18 are discharged for the Bernoulli case.

Let \(\mu \) be a probability measure on a finite-dimensional real inner-product space \(E\), absolutely continuous with respect to a Haar measure. Then every set \(s\) of full \(\mu \)-measure has Hausdorff dimension equal to the ambient dimension: \(\dim _H s = \operatorname {finrank}_{\mathbb {R}} E\).

Let \(\mu \) be a shift-invariant probability measure on the full shift \(\Sigma \) such that the left shift \(\sigma \) is ergodic for \(\mu \), and suppose the Kolmogorov–Sinai entropy of the coordinate partition is positive. Then there is a full-measure carrier set \(s \subseteq \Sigma \) (\(\mu (s^{\mathsf c}) = 0\)) with

where \(h_\mu (\sigma )\) is the partition-independent system entropy (ksEntropy, Definition 10.13).

Let \(\mu \) be a probability measure on a Borel second-countable metric space \(E\), let \(\alpha {\gt} 0\), and let \(s\) be a set of full \(\mu \)-measure (\(\mu (s^{\mathsf c}) = 0\)) such that for every \(x \in s\) the local-dimension quotient \(\log \mu (\bar B(x,r))/\log r\) tends to \(\alpha \) as \(r \to 0^+\). Then \(\dim _H s = \alpha \).

For any finite measurable partition \(P\) of the circle whose \(n\)-fold refinement \(\bigvee _{k=0}^{n-1}T^{-k}P\) under the doubling map eventually has at most \(C\cdot e^{n\log 2}\) non-empty atoms (\(C\ge 1\)),

This is the per-partition bound \(h(\alpha ,T)\le \sum \lambda ^{+}\), not the system Margulis–Ruelle inequality; the right-hand side is the computed Lyapunov datum (Corollary 13.14), not an abstract constant. The atom-count hypothesis is automatic for the binary partition, where the bound is attained (Theorem 13.17).

The sum of the strictly positive Lyapunov exponents of the doubling-map cocycle is \(\log 2\), with every spectrum hypothesis (invertibility, measurability, both log-integrability conditions) discharged unconditionally from the constant-cocycle API.

The constant cocycle with generator \(M = (2)\) over the ergodic doubling map has top Lyapunov exponent \(\log 2\).

The set of all \(n\)-fold-join cells of the binary partition under the doubling map, \(\bigcap _{k{\lt}n}T^{-k}\bigl(\operatorname {binCell}(f(k))\bigr)\) over all lengths \(n\) and digit strings \(f:\operatorname {Fin}n\to \operatorname {Fin}2\). Each such set is a dyadic arc of length \(2^{-n}\).

A reusable spectral interface: for a strongly mixing measure-preserving map on a probability space, any measurable eigenfunction \(g\) with \(g\circ f = l\cdot g\), \(\left\lvert l \right\rvert = 1\), \(l\ne 1\), vanishes a.e.

Under \(\operatorname {diam} X \le 1\) the Kuratowski embedding \(\operatorname {kur} a = d(a, \cdot )\) () isometrically embeds \(X\) into the bounded continuous functions \(X \to ^{b} \mathbb {R}\). Two test bundles \(\operatorname {muFun}\), \(\operatorname {nuFun}\) (, ) interpolate the Kuratowski images of \(x\) and \(Tx\) with two distinct height weightings, and \(\operatorname {hgt}\) () measures the vertical \(\mathbb {R}/\mathbb {Z}\) coordinate. Evaluated at the canonical fundamental-domain representatives, the sum

is the embedding metric \(\operatorname {embDist}\) on the constant-roof-\(1\) suspension space.

The suspension flow is \(5\)-Lipschitz in the time parameter along each orbit: \(\operatorname {embDist}(\zeta _a q, \zeta _b q)\le 5\, \left\lvert a - b \right\rvert \).

\(\operatorname {embDist}\) satisfies the triangle inequality and separates points (): the two test parts descend from the sup-norm triangle inequality of \(X \to ^{b} \mathbb {R}\), the height part from the \(\mathbb {R}/\mathbb {Z}\) quotient metric, and a \(2\times 2\) linear solve on the two weightings recovers the base point from a vanishing distance. It is bi-Lipschitz to the route gauge \(\operatorname {routeDist}\).

For a roof \(\tau \ge \rho _{\min } {\gt} 0\), the canonical box representative \(\operatorname {suspensionRepVar}\) () is the unique orbit representative in \(\{ (x,s) : 0\le s {\lt} \tau x\} \) (the roof-cocycle is strictly increasing with gaps \(\ge \rho _{\min }\), so it partitions \(\mathbb {R}\)), and \(\operatorname {normHeightVar}\) () is its normalized height \(s/\tau x\). The metric \(\operatorname {embDistVar}\) is the constant-roof sum evaluated at the normalized representatives; it is a genuine metric ().

The realisation cost of the variable roof appears only in the flow-Lipschitz constant: the flow moves the raw height at unit speed, hence the normalized height at speed \(1/\tau \le 1/\rho _{\min }\), giving \(\operatorname {embDistVar}(\zeta _a q, \zeta _b q)\le (5/\rho _{\min })\, \left\lvert a - b \right\rvert \).

The Shannon entropy of a finite family of cells \(s:\iota \to \mathcal P(\alpha )\) with respect to a measure \(\mu \) is

the average information gained by learning which cell a \(\mu \)-random point lies in. The definition is on loose data — an arbitrary Fintype-indexed family of sets — so it applies both to genuine partitions and to intermediate non-partition families.

For the pure state \(\rho \), the length-\(1\) correlation matrix is the classical pure state \(\operatorname {diag}(1,0)\), so \(S(\rho [X^{(1)}]) = 0\).

For the same pure state, the length-\(2\) correlation matrix is not idempotent (its diagonal \(w_{00}\)-entry violates \(M^{2} = M\): the computation yields \(3/8\ne 1/2\)), hence \(S(\rho [X^{(2)}]) {\gt} 0\).

For two finite measurable partitions \(\alpha =(A_i)\) and \(\beta =(B_j)\) of a probability space, the entropy of the join (common refinement) \(\alpha \vee \beta \), whose cell family at \((i,j)\) is the intersection \(A_i\cap B_j\), satisfies

A finite measurable partition of a probability space into \(k\) cells (with \(k\ge 1\)) has Shannon entropy at most \(\log k\).

For a measure-preserving \(T\) and a finite measurable partition \(\beta \), the pullback partition \(T^{-1}\beta \) with cells \(T^{-1}B_j\) has the same Shannon entropy as \(\beta \): \(H(T^{-1}\beta )=H(\beta )\).

Let \(S\) be orthogonal (\(S S^{\top }=1\)) with the graded forward decay \(\left\lvert S_{ab} \right\rvert \le c\cdot e^{-\max (g_b-g_a,0)}\). Then every entry obeys the reverse bound at the full pairwise rate:

Assume the base shift \(T\) is ergodic for \(\operatorname {bernZ}\nu \). Then every measurable set \(A\) invariant under all time-\(t\) maps of the suspension flow (\(\zeta _t^{-1}(A) = A\) for every \(t \in \mathbb {R}\)) is null or conull: \(\hat\mu (A) = 0\) or \(\hat\mu (A) = 1\). The base hypothesis is discharged unconditionally by the two-sided Bernoulli ergodicity theorem (ergodic_biShiftEquiv_bernZ); the companion ergodic_bernSuspensionFlow_uncond records the unconditional statement. By contrast, the time-\(1\) map alone is never ergodic (not_ergodic_bernSuspensionFlow_one): the saturated section set \(\{ [x,s] : \operatorname {fract} s {\lt} 1/2\} \) is invariant of mass \(1/2\) — the constant-roof special-flow dichotomy of Cornfeld–Fomin–Sinai.

\(\operatorname {catTorus}\) is ergodic for the Haar probability measure on \(\mathbb {T}^2\).

Define \(E_i = V_i \sqcap W_{\mathrm{sidx}\, i}\). Then \(\mathrm{finrank}\, E_i \ge 1\), the telescoping identities \(V_i = E_i \sqcup V_{i+1}\) and \(E_i \sqcap V_{i+1} = \bot \) hold, and consequently \((E_i)\) is independent with \(\bigsqcup _i E_i = V_0 = \top \).

For \(T\) measure-preserving on a probability space and a conull measurable set \(S\), there is a conull measurable \(S' \subseteq S\) invariant under both \(T\) and \(T^{-1}\).

Let \((\alpha ,\mu )\) be standard Borel with automorphism \(e\), and let \(Q : \mathbb {N}\to \mathrm{Set}\, \alpha \) be a countable family of measurable cells such that (i) the two-sided \(Q\)-itinerary recovers each set of a fixed standard-Borel generating sequence up to a \(\mu \)-null set, and (ii) the entropy terms \(i \mapsto \operatorname {negMulLog}\, \mu (Q_i)\) are summable. Then there is a countable family that two-sidedly generates \((\alpha ,e,\mu )\) mod \(0\) and has finite static Shannon entropy — sub-problem A of Krieger’s theorem (Downarowicz, Thm. 4.2.3, first half). The structural reduction from the inductive Keane–Serafin data — an increasing sequence of finite partitions with per-level recovery and a summable geometric mass envelope — is ; the construction of that data from the raw dynamical hypotheses (the SMB-driven Rokhlin-tower marker painting of Keane–Serafin §2) is the disclosed residual, isolated as an explicit hypothesis bundle rather than formalized.

Let \(T\) be measure preserving and assume the in-measure SMB upper bound of Theorem 11.10. Write \(h = h(P,T)\). For every \(\varepsilon {\gt} 0\) and all sufficiently large \(N\), there is a finite set \(S\) of rank-\(N\) names \(g : \operatorname {Fin}N \to \iota \) with

The set \(S\) is the set of good names () — names whose join cell has measure at least \(e^{-N(h+\varepsilon )}\); the cardinality bound is the unconditional pigeonhole .

For merely measure-preserving \(T\) (no ergodicity) and invertible measurable \(A\) with the standing integrability, there is a family of \(T\)-invariant integrable functions \(\lambda _i : X \to \mathbb {R}\) such that for each \(i {\lt} d\) and \(\mu \)-a.e. \(x\), \(\tfrac 1n\log \sigma _i(A^{(n)}(x)) \to \lambda _i(x)\). The exponents become invariant functions rather than constants.

For a measure-preserving \(T\), an integrable subadditive cocycle \(g\) whose normalized integrals are bounded below, the sequence \((\int g\, (n+1)\, d\mu )/(n+1)\) converges to the Fekete constant \(\gamma \).

For ergodic \(T\) and invertible \(A\), every Oseledets filtration has a deterministic dimension profile: there is a strictly decreasing \(m : \mathrm{Fin}(k+1) \to \mathbb {N}\) with \(m_0 = d\), \(m_k = 0\) and, \(\mu \)-a.e., \(\mathrm{finrank}\, V_i(x) = m_i\).

Over the quotient, Definition 9.53 is cohomologous to the cover cocycle (Definition 9.4): at a general representative \((x,s)\) with \(0 \le s\) the two differ by the measurable rep-level frame \(C(x,s) = \operatorname {cocycleZ} A\, T\, \lfloor s\rfloor x\). Because different representatives of one class carry different \(\lfloor s\rfloor \), the issue’s schematic class-level conjugacy is provably impossible; the honest general statement is this rep-level frame, collapsing to the conjugation-free canonical identity at \(\operatorname {unitFwd}\).

Let \( \mu \) be a probability measure, \( \varphi \) a measure-preserving flow with \( \varphi (1) \) ergodic, and \( A \) a flow cocycle with the two uniform integrable dominators \( g, g' \). Then there exist \( k \), exponents \( \lambda : \mathrm{Fin}\, k \to \mathbb {R}\), and a subspace family \( V \) forming an Oseledets filtration for the generator \( A(1,\cdot ) \) over the dynamics \( \varphi (1) \).

There is an antitone sequence \(\lambda ^0 : \mathbb {N}\to \mathbb {R}\) (antitone on \([0,d)\)) such that, for every \(i{\lt}d\) and \(\mu \)-a.e. \(x\),

where \(\sigma _i\) is the \(i\)-th sorted singular value.

For ergodic \(T\) and invertible \(A\), each exponent \(\lambda _i\) carries a positive deterministic multiplicity \(m_i = \dim V_i - \dim V_{i+1}\) with \(\sum _i m_i = d\).

Over an alphabet \(\operatorname {Fin}l\) with a reserved sentinel letter \(s\), any finite name set with \(\# \mathrm{Name} \le (l-1)^m\) admits an injection \(\mathrm{enc} : \mathrm{Name} \hookrightarrow \mathrm{List}(\operatorname {Fin}l)\) into blocks of length \(m+1\), each containing the sentinel exactly once (at its end). The blocks form a prefix-free family, so arbitrary concatenations are uniquely decodable (); when \(e^{N(h+\varepsilon )} {\lt} k^{N}\), the good names of Theorem 11.12 embed into \(\operatorname {Fin}k\)-blocks of length \(O(N)\) — the symbolic code from which the coding partition is read off along a Rokhlin tower.

Summing the positive parts \(\max (\lambda _i(x),0)\) over \(i {\lt} d\) yields a single \(T\)-invariant integrable function \(G_+ : X \to \mathbb {R}\), the non-ergodic analogue of the positive-exponent sum.

\(\mu \)-a.e., \(\exp (\mathrm{exponents}_i)\) is the \(i\)-th sorted eigenvalue of the Oseledets limit matrix \(\Lambda (x)\).

Under the same hypotheses, every singular value of the cocycle iterate \(A^{(n)}(x) = \operatorname {cocycle}(\operatorname {derivativeCocycle} T)\, T\, n\, x\) is at least \(K^n\): \(\ K^n\le \sigma _i\bigl(A^{(n)}(x)\bigr)\) for every \(i {\lt} d\).

The full Lyapunov spectrum with multiplicity is the total function \(\mathrm{exponents} : \mathrm{Fin}\, d \to \mathbb {R}\), whose \(i\)-th entry is the deterministic limit of \(\tfrac 1n\log \sigma _i(A^{(n)})\), sorted non-increasingly. The top entry is \(\mathrm{topExponent}\).

Every Lyapunov exponent of a uniformly expanding map is strictly positive: \(0 {\lt} \log K \le \lambda _i\).

For each sorted index \(i\) and \(\mu \)-a.e. \(x\), \(\tfrac 1n\log \sigma _i(A^{(n)}(x)) \to \mathrm{exponents}_i\); moreover \(\mathrm{exponents}\) is antitone.

The \(k\)-th exterior generator \(\mathrm{extGen}\, k\, A\) sends \(x\) to the \(k\)-th compound matrix \(C_k(A x)\) of \(k\times k\) minors. Its iterated cocycle is the compound of the iterate: \(\mathrm{cocycle}(\mathrm{extGen}\, k\, A)\, T\, n\, x = C_k(A^{(n)}(x))\).

If \(E\) is finite-dimensional and \(g\) is an ultrametric growth function, then \(\{ g(v) : v\neq 0\} \) is finite, with at most \(\dim _{\mathbb {R}} E\) elements.

A descending flag \(V_0 \supseteq \cdots \supseteq V_k = \bot \) in a modular lattice, with complements \(E_i\) satisfying \(V_i = E_i \sqcup V_{i+1}\) and \(E_i \sqcap V_{i+1} = \bot \), telescopes into an independent family \((E_i)\) with \(\bigsqcup _i E_i = V_0\).

A continuous-time linear cocycle over a measure-preserving flow \( \varphi \), valued in invertible \( d \times d \) real matrices, is a family \( A : \mathbb {R}\to X \to \mathrm{Mat}_d(\mathbb {R}) \) with \( A(0,x) = 1 \), the cocycle identity (newest factor on the left) \( A(t+s,x) = A(t, \varphi (s)x)\, A(s,x) \), a proof that \( \det A(t,x) \ne 0 \) for all \( t, x \), and measurability of each time-\( t \) map \( A(t,\cdot ) \).

Along the \(\operatorname {atTop}\) half-line \(0 \le t\), the growth rate of Definition 9.53 equals the descended flow exponent \(\operatorname {flowExponentAt}\) (Definition 9.16).

Writing \(\langle n\rangle = \max \{ 1, \left\lvert n_0 \right\rvert , \left\lvert n_1 \right\rvert \} \) for the Japanese bracket of a frequency \(n\in \mathbb {Z}^2\) (), a function \(f:\mathbb {T}^2\to \mathbb {C}\) lies in \(\mathcal C_s = \operatorname {FourierDecay} s\) when its multivariate Fourier coefficients decay at least like \(\langle n\rangle ^{-s}\): \(\ \left\lVert \hat f(n) \right\rVert \le K\, \langle n\rangle ^{-s}\) for some constant \(K\). Every character \(\texttt{mFourier}\, m\) lies in every class (, its coefficients a Kronecker delta), and the class is closed under integrable sums and scalar multiples (), so every trigonometric polynomial belongs to it.

Let \(T\) be ergodic for a probability measure \(\mu \), let \(A\) be measurable and everywhere invertible with \(\log ^{+}\left\lVert A \right\rVert , \log ^{+}\left\lVert A^{-1} \right\rVert \in L^{1}(\mu )\). Then there is a constant \(\lambda _1 \in \mathbb {R}\) (the top Lyapunov exponent) with

Under the same hypotheses there is a constant \(\lambda \in \mathbb {R}\) with

so the bottom Lyapunov exponent \(\lambda _k = -\lambda \) exists and is finite.

The partial-sum growth rate is the infimum over \(n\) of the normalized integrals:

For \(k \le d\), the ergodic growth rate \(\Gamma _k\) of the product of the top-\(k\) singular values equals the sum of the top-\(k\) exponents: \(\Gamma _k = \sum _{i{\lt}k} \mathrm{exponents}_i\).

Along a filter of generators \(B_i \to A\) for which each fixed-\(n\) integral is continuous, the partial-sum rate is upper semicontinuous: \(\limsup _i \Gamma _k(B_i) \le \Gamma _k(A)\); specializing \(k=1\) gives the same for the top exponent. This is USC, not continuity — equality can fail when a spectral gap closes.

Fix a real time \( t_0 \). For almost every \( x \) and every test vector \( u \), the discrete-time growth \( \limsup \) of the cocycle applied to \( u \) at \( x \) equals that at \( \varphi (t_0)x \) applied to the pushed-forward vector \( A(t_0,x)\, u \).

For a dynamics \(\Phi \), a state \(\rho \) on \(\mathbb {C}^{d}\) and an operational partition \(X\), the depth-\(n\) Gram vectors are the Hilbert–Schmidt vectors \(w_f = (\mathrm{refine}\, \Phi \, X\, n\, f)\cdot \sqrt{\rho }\), reshaped into the matrix \(\mathrm{gramVec}\, \Phi \, \rho \, X\, n\) with rows indexed by \(\operatorname {Fin}d\times \operatorname {Fin}d\) and columns by the classical index set \((\operatorname {Fin}k)^{\operatorname {Fin}n}\); here \(\sqrt{\rho } = \texttt{CFC.sqrt}\, \rho \) is the continuous-functional-calculus positive-semidefinite square root.

A growing quantum world bundles, for one concrete faithful local state \(\rho _r\) and one diagonal reference \(\operatorname {diagState} s\), the three faces of the growing-finite tower:

1. aliveness — the single-qubit entropy \(S(\rho _r) {\gt} 0\) is positive and the per-step spatial rate \(\mathrm{blockEntropy}\, \rho _r\, n / n\) converges to it (Theorem 17.70);

2. the per-stage dephasing seal at the world’s own block states — at every level \(n\), no faithful-ancilla Stinespring recovery inverts the partial dephasing on both \(\rho _r\otimes \rho _r^{\otimes n}\) and \(\operatorname {diagState} s\otimes \rho _r^{\otimes n}\), the block being the world’s own length-\(n\) block \(\rho _r^{\otimes n}\) (Theorem 17.74);

3. a base-factor non-commutativity certificate — the single-qubit dynamics does not preserve the canonical MASA and the seal’s eigenprojection is off-diagonal (Theorem 17.57).

A concrete alive-and-sealed growing quantum world exists. Take the balanced faithful local state \(\rho _{1/2}\) and the balanced diagonal reference \(\operatorname {diagState}(1/2)\): aliveness holds because \(h_2(3/4) {\gt} 0\), the per-stage seal is Theorem 17.74 instantiated at the world’s own block state \(\rho _{1/2}^{\otimes n}\) (positive definite), and the base-factor certificate is Theorem 17.57.

For \(x\in X\) and \(v\in \mathbb {R}^d\) the growth sequence is

where \(A^{(n)}(x)=\mathtt{cocycle}\, A\, T\, n\, x\) is the \(n\)-step cocycle iterate acting on \(v\) via \(\mathtt{toEuclideanCLM}\).

For continuous \(f, g:\mathbb {T}^2\to \mathbb {C}\), Parseval turns the time-\(k\) correlation into a bilinear character sum, and the Koopman action \(\texttt{mFourier}\, n\circ \operatorname {catTorus}^{[k]} = \texttt{mFourier}(A^k n)\) shifts the index on one factor: the sum over \(b\ne 0\) of \(\overline{\hat f(A^k b)}\, \hat g(b)\) converges to the centred correlation \(\int \overline f\, (g\circ \operatorname {catTorus}^{[k]}) - (\int \overline f)(\int g)\).

If \(F\) is \(\operatorname {embDist}\)-\(r\)-Hölder and \(T\) is Lipschitz, the induced base observable \(\operatorname {inducedBaseCocycle} T\, \tau \, F\) is \(r\)-Hölder on the base — the regularity input the base Livšic converse consumes.

For an \(\operatorname {embDist}\)-\(r\)-Hölder, bounded \(F\) and a base transfer function \(u_0\) cobounding the induced base observable, the fundamental-domain candidate \(\operatorname {uCover}(x, s) = u_0(x) + \int _0^{s} F[x, \sigma ]\, d\sigma \) descends to a transfer function that is Hölder for the embedding metric. This cross-seam gluing is the substantive content of the theorem.

Let \(f\) be operator convex on an interval \(I\), let \(A,B\) be a contraction pair (\(A^{*}A + B^{*}B = 1\)), and let \(X,Y\) be self-adjoint with spectra in \(I\). Then

If \(x \mapsto V(x)\) and \(x \mapsto W(x)\) are measurable subspace families, then so is \(x \mapsto V(x) \sqcap W(x)\).

Fix \(x\). If the forward envelope for \(V_x\) holds at rate \(a\), \(U_x\) decays backward at rate \(\le b\), and \(a + b {\lt} 0\), then \(V_x \sqcap U_x = \bot \).

Every point \(x\) has an \(n\)-step itinerary \(f : \operatorname {Fin}n \to \iota \), recording for each \(k {\lt} n\) which cell of \(P\) contains \(T^k x\); among the admissible codes the least one in a fixed enumeration is chosen, which makes the selector genuinely measurable (the cells may overlap on a null set, so a bare choice function would not be). The atom of \(x\) is the cell \(\bigcap _{k{\lt}n} T^{-k}(P_{f(k)})\) of the iterated join containing \(x\), and the information function is

the surprise of learning the \(n\)-name of \(x\). It is nonnegative and measurable (): in its simple-function form it is a finite sum of indicators of the itinerary fibers, each the difference of a join cell and the finitely many strictly smaller cells.

Matching two-sided itineraries force equality of points (the Adler–Weiss contraction estimate), so \(\operatorname {awSymbFull}\) is injective on the nose — again no boundary null set is discarded. It is therefore a measurable embedding (, \(\mathbb {T}^2\) standard Borel), and the induced measurable equivalence onto its range is measure preserving (): a measure conjugacy of the cat map onto the golden SFT.

For self-adjoint \(Q\), a band indicator \(\chi =\mathbf1_{(c,\infty )}\), and a continuous \(f\ge 0\) on \(\mathrm{spec}(Q)\) with \(a\le f(t)\) whenever \(c{\lt}t\),

A matrix is a function \(m \to n \to \alpha \), and we equip \(\operatorname {Matrix}\, m\, n\, \alpha \) with the Pi (product) \(\sigma \)-algebra induced from the entry type \(\alpha \). For finitely many entries over a Borel \(\alpha \) this agrees with the Borel \(\sigma \)-algebra. Mathlib does not register this automatically because \(\operatorname {Matrix}\) is a def rather than reducibly the underlying Pi type, so the ambient Pi instance does not transfer; we install it explicitly.

The entrywise (Pi) measurable structure on \(\operatorname {Matrix}(\operatorname {Fin}d)(\operatorname {Fin}d)\, \mathbb {R}\) is an OpensMeasurableSpace for the L2 operator-norm topology, since that topology is installed (via replaceTopology) to be definitionally the Pi product topology, of which the Pi \(\sigma \)-algebra is exactly the Borel structure.

Under the same hypotheses, the limsup envelope \(f_+\) is integrable.

Let \(T\) be measure-preserving for a finite measure \(\mu \), \(A\) measurable and everywhere invertible, \(d \neq 0\), with both \(\log ^{+}\left\lVert A \right\rVert \) and \(\log ^{+}\left\lVert A^{-1} \right\rVert \) integrable. Then each \(x \mapsto \log \left\lVert A^{(n)}(x) \right\rVert \) is integrable. The companion \(\texttt{integrable\_ logNorm\_ inv\_ cocycle}\) gives the same for the inverse iterates.

If \(T\) is measure-preserving and \(g\) is integrable, then each Birkhoff partial sum and the maximal function \(\texttt{maxBirkhoff}\, T\, g\, N\) are integrable; if moreover \(g\) and \(T\) are measurable, then \(\texttt{maxBirkhoff}\, T\, g\, N\) is measurable.

The hypothesis \(\texttt{IntegrableLogNorm}\, A\, \mu \) asserts that the positive part of the log-norm of the generator is integrable: \(\log ^{+}\left\lVert A(\cdot ) \right\rVert \in L^{1}(\mu )\), where \(\log ^{+} t = \max (\log t, 0)\). This is the standard integrability assumption of the Furstenberg–Kesten and Oseledets theorems; combined with the same hypothesis for the inverse generator \(A^{-1}\) it pins both extremal Lyapunov exponents in \(\mathbb {R}\).

If \( g \in L^1(\mu ) \) dominates \( \log ^{+}\left\lVert A(s,x) \right\rVert \) for all \( s \in [0,1] \) and all \( x \), then the time-one map \( A(1,\cdot ) \) has integrable positive log-norm.

If \( g' \in L^1(\mu ) \) dominates \( \log ^{+}\left\lVert (A(s,x))^{-1} \right\rVert \) for all \( s \in [0,1] \) and all \( x \), then the inverse \( (A(1,\cdot ))^{-1} \) has integrable positive log-norm.

For measure-preserving \(T\) and integrable \(f\), \(\int \sum _{k{\lt}n} f(T^{k}x)\, d\mu = n\int f\, d\mu \).

For every \(n\),

the Shannon entropy of the \(n\)-fold iterated join (the \(n\)-th term of the Kolmogorov–Sinai entropy sequence).

Let \(T\) be a measure-preserving, differentiable self-map of \(\mathbb {R}^d\) with \(\mu \ll \operatorname {volume}\) and everywhere-nonsingular derivative, let \(\xi \) be a nonempty injectivity partition (Definition 13.10), and assume \(\log \rho \in L^1(\mu )\) for the density \(\rho =d\mu /d\operatorname {vol}\) and \(\log \left\lvert \det DT \right\rvert \in L^1(\mu )\). Then, with no generating-partition hypothesis,

This is the generator-free lower bound on entropy (Rokhlin 1967, §9; Parry 1969; Coudène, Ergodic Theory and Dynamical Systems, Cor. 12.1): the integrated log volume distortion never exceeds the Kolmogorov–Sinai entropy. It needs no ergodicity and no expansion — those enter only for Rokhlin’s equality. Here \(\mu \ll \operatorname {volume}\) enters solely through Coudène’s Remark \(\left\lvert T'_\mu \right\rvert =\left\lvert \det DT \right\rvert \) a.e., identifying the Radon–Nikodym distortion with the Jacobian determinant. The hypotheses are non-vacuous on \(\mathbb {R}^d\): e.g. \(T=\operatorname {id}\) with any a.c. probability \(\mu \) gives \(0\le h_\mu (\operatorname {id})\).

The channel adjoint intertwines the modular \(it\)-flows if, for all \(t\),

If IntertwinesIt holds (all four states faithful), then the Petz map recovers the input: \(P_{\sigma ,\Lambda }(\Lambda \rho )=\rho \).

An invariant subbundle is a measurable family of fibre subspaces \(W(x) \le \mathbb {R}^d\) that is \(A\)-equivariant a.e.: \(A(x)\, W(x) = W(Tx)\).

\(\operatorname {awSymbFull}\) is a measure-theoretic factor map from \((\mathbb {T}^2, \operatorname {catTorus}, \mathrm{vol})\) onto the two-sided full shift, with the intertwining \(\operatorname {awSymbFull}\circ \operatorname {catTorus} = \operatorname {biShiftMap} \circ \operatorname {awSymbFull}\) holding everywhere (not merely a.e.): the empty junk cell of Lemma 13.39 gives every point a well-defined symbol at every time. The pushforward concentrates on the golden subshift-of-finite-type carrier, which has full measure \(1\) ().

\(\operatorname {suspensionFactorMap}\) intertwines the two suspension flows () and pushes the invariant measure of the source suspension to that of the target (), so it is measure preserving; packaged at time \(1\) it is an \(\operatorname {Entropy.IsFactorMap}\) of the time-\(1\) flow maps. When the base map \(\pi \) is injective, so is the lift (), giving a conjugacy stage of the tower.

A \(1\)-periodic fibre profile \(h\) descends to a flow observable \(F[x,s] = h(s)\) on the cat suspension (, a \(\texttt{Quotient.lift}\) whose well-definedness is the profile’s periodicity). For the concrete choice \(h(s) = \sin (2\pi s)\) (), the induced base observable vanishes identically — its one-lap integral is \(\int _0^{1}\sin (2\pi s)\, ds = 0\) — so all periodic sums vanish and \(\texttt{sinFibreObservable}\) is a flow coboundary of the cat-map suspension flow.

A finite measurable partition \(P\) is (one-sided) generating for \((\alpha ,T,\mu )\) if the forward-saturated \(\sigma \)-algebra it generates is the ambient one:

where \(\sigma (P)\) is the \(\sigma \)-algebra generated by the cells of \(P\). This is the standard generator condition for a (possibly non-invertible) endomorphism: the whole measurable structure is recovered from the observable \(P\) and its forward time-translates.

For a measure-preserving automorphism \(e:\alpha \simeq \alpha \), a finite partition \(P\) is two-sided generating if the \(\sigma \)-algebra saturated under both time directions is the ambient one:

This is the correct condition for invertible systems: forward one-sided generation is strictly stronger and provably fails for genuine automorphisms such as the two-sided Bernoulli shift.

\(\operatorname {catAWPartition}\) is two-sided generating for \(\operatorname {catTorus}\). Each point has a unique admissible two-sided itinerary (consecutive symbols satisfy \(\operatorname {tgt} = \operatorname {src}\), by the geometric branch-step keystone and injectivity of the covering projection on \(R_1 \cup R_2\)); two points with the same itinerary have difference propagating linearly, so the contraction estimate forces them equal. Separation then yields generation through Theorem 13.40.

For a measurable automorphism of a standard Borel space and a finite measurable partition \(P\), if the two-sided family of cell-preimages separates points then \(P\) is two-sided generating (Definition 10.16). The identity map from the ambient Borel structure to the coarser saturated \(\sigma \)-algebra is an injective measurable map into a countably separated space, hence a measurable embedding by Blackwell’s theorem.

The two-sided saturation of a finite partition \(P\) under an automorphism \(e\) is the \(\sigma \)-algebra \(\bigvee _{n\in \mathbb {Z}} e^{-n}\sigma (P)\) generated by the two-sided \(P\)-itinerary. The partition two-sidedly generates \((\alpha , e, \mu )\) mod \(0\) when the ambient \(\sigma \)-algebra is contained in the \(\mu \)-completion of the saturation — every measurable set is recovered from the itinerary up to a null set. This mod-\(0\) form (not the literal equality of \(\sigma \)-algebras) is the faithful conclusion of Krieger’s theorem. The \(\mathrm{Countable}\)-indexed analogue for a family of cells \(Q : \kappa \to \mathrm{Set}\, \alpha \) is .

If \(\nu \) charges two distinct symbols \(i \ne j\) with different masses (\(\nu \{ i\} \ne \nu \{ j\} \)), then the witness partition is heterogeneous: two of its cells carry distinct \(\hat\mu \)-mass (IsHeterogeneous, the honest non-uniformity predicate whose negation is exactly the equal-measure hypothesis of the monofractal degeneracy \(D_q \equiv \log N / (-\log \varepsilon )\)).

On the Bowen–Walters metric space \(\operatorname {SuspensionSpace} T\, \tau \) (constant roof \(1\)), a flow observable \(F\) is a Hölder flow coboundary if it is a flow coboundary (Definition 15.44) whose transfer function \(u\) is \(\operatorname {embDist}\)-Hölder: \(\left\lvert u\, p - u\, q \right\rvert \le C\, \operatorname {embDist}(p, q)^{r}\) for some \(0 {\lt} r\le 1\), \(C\). Forgetting the modulus () shows a Hölder flow coboundary is a flow coboundary.

The equivalence has content on both sides. The \(\sin (2\pi \cdot )\) fibre profile \(\operatorname {sinFibreObservable}\) is embedding-Hölder and is a Hölder flow coboundary of the cat suspension (its one-lap integral \(\int _0^{1}\sin (2\pi s)\, ds = 0\) makes the induced base observable vanish); by contrast the constant observable \(1\) is embedding-Hölder yet not a Hölder flow coboundary (), since a Hölder flow coboundary is in particular a flow coboundary and \(1\) is defeated by the closed-orbit obstruction Theorem 15.50.

For a self-map \(T\) of \(\operatorname {EuclideanSpace}\, \mathbb {R}\, (\operatorname {Fin}d)\) and a finite measurable partition \(\xi \), the predicate \(\texttt{IsInjectivityPartition}\, \mu \, T\, \xi \) packages the two hypotheses the conditional-expectation proof needs: \(T\) is injective on each cell of \(\xi \), and each cell is measurable. (These are literally the hypotheses of Mathlib’s change-of-variables lemma; no Markov condition and no generating condition are baked in.)

For a measure \(\mu \), map \(T\), generator \(A\), count \(k\), exponent list \(\lambda : \mathrm{Fin}\, k \to \mathbb {R}\) and flag \(V : \mathrm{Fin}(k+1) \to X \to \mathrm{Submodule}\), the predicate \(\mathrm{IsOseledetsFiltration}\ \mu \ T\ A\ k\ \lambda \ V\) asserts: \(\lambda \) is strictly antitone; each level \(V_i\) is a measurable subspace family; and \(\mu \)-a.e. \(x\) carries the strictly decreasing \(A\)-equivariant flag \(\mathbb {R}^d = V_0(x) \supsetneq \cdots \supsetneq V_k(x) = 0\) with exact growth rate \(\lambda _i\) on the stratum \(V_i \setminus V_{i+1}\). This is byte-identical to the conclusion of the main theorem.

A sequence \(g : \mathbb {N}\to X \to \mathbb {R}\) is a subadditive cocycle over \(T\) when

For \(g_n = \log \left\lVert A^{(n)} \right\rVert \) this is submultiplicativity of the operator norm composed with the cocycle identity.

Assume \(\det A \neq 0\) everywhere and \(d \neq 0\). Then \(g_n(x) = \log \left\lVert A^{(n)}(x) \right\rVert \) is a subadditive cocycle over \(T\):

Under the same hypotheses, \(g_n(x) = \log \left\lVert (A^{(n)}(x))^{-1} \right\rVert \) is a subadditive cocycle over \(T\).

\(\mathrm{restLog}\, A\, V\, T\) is an everywhere subadditive cocycle: for all \(m,n,x\), \(\mathrm{restLog}\, (m+n)\, x \le \mathrm{restLog}\, m\, x + \mathrm{restLog}\, n\, (T^{[m]}x)\).

Let \(E\) be a real vector space. A function \(g\colon E\to \mathbb {R}\) is an ultrametric growth function when it is scaling-invariant,

and non-Archimedean (strong triangle inequality),

The value \(g(0)\) is never used; the side conditions \(v\neq 0\) are carried explicitly to avoid extended-real arithmetic.

Under the hypotheses of 4.10, for a.e. \(x\) the function \(v\mapsto \bar\lambda (v)\) is an ultrametric growth function.

The level sets \(\{ x : 0 {\lt} \texttt{maxBirkhoff}\, T\, g\, N\, x\} \) are monotone in \(N\), and their union over all \(N\) is exactly \(\{ x : \exists n,\ 0 {\lt} \texttt{birkhoffSum}\, T\, g\, (n+1)\, x\} \).

On a probability space with \(0 {\lt} d\), any Oseledets filtration has \(0 {\lt} k\).

Let \(D = (D_{km})\) be a doubly stochastic \(K\times M\) matrix (\(D_{km}\ge 0\), all row sums and all column sums equal to \(1\)), let \(p\ge 0\) on \(K\) and \(s\ge 0\) on \(M\) have equal total mass \(\sum _k p_k = \sum _m s_m\), and assume the support condition \(s_m = 0\Rightarrow D_{km}p_k = 0\). Writing \(a_m = \sum _k D_{km}p_k\) for the column marginal,

For faithful \(\rho \), with the Bratteli–Robinson II §5.3 convention \(\sigma _{-i}(y)=\rho \, y\, \rho ^{-1}\),

A Kraus channel on \(\operatorname {Matrix}_n(\mathbb {C})\) is a finite family of Kraus operators \(K : \iota \to \operatorname {Matrix}_n(\mathbb {C})\) satisfying the completeness relation \(\sum _i K_i^{\dagger }K_i = 1\). Its Schrödinger action is \(\Lambda (X)=\sum _i K_i\, X\, K_i^{\dagger }\) (Lean: toMat, restricted to states as toDM), and its Heisenberg (Hilbert–Schmidt) adjoint is \(\Lambda ^{\dagger }(X)=\sum _i K_i^{\dagger }X\, K_i\) (Lean: adj).

A Kraus channel is trace preserving, \(\operatorname{tr}(\Lambda X)=\operatorname{tr}X\) for every \(X\); consequently the adjoint is unital, \(\Lambda ^{\dagger }(1)=1\) (Lean: adj_unital).

The conjugation \(E_j M E_j^{*}\) is exactly the \((j,j)\)-diagonal block of \(M\), the submatrix \(M_{(a,j),(a',j)}\) on the \(A\)-index.

For each \(j\in n_B\), the right block-inclusion operator \(E_j\in \operatorname {Matrix}_{n_A,\, n_A\times n_B}(\mathbb {C})\) is the co-isometry onto the \(A\)-factor picking out the \(j\)-th \(B\)-slice,

The left block-inclusion operator \(F_i\in \operatorname {Matrix}_{n_B,\, n_A\times n_B}(\mathbb {C})\), \((F_i)_{b,\, (i',b')} = [\, (i',b') = (i,b)\, ]\), is defined symmetrically for the \(A\)-factor.

Each block inclusion is a co-isometry onto its factor: for every \(j\), \(E_j E_j^{*} = 1\) on \(\mathbb {C}^{n_A}\), and symmetrically \(F_i F_i^{*} = 1\) on \(\mathbb {C}^{n_B}\) .

Let \(e\) be an ergodic, aperiodic (: every set of \(n\)-periodic points, \(n \ne 0\), is \(\mu \)-null), measure-preserving automorphism of a probability space, and let \(k \in \mathbb {N}\) satisfy

where \(h(e)\) is the Kolmogorov–Sinai entropy of the system. If the Krieger coding construction supplies a coding datum (Definition 11.17), then \(e\) admits a finite two-sided generator of size \(\le k\), mod \(0\): a partition \(P\) indexed by \(\operatorname {Fin}k\) with \(\mathrm{IsGeneratingTwoSidedMod0}\ e\ P\). The entropy threshold is pinned to the genuine \(h(e)\), so the hypothesis is a real entropy constraint. Interface disclosure: the coding datum has no in-repo constructor — the tower-painting combinatorics that would discharge it from the dynamical hypotheses alone are not formalized — so this is the honest conditional form of Krieger’s theorem, with the dynamical hypotheses retained as the faithful classical interface that the upstream layers (Theorem 11.11, Theorem 11.12, Theorem 11.13 and Theorem 11.16) are designed to feed.

Given a Krieger coding datum \(D\) for \((e, \mu , k)\), there exists a partition \(P\) into at most \(k\) cells that two-sidedly generates \((\alpha , e, \mu )\) mod \(0\). No entropy, ergodicity or aperiodicity hypotheses enter: those are consumed upstream, in producing \(D\). This is the pure recovery content of Krieger’s theorem.

For an automorphism \(e\) and \(k \in \mathbb {N}\), a Krieger coding datum bundles: the measure-preservation of \(e\); a countable index type \(\kappa \) with measurable cells \(Q : \kappa \to \mathrm{Set}\, \alpha \) two-sidedly generating mod \(0\); and a \(\operatorname {Fin}k\)-valued partition \(P\) that codes \(Q\) two-sidedly mod \(0\) (): the two-sided \(P\)-itinerary recovers each \(Q\)-cell up to a \(\mu \)-null set. This is exactly the object the Krieger construction produces when \(\log k\) exceeds the entropy: \(Q\) from Theorem 11.16, and \(P\) from the column coding of the \(\le k^{N}\) good names (Theorem 11.12 and Theorem 11.13) along refining Rokhlin towers (Theorem 11.11).

The maximally mixed state is closed under the Kronecker product, \(\mathrm{mm}\otimes \mathrm{mm}=\mathrm{mm}\): concretely \((a^{-1}\cdot 1)\otimes (b^{-1}\cdot 1)=(a\, b)^{-1}\cdot 1\) with \(a,b\) the factor cardinalities.

For faithful states \(\rho ,\sigma \), with (vectorised) relative modular operator \(\Delta =\sigma \otimes (\rho ^{-1})^{\top }\) and cyclic vector \(\xi =\operatorname {vec}(\rho ^{1/2})\),

The Kolmogorov–Sinai entropy of the system is the supremum of the partition-relative entropies over all finite measurable partitions,

taken in the complete lattice \(\overline\mathbb {R}=[-\infty ,+\infty ]\) (EReal) so that the definition is total even when the entropy is infinite. The supremum ranges over \(\operatorname {Fin}n\)-indexed partitions for every \(n\in \mathbb {N}\), which by reindexing invariance realizes the full partition supremum; each \(h(T,\alpha )\) is below \(h(T)\) (), and \(h(T)\ge 0\), witnessed by the trivial one-cell partition.

The Kolmogorov–Sinai entropy (Definition 10.13) of the time-\(1\) map of the Bernoulli suspension flow equals the single-symbol Shannon entropy:

In particular the flow’s metric entropy (defined as the entropy of its time-\(1\) map) is \(H(\nu )\), strictly positive for a genuinely biased \(\nu \).

The Kolmogorov–Sinai entropy of the doubling map is \(h_\mu (T)=\log 2\).

Under the same positivity hypothesis, the classical Kolmogorov–Sinai entropy of \(\sigma \) equals the full CNT dynamical entropy — the supremum over all operational partitions, not merely the diagonal ones:

This upgrades the one-sided bound Theorem 17.34 to an equality. Both sides are \(0\): the KS entropy of a finite-set permutation, and (by Theorem 17.41) the finite-dimensional CNT dynamical entropy — so the equality, while honest, is degenerate.

Let \(T\) be a measure-preserving transformation of a standard Borel probability space and \(P\) a one-sided generating finite partition. Then the entropy of the system is attained on \(P\):

No ergodicity is required; standard Borelness and the probability hypothesis feed the kernel disintegration, and the saturation supplied by the generating hypothesis is load-bearing.

Let \(e\) be a measure-preserving automorphism of a standard Borel probability space and \(P\) a two-sided generating finite partition. Then

Under the same positivity hypothesis, the full CNT dynamical entropy of \(\mathrm{adPerm}\, \sigma \) in the state \(\rho _\mu \) — the supremum over all operational partitions — dominates the classical Kolmogorov–Sinai entropy:

The golden-SFT image system \(\operatorname {Measure.map}\, \operatorname {awSymbFull}\, \mathrm{vol}\) under the shift has Kolmogorov–Sinai entropy exactly \(\log \! \big((3+\sqrt5)/2\big)\): being an injective factor, the Adler–Weiss coding is an entropy-preserving conjugacy, so the issue’s expectation of a strict drop at every step is impossible on this stage and is disclosed as such.

The \(1\)-block source merge \(\operatorname {mergeSrc} : (\operatorname {Fin} 5)^{\mathbb {Z}}\to (\operatorname {Fin} 2)^{\mathbb {Z}}\), \(y\mapsto (k\mapsto \operatorname {src}(y\, k))\), composed with the coding is the coarse itinerary \(\operatorname {coarseSymb}\); its image two-symbol system has entropy \(= h(\operatorname {catTorus}, \operatorname {coarseAWPartition})\le \log 2 {\lt} \log \lambda _+\) ( for the exact value): a genuine lumping with a strict entropy drop.

The Kolmogorov–Sinai entropy of \(T\) relative to a finite partition \(\alpha \) is the Fekete limit of the subadditive iterated-join entropy sequence,

packaged through Mathlib’s \(\texttt{Subadditive.lim}\) (an infimum of the averaged terms, so the definition is total).

For the binary partition \(\alpha = \{ [0,\tfrac 12), [\tfrac 12,1)\} \) of the circle (), the partition-relative Kolmogorov–Sinai entropy under the doubling map is exactly \(\log 2\).

On a standard Borel space (with \(\iota \) nonempty), the Kolmogorov–Sinai entropy of the pair equals the conditional entropy of \(P\) given the \(\sigma \)-algebra of the strict future:

This is the integral-level statement of the sharp SMB rate: the precise constant the pointwise theorem converges to. No ergodicity is needed.

Let \(T\) be a measure-preserving, differentiable self-map of \(\mathbb {R}^d\) with \(\mu \ll \mathrm{vol}\), everywhere-nonsingular derivative, a one-sided generating finite partition \(\xi \) on whose cells \(T\) is injective, and \(\mu \)-integrable \(\log \rho \) (\(\rho \) the density \(d\mu /d\mathrm{vol}\)) and \(\log \left\lvert \det DT \right\rvert \). Then the per-partition entropy is the integrated volume distortion:

Together with the determinant identity of Theorem 6.20, the right-hand side is the sum \(\sum _i\lambda _i\) of all Lyapunov exponents of the derivative cocycle — so for expanding maps (all exponents positive) this saturates the Margulis–Ruelle bound, giving Pesin’s entropy formula in that regime.

Let \(T\) preserve a probability measure and let \(P\) be a finite partition indexed by a type of cardinality \(b\) such that for every \(n\), every cell of the \(n\)-fold join \(\bigvee _{k=0}^{n-1}T^{-k}P\) has measure exactly \(b^{-n}\). Then \(h(P,T) = \log b\).

The iterated-join entropy sequence of \((T,\alpha )\) is

the Shannon entropy of the flat \(n\)-fold join. Its term at \(0\) is \(0\), and every term is nonnegative.

On a standard Borel space, the \(n\)-step join entropy telescopes into conditional entropies:

where the \(k\)-th conditioning \(\sigma \)-algebra is generated by the cells of the \(T\)-pullback of the \(k\)-step join.

For a measure-preserving \(T\) on a probability space and all \(n,m\),

For a measure-preserving \(T\) and a finite measurable partition \(\alpha \), the iterated join \(\bigvee _{k=0}^{n-1}T^{-k}\alpha \) is realized as the partition indexed by the flat function type \(\operatorname {Fin}n\to \iota \), whose cell at \(f\) is the intersection \(\bigcap _{k{\lt}n} T^{-k}\bigl(\alpha _{f(k)}\bigr)\). Cells are measurable as finite intersections of preimages under the iterates \(T^k\), pairwise a.e. disjoint (two distinct indices disagree at some coordinate \(k\), where measure-preservation of \(T^k\) transports the disjointness of \(\alpha \)), and cover the space. For \(n=0\) this is the trivial one-cell partition.

The upper Lyapunov growth function is the \(\limsup \) of the growth sequence,

For nonzero \(v,w,v+w\), if the three growth sequences are bounded, then \(\bar\lambda (v+w)\le \max (\bar\lambda (v),\bar\lambda (w))\).

On the good set, if \(v\in \mathrm{vflag}(A,T,x)(i)\) but \(v\notin \mathrm{vflag}(A,T,x)(i{+}1)\), then \(\bar\lambda _x(v)=\mathrm{specList}(i)=\lambda _i\).

Under the same hypotheses, for a.e. \(x\) and every \(v\neq 0\),

Assume \(T\) ergodic on a probability space, \(A\) measurable and invertible, with \(\log \left\lVert A \right\rVert \) and \(\log \left\lVert A^{-1} \right\rVert \) integrable. Then there exist \(\lambda _{\mathrm{bot}}\le \lambda _{\mathrm{top}}\) such that for a.e. \(x\) and every \(v\neq 0\), \(\bar\lambda (v)\in [\lambda _{\mathrm{bot}},\lambda _{\mathrm{top}}]\).

For \(c\neq 0\) and \(v\neq 0\), \(\bar\lambda (c\cdot v)=\bar\lambda (v)\).

The sublevel subspace at threshold \(t\),

is the submodule 4.6 of \(\bar\lambda _x\) at \(t\) on the good set, and \(\bot \) off it.

For a subadditive cocycle and \(n : \mathbb {N}\), one has \(g\, (n+1)\, x \le \texttt{birkhoffSum}\, T\, (g\, 1)\, (n+1)\, x\); more generally, for any decomposition of \([0,N)\) into \(k+1\) consecutive blocks of lengths \(\ell _0, \dots , \ell _k\), the cocycle is dominated by the sum of the block values along the orbit at the frontiers \(T^{[\sum _{j{\lt}i}\ell _j]}x\).

Let \(\varphi \) be \(r\)-Hölder with constant \(C_\varphi \), let \(T\) satisfy \(\texttt{ExpClosing}\, T\, r\, \delta \, K\), and let \(\varphi \) have vanishing periodic sums. Then any almost-\(n\)-return is controlled by its gap,

The bilateral shift space splits measurably as \(\text{past}\otimes \text{future}\) via \(\texttt{joinPF}\), under which the two-sided Bernoulli measure is the product of the past and future Bernoulli laws (measure-preservingly).

A real-valued function that is \(\texttt{HolderOnWith}\ C\ r\) (exponent \(r\le 1\)) on a subset \(s\) of a metric space extends to a global \(\texttt{HolderWith}\ C\ r\) function agreeing with it on \(s\), with the same constant and exponent.

For \(0\le \theta {\lt}1\), \(\ \sum _{i{\lt}n}\theta ^{\min (i,\, n-i)} \le 2\, (1-\theta )^{-1}\).

For a measurable a.e. transfer function of an \(r\)-Hölder \(\varphi \) over the two-sided Bernoulli measure, and for a.e. stable pair (two pasts, one shared future) the values of \(u\) differ by at most \(C\theta /(1-\theta )\); symmetrically for a.e. unstable pair (two futures, one shared past), .

If \(x\) almost \(n\)-returns within radius \(1/2\), then \(x_0 = x_n\).

The left shift is \(2\)-Lipschitz for the \(\texttt{PiNat}\) ultrametric (), hence continuous; the full shift over a finite discrete alphabet is compact by Tychonoff.

If \(\varphi \, x = u(T x) - u\, x\) for all \(x\), then the Birkhoff sum collapses to the endpoints, \(S_n\varphi \, x = u(T^n x) - u\, x\). Pure algebra — no metric, no regularity, valid for a bare \(T\).

The restriction \(\texttt{toShift}:\texttt{BiShift}\ \alpha _0\to \texttt{Shift}\ \alpha _0\) to the non-negative coordinates is a measure-preserving factor map () intertwining the two-sided shift with the one-sided one: the one-sided Bernoulli system is a factor of the invertible two-sided one. Hölder data, measurability, and the a.e. cohomological equation all transport up this factor.

Factoring \(Q\) over the eigen-line coordinates \(a_{\pm }\), with \(a_+\) scaling by \(\lambda _+^k\) under \(A^k\) () and \(a_+\cdot a_- = Q\), the lower bound \(\left\lvert Q \right\rvert \ge 1\) combines with the elementary sup-norm bounds \(\left\lvert a_- \right\rvert \le \lambda _+\left\lVert \cdot \right\rVert \), \(\left\lvert a_+ \right\rvert \le (\lambda _+-1)\left\lVert \cdot \right\rVert \) into the quantitative expansion estimate: for every nonzero \(n\in \mathbb {Z}^2\) and every \(k\),

For thresholds \(c\le c'\) with limit band projectors \(P,P'\), if \(Pv=0\) then \(P'v=0\).

The pointwise limsup \(x \mapsto \limsup _n \texttt{birkhoffAverage}\, \mathbb {R}\, T\, g\, n\, x\) is a.e. \(T\)-invariant.

For a finite measure, measure-preserving \(T\), an integrable subadditive cocycle with normalized integrals bounded below, the limsup envelope \(f_+(x) = \limsup _n \texttt{cdiv}\, g\, n\, x\) (and likewise the liminf envelope \(f_-\)) is a.e. \(T\)-invariant.

There is a forward top value \(\lambda _1^+\) (the a.e. limit of \(\tfrac 1n\log ^+\left\lVert A^{(n)} \right\rVert \)) such that, whenever \(\lambda _1^+ {\gt} 0\), \(\mu \)-a.e. the genuine log-norm \(\limsup \) is exactly \(\lambda _1^+\):

The positivity hypothesis is essential; in the contracting case \(\lambda _1^+ = 0\) the genuine growth may tend to \(-\infty \) and equality fails.

Under the same singular hypotheses there is \(\lambda _1^+\) with, \(\mu \)-a.e.,

The \(\limsup \) is taken in \(\mathrm{EReal}\) so the bound is unconditional even when the growth tends to \(-\infty \); this is one-sided only.

For ergodic \(T\) and a possibly-singular generator with \(\log ^+\left\lVert A \right\rVert \in L^1\), there is a constant \(\Gamma _k^+\) with, \(\mu \)-a.e.,

The top-\(k\) volume growth is bounded above unconditionally (the \(\mathrm{EReal}\) \(\limsup \) allows volume collapse), again one-sided only.

This lemma belonged to a superseded determinant/volume-squeeze route and has no counterpart in the current Lean development; the statement is kept for exposition and the label only so that cross-references resolve. Given a volume cocycle whose top, slow and remaining log-exponent sequences satisfy a sum law \(\mathrm{vol} = \mathrm{slow} + \mathrm{rest}\) with the appropriate limits, the slow restricted-operator-norm exponent obeys \(\limsup _n \tfrac 1n\log (\mathrm{slow}_n) \le \lambda _i\).

If \(g\) is an ultrametric growth function and \(v\colon \iota \to E\) is a family of nonzero vectors with \(g\circ v\) injective, then \(v\) is linearly independent over \(\mathbb {R}\).

For the suspension flow of \(\operatorname {catTorus}\) under the unit roof \(\tau \equiv 1\), a flow observable \(F\) whose induced base observable is Hölder (exponent \(0 {\lt} r\le 1\)) and whose fibre restrictions are interval-integrable is a flow coboundary iff every periodic Birkhoff sum of its induced base observable vanishes; equivalently (flow-native form ) iff every closed-orbit integral of \(F\) vanishes.

For the suspension flow of \(\operatorname {catTorus}\) under the unit roof \(\tau \equiv 1\), a flow observable \(F\) that is \(\operatorname {embDist}\)-Hölder (\(0 {\lt} r\le 1\)) and bounded, with interval-integrable fibre restrictions, is a Hölder flow coboundary iff every periodic Birkhoff sum of its induced base observable vanishes — equivalently () iff every closed-orbit integral of \(F\) vanishes.

Let \(T\) be a Lipschitz homeomorphism of a compact metric base with \(\operatorname {diam} X\le 1\), the summed exponential closing property and a dense forward orbit, and let \(F\) be an \(\operatorname {embDist}\)-\(r\)-Hölder (\(0 {\lt} r\le 1\)), bounded flow observable with per-fibre interval-integrable restrictions. Then

the flow-native (closed-orbit) form being . This is simultaneously tier 2 (the Hölder-regularity upgrade) and the issue’s tier-4 Anosov statement over the suspension presentation.

The equivalence extends to a variable roof \(\tau \) that is bounded below by \(\rho _{\min } {\gt} 0\), bounded above, and Lipschitz, measured against the variable-roof Bowen–Walters metric \(\operatorname {embDistVar}\) of Definition 9.63: an \(\operatorname {embDistVar}\)-\(r\)-Hölder bounded \(F\) is a Hölder flow coboundary iff its induced base observable has vanishing periodic sums.

Let \(\Phi = \zeta \) be the suspension flow of \(T\) under a constant roof \(\tau \equiv c\), let \(F\) be a flow observable with per-fibre interval-integrable restrictions, and suppose the base map \(T\) discharges the discrete Livšic converse for the induced base observable — vanishing periodic Birkhoff sums of \(\texttt{inducedBaseCocycle}\, F\) imply it is a discrete coboundary. Then

Under the same hypotheses, the obstruction may be phrased directly in flow-native terms: \(F\) is a flow coboundary iff every closed-orbit integral of \(F\) vanishes — for every base \(n\)-periodic point \(p\), the integral of \(F\) around the corresponding closed flow orbit (of period \(S_n\tau \, p\)) is zero.

For a measure \(\mu \) on a (pseudo-)metric measurable space \(E\) and a point \(x\), the upper local (pointwise) dimension is

the \(\limsup \) along the filter \(r \to 0^+\) of the closed-ball mass quotient. Where the genuine limit exists (as in the absolutely-continuous case below), this \(\limsup \) is the honest local dimension.

For an ergodic, log-integrable, differentiable uniformly expanding map with constant \(K {\gt} 1\) (and \(d\ge 1\)), each Lyapunov exponent of the tangent cocycle satisfies \(\log K \le \lambda _i\).

If \(P^{c}_n(x)\to P\) with \(c{\gt}0\) and \(Pv\neq 0\), and the cocycle growth sequence is cobounded, then

For \(\det A \neq 0\) everywhere and \(d \neq 0\),

Under the same hypotheses,

Let \(p : \iota \to \mathbb {R}\) satisfy \(p_i \ge 0\) for all \(i\) and \(p_i {\gt} 0\) for some \(i\). Then \(q \mapsto \log Z_q\) is convex on all of \(\mathbb {R}\). This is the cumulant-convexity / Hölder property, the mathematical core of the multifractal theory.

The Lyapunov spectrum at \(x\) is the finite set of realized values

defined as a Finset via 4.5 on the good set where \(\bar\lambda _x\) is an ultrametric growth function, and as \(\emptyset \) off it. A value lies in it iff it is realized by some nonzero vector.

Given a.e. that every realized value of the upper growth function is a deterministic exponent (upper inclusion) and every deterministic exponent is attained (lower inclusion), for \(\mu \)-a.e. \(x\),

If at \(x\) every realized exponent is a deterministic one and every deterministic exponent is attained, then \(\mathrm{lyapunovSpectrum}(x) = \mathrm{distinctExp}\, \lambda ^0\, d\).

Let \(T\) be an ergodic self-map of \(\mathbb {R}^d\) with everywhere-nonsingular derivative cocycle and one-sided log-integrable derivative data (Definition 2.4 for \(DT\) and \(DT^{-1}\)), and suppose the geometric atom-counting bound holds: every finite partition \(\alpha \) satisfies \(h(T,\alpha )\le \sum _{\lambda _i{\gt}0}\lambda _i\), the sum of the strictly positive Lyapunov exponents of the derivative cocycle (a deterministic constant, the spectrum being a.e. constant by ergodicity). Then the system entropy obeys

Same setting (ergodic \(T\) on \(\mathbb {R}^d\), nonsingular log-integrable derivative cocycle). Assume only the honest non-compactness input — the Mañé/Katok geometric atom count: for every finite partition \(P\) there are \(\varepsilon {\gt}0\) and \(C\ge 0\) such that, \(\mu \)-a.e., the number of atoms of the \(n\)-fold refinement meeting an orbit eventually satisfies

the \(\varepsilon \)-covering number of the image ellipsoid. Then

Some such distortion hypothesis is unavoidable on the noncompact phase space: the bare inequality is false in general (Riquelme 2017).

The mass exponent of the family \(p\) at scale \(\varepsilon \) is

defined for every \(q\) with no case split. For a probability family \(\tau (1) = 0\), since \(Z_1 = 1\).

Under the same hypotheses on \(p\), for a scale \(0 {\lt} \varepsilon {\lt} 1\) the mass exponent \(q \mapsto \tau (q) = \log Z_q / \log \varepsilon \) is concave on \(\mathbb {R}\).

For a measurable \(T\), a function \(g : X \to \mathbb {R}\), \(N : \mathbb {N}\) and \(x\), the maximal function is

the nonempty Finset.sup’ of the Birkhoff partial sums over \(k \in \texttt{range}\, (N+1)\). Since the \(k = 0\) term equals \(\texttt{birkhoffSum}\, T\, g\, 0\, x = 0\), the maximal function is its own positive part.

On the set where the maximal function is positive, for every \(x\) with \(0 {\lt} \texttt{maxBirkhoff}\, T\, g\, N\, x\),

For all \(g, N, x\) one has \(0 \le \texttt{maxBirkhoff}\, T\, g\, N\, x\).

For all \(g, N, x\),

For a fixed real polynomial \(q\), the map \(a\mapsto q(a)\) on \(\mathrm{Mat}_{d\times d}(\mathbb {R})\) is measurable.

Let \(M\) be measurable with each \(M x\) self-adjoint, and \(f\colon \mathbb {R}\to \mathbb {R}\) continuous. Then \(x\mapsto \mathrm{cfc}\, f\, (M x)\) is measurable.

Let \(M\colon X\to \mathrm{Mat}_{d\times d}(\mathbb {R})\) be measurable with each \(M x\) self-adjoint, and let \(g\colon \mathbb {R}\to \mathbb {R}\). If a fixed polynomial \(q\) agrees with \(g\) on the spectrum of every \(M x\), then \(x\mapsto \mathrm{cfc}\, g\, (M x)\) is measurable.

If \(A\) and \(T\) are measurable then for each \(n\) the iterate \(x \mapsto A^{(n)}(x)\) is measurable for the entrywise structure 2.1.

The determinant \(M \mapsto \det M\) is measurable.

The inverse \(M \mapsto M^{-1}\) is measurable on the entrywise structure.

The map \(M \mapsto \left\lVert M \right\rVert \) on \(\operatorname {Matrix}(\operatorname {Fin}d)(\operatorname {Fin}d)\, \mathbb {R}\) is measurable.

For fixed \(v\), the map \(x\mapsto \bar\lambda _x(v)\) is measurable.

\(\operatorname {rep}\) is \(\operatorname {generateFrom}(\operatorname {dyadicArcSet})\)-measurable.

A subspace-valued map \(V\colon X\to \mathrm{Submodule}\, \mathbb {R}\, \mathbb {R}^d\) is a measurable family of subspaces when \(x\mapsto \mathrm{orthProjMatrix}(V x)\) is measurable. Equivalently (measurable_orthProjMatrix_iff), for each standard basis index \(j\) the \(\mathbb {R}^d\)-valued map \(x\mapsto (V x).\mathtt{starProjection}(e_j)\) is measurable.

For a finite measure, measure-preserving \(T\), integrable \(g\) and \(\varepsilon {\gt} 0\), the superlevel set where the limsup of the Birkhoff averages exceeds \(\mu [g\mid \mathcal I] + \varepsilon \) is null.

A finite measurable partition of \((\alpha ,\mu )\) is a structure bundling a Fintype-indexed family of cells \(A_i\subseteq \alpha \) together with proofs that each cell is measurable, that the cells are pairwise almost-everywhere disjoint, and that they cover the whole space, \(\bigcup _i A_i=\alpha \).

\(\operatorname {catTorus}\) preserves the Haar probability measure on \(\mathbb {T}^2\).

A measure-preserving one-parameter flow on a measurable space \( X \) for a measure \( \mu \) is a family \( \varphi : \mathbb {R}\to X \to X \) together with the data \( \varphi (0) = \mathrm{id} \), \( \varphi (s+t) = \varphi (s) \circ \varphi (t) \) for all \( s, t \in \mathbb {R}\), and a proof that each time-\( t \) map \( \varphi (t) \) preserves \( \mu \). No topology on \( X \) is assumed; in particular no joint continuity in \( (t, x) \) is required.

For a finite weight family \(p : \iota \to \mathbb {R}\) and a merge map \(f : \iota \to \kappa \), the merged weights \(\mathrm{mergedWeights}\, f\, p : \kappa \to \mathbb {R}\) push \(p\) forward along \(f\) by summing each fibre: \((\mathrm{mergedWeights}\, f\, p)(b) = \sum _{a \, :\, f a = b} p_a\).

For a faithful (positive-definite) density matrix \(\rho \), the modular automorphism \(\sigma _t(a)=\rho ^{it}\, a\, \rho ^{-it}\) is built from the unitary power \(\rho ^{it}\) of the Tomita–Takesaki / Petz-equality infrastructure. It is the finite-dimensional shadow of the type-III modular flow of the qubit chain.

The modular flow is a one-parameter group, \(\sigma _s\circ \sigma _t=\sigma _{s+t}\):

Together with \(\sigma _0=\mathrm{id}\), multiplicativity \(\sigma _t(a\, b)=\sigma _t(a)\, \sigma _t(b)\) and the adjoint law \(\sigma _t(a^{\dagger })=\sigma _t(a)^{\dagger }\), this makes \((\sigma _t)_t\) a one-parameter \(*\)-automorphism group.

A faithful non-tracial (Powers-type) product state \(\operatorname {diagState} s\) (\(0{\lt}s{\lt}1\), \(\rho =\operatorname {diag}((1+s)/2,(1-s)/2)\)) has a nontrivial modular flow: the modular group is not the trivial action. Concretely, at \(t_0=\pi /\log \bigl((1+s)/(1-s)\bigr)\) the flow sends the off-diagonal unit \(E_{01}\) to \(-E_{01}\).

The maximally mixed (tracial) state has trivial modular flow, \(\sigma _t=\mathrm{id}\) for all \(t\):

In particular this holds at every level \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\) of the chain — the type II\(_1\) shadow.

The intrinsic clock is consistent along the chain: on the embedded block \(\mathrm{shiftAdjoinQubit}\, a=1\otimes a\), the modular flow of the \((n+1)\)-fold product state acts by the modular flow of the \(n\)-fold state on \(a\),

because \(\rho ^{it}\) acts trivially on the fresh qubit.

Unconditionally, for a faithful ancilla \(\alpha \), a unitary dilation \(U\), and faithful \(\sigma \),

This is the data-processing inequality for the faithful-ancilla mixed-Stinespring family; that family does not contain every CPTP channel (amplitude damping lies outside it, since its exact dilation needs a pure, non-faithful ancilla), so it is not an in-repo DPI for a general Kraus channel.

If a nonnegative sequence \(a\) obeys a one-step recursion fed by finitely many source sequences each decaying geometrically with ratio \(\rho {\lt}1\), then \(a_n\) is bounded by a fixed multiple of the summed source envelopes for all \(n\).

For a measure-preserving flow \( \varphi \) and \( n \in \mathbb {N}\), the integer-time map of the flow is the \( n \)-fold iterate of its time-one map: \( \varphi (n) = (\varphi (1))^{[n]} \).

Under the same hypotheses,

If a map \(R\) satisfies the faithful data-processing inequality and inverts a coarse-graining \(\Lambda \) on the pair \(\rho ,\sigma \) (\(R(\Lambda \rho ) = \rho \), \(R(\Lambda \sigma ) = \sigma \) with \(\Lambda \sigma \) faithful), then a strict drop \(S(\Lambda \rho \| \Lambda \sigma ) {\lt} S(\rho \| \sigma )\) is impossible.

Unconditionally: a strict relative-entropy drop under a coarse-graining \(\Lambda \) (an endomorphism of the input space) rules out any Stinespring recovery channel \(R\, x = \operatorname{Tr}_E(U(x\otimes \alpha )U^{*})\) inverting \(\Lambda \) on \(\rho ,\sigma \).

Assume \(\det A(x) \neq 0\) for all \(x\) and \(d \neq 0\). Then \(0 {\lt} \left\lVert A^{(n)}(x) \right\rVert \) for every \(n, x\). The analogous statement \(\texttt{norm\_ inv\_ cocycle\_ pos}\) holds for the inverse iterates \(\left\lVert (A^{(n)}(x))^{-1} \right\rVert \).

When \(d \neq 0\), \(\left\lVert (1 : \operatorname {Matrix}(\operatorname {Fin}d)(\operatorname {Fin}d)\, \mathbb {R}) \right\rVert = 1\) for the L2 operator norm.

For a self-adjoint matrix \(M\) and any function \(g\),

where \(\mu _j\) ranges over the sorted eigenvalues of \(M\).

The CNT/ALF entropy sequence \(n\mapsto S(\rho [X^{(n)}])\) of the partition above, under the identity dynamics and the invariant pure state, is not subadditive: it violates \(u_2\le u_1 + u_1\), since

This is a feature, not a defect: it is precisely why the CNT rate is an infimum (an \(\texttt{sInf}\)), not a Fekete limit of a subadditive sequence.

\((P_K\, P_L\, P_K)\, v = v\) if and only if \(v \in K \sqcap L\).

An operational partition of unity of size \(k\) is a family \((x_i)_{i{\lt}k}\) of operators in \(\operatorname {Matrix}_d(\mathbb {C})\) satisfying the partition-of-unity relation \(\sum _i x_i^{\dagger }x_i=1\). This is the noncommutative analogue of a measurable partition of the state space.

The function \(x\mapsto -\log x\) is operator convex on \((0,\infty )\): for every matrix dimension its continuous functional calculus is convex in the Loewner order on the self-adjoint matrices with spectrum in \((0,\infty )\).

For \(f:\mathbb {R}\to \mathbb {R}\) and matrices \(L,R\) with \(R\) positive definite, the operator perspective is

with the functional calculus and real powers supplied by the continuous functional calculus.

Let \(f\) be operator convex on \(I\), let \(R_1,R_2\) be positive definite with the sandwiched arguments \(R_i^{-1/2}L_iR_i^{-1/2}\) self-adjoint and with spectra in \(I\), and let \(c\in [0,1]\). Then

The same formula \(P_f(L,R) = R^{1/2}f(R^{-1/2}LR^{-1/2})R^{1/2}\) for matrices indexed by an arbitrary finite type \(m\) (rather than \(\operatorname {Fin}N\)).

Under the same hypotheses as Theorem 16.29, joint convexity of \(\mathrm{opPersp}\, f\) holds over an arbitrary finite index type \(m\).

For positive-definite \(\rho ,\sigma \), the perspective of \(-\log \) at the commuting pair \(L = 1\otimes \sigma ^{\top }\), \(R = \rho \otimes 1\) has the closed form

For \(K\le \mathbb {R}^d\), \(\mathrm{orthProjMatrix}(K)\) is the matrix of the orthogonal projection onto \(K\), namely the preimage of \(K.\mathtt{starProjection}\) under the star-algebra isomorphism \(\mathtt{toEuclideanCLM}\). A subspace is determined by this matrix, which lives in a space carrying the Borel/Pi measurable structure. Its \((i,j)\) entry equals the \(i\)-th coordinate of the projection applied to the standard basis vector \(e_j\) (orthProjMatrix_apply).

Let \(\mu \) be a probability measure, \(T:X\to X\) ergodic measure-preserving, and \(A:X\to \mathrm{Mat}_{d\times d}(\mathbb {R})\) measurable with \(\det (A x)\neq 0\) and \(\log ^{+}\left\lVert A \right\rVert ,\, \log ^{+}\left\lVert A^{-1} \right\rVert \in L^1(\mu )\). Then there are \(k\) distinct Lyapunov exponents \(\lambda :\mathrm{Fin}\, k\to \mathbb {R}\), strictly decreasing, and a measurable family of subspaces

with each \(x\mapsto V_i\, x\) measurable, such that for \(\mu \)-a.e. \(x\): \(V_0\, x=\top \), \(V_k\, x=\bot \); the flag is strictly decreasing, \(V_{i+1}\, x {\lt} V_i\, x\); it is \(A\)-equivariant, \(A(x)\, V_i\, x = V_i\, (Tx)\); and along it the cocycle grows at the exact rate \(\lambda _i\):

Under the standing hypotheses (ergodic \(T\), invertible measurable \(A\) with \(\log ^+\left\lVert A \right\rVert , \log ^+\left\lVert A^{-1} \right\rVert \in L^1\)), there exist \(k, \lambda , V\) with \(\mathrm{IsOseledetsFiltration}\ \mu \ T\ A\ k\ \lambda \ V\).

Let \(\mu \) be a probability measure on \(E = \operatorname {EuclideanSpace}\, \mathbb {R}\, (\operatorname {Fin}d)\) and let \(T\) be ergodic for \(\mu \), differentiable, with everywhere nonvanishing Jacobian determinant \(\det (\operatorname {derivativeCocycle}\, T\, x)\ne 0\) and with the log-integrability \(\log ^{+}\left\lVert D_xT \right\rVert ,\ \log ^{+}\bigl\lVert (D_xT)^{-1} \bigr\rVert \in L^1(\mu )\) (stated for the matrix generator and its matrix inverse; by the isometry these are exactly the \(\operatorname {fderiv}\) conditions). Then:

1. for every \(n\) and \(x\), the cocycle iterate represents \(D_x(T^{[n]})\) — the cocycle is the genuine tangent cocycle; and

2. there exist \(k\le d\), strictly decreasing exponents \(\lambda _1 {\gt} \dots {\gt} \lambda _k\), and a measurable family of subspaces \(E = V_0(x) \supsetneq V_1(x) \supsetneq \dots \supsetneq V_k(x) = 0\), a.e. equivariant under \(D_xT\), such that for every \(v\in V_{i-1}(x)\setminus V_i(x)\),

   \[ \frac1n\, \log \bigl\lVert D_x\bigl(T^{[n]}\bigr)v \bigr\rVert \; \longrightarrow \; \lambda _i . \]

Under the one-sided hypotheses (with \([\mathrm{NeZero}\, d]\)) there exist a decreasing exponent sequence \(\mathrm{lam0} : \mathbb {N}\to \mathbb {R}\) realizing the a.e. per-index singular-value limits, and a measurable filtration \(V : \mathrm{Fin}(k+1) \to X \to \mathrm{Submodule}\, \mathbb {R}\, (\mathbb {R}^d)\) where \(k = \mathrm{numExp}\, \mathrm{lam0}\, d\), such that for a.e. \(x\): \(V_0 x = \top \), \(V_k x = \bot \), \(V\) is strictly decreasing and \(A\)-equivariant, every \(v \in V_{i}x \setminus V_{i+1}x\) grows at rate \(\mathrm{expEnum}\, \mathrm{lam0}\, d\, i\), and

Under the standing ergodic, invertible, log-integrable hypotheses, and assuming the top-gap envelope \(\mathrm{TopGapMassEnvelope}\) quantified over \(\lambda ^0\), the full Oseledets filtration conclusion holds.

Assume the per-vector spectral upper bound on the slow flag, the reverse slow-flag inclusion, the two spectrum inclusions, and the band-projector convergence datum. Then there exist \(k\), strictly decreasing \(\lambda :\mathrm{Fin}\, k\to \mathbb {R}\), and a measurable family \(V\) forming a.e. a strictly decreasing \(A\)-equivariant flag along which \(\tfrac 1n\log \left\lVert A^{(n)}(x)v \right\rVert \to \lambda _i\) on each stratum.

Under the standing hypotheses there exist \(k, \lambda , V\) and a strictly decreasing \(m\) with \(m_0 = d\), \(m_k = 0\), such that \(\mathrm{IsOseledetsFiltration}\ \mu \ T\ A\ k\ \lambda \ V\) holds and \(\mu \)-a.e. \(\mathrm{finrank}\, V_i(x) = m_i\).

Let \( \mu \) be a probability measure on \( X \), let \( \varphi \) be a measure-preserving one-parameter flow whose time-one map \( \varphi (1) \) is \( \mu \)-ergodic, and let \( A \) be a continuous-time linear cocycle over \( \varphi \) valued in invertible \( d \times d \) real matrices. Suppose \( g, g' \in L^1(\mu ) \) satisfy, for all \( s \in [0,1] \) and all \( x \),

Then there exist \( k \in \mathbb {N}\), a strictly decreasing sequence of exponents \( \lambda : \mathrm{Fin}\, k \to \mathbb {R}\), and a measurable family of subspaces \( V : \mathrm{Fin}(k+1) \to X \to \mathrm{Submodule}\, \mathbb {R}\, (\mathrm{EuclideanSpace}\, \mathbb {R}\, (\mathrm{Fin}\, d)) \) such that:

• \( \lambda \) is strictly decreasing and each \( V_i \) is a measurable subspace family;

• (full flow equivariance) for every \( t \in \mathbb {R}\), almost every \( x \) has \( A(t,x) \cdot V_i\, x = V_i\, (\varphi (t)x) \) for all \( i \);

• almost every \( x \) carries the strict flag \( \top = V_0\, x \supsetneq \cdots \supsetneq V_k\, x = \bot \), and on each stratum \( v \in V_i\, x \setminus V_{i+1}\, x \) the continuous-time growth rate is exactly \( \lambda _i \):

  \[ t^{-1}\log \left\lVert A(t,x)v \right\rVert \to \lambda _i \quad (t \to \infty ). \]

Let \(T : X \simeq _m X\) be an invertible ergodic measure-preserving transformation of a probability space, and let \(A : X \to \mathrm{Matrix}\, (\mathrm{Fin}\, d)\, (\mathrm{Fin}\, d)\, \mathbb {R}\) be measurable with \(\det A(x) \neq 0\) for all \(x\) and \(\log ^+\left\lVert A \right\rVert , \log ^+\left\lVert A^{-1} \right\rVert \in L^1(\mu )\). Then there exist \(k \in \mathbb {N}\), a strictly decreasing \(\lambda : \mathrm{Fin}\, k \to \mathbb {R}\), and measurable subspace families \(E : \mathrm{Fin}\, k \to X \to \mathrm{Submodule}\, \mathbb {R}\, (\mathbb {R}^d)\) such that for \(\mu \)-a.e. \(x\):

• \(\mathbb {R}^d = \bigoplus _{i} E_i(x)\) is an internal direct sum with every \(E_i(x) \neq \bot \);

• each \(E_i\) is \(A\)-equivariant: \(\bigl(A(x)\bigr)\bigl(E_i(x)\bigr) = E_i(Tx)\);

• for every nonzero \(v \in E_i(x)\),

  \[ \tfrac {1}{n}\log \bigl\lVert A^{(n)}(x)\, v \bigr\rVert \to \lambda _i \qquad \text{and}\qquad \tfrac {1}{n}\log \bigl\lVert \bigl(A^{(n)}(T^{-n}x)\bigr)^{-1}\, v \bigr\rVert \to -\lambda _i. \]

Any constant \(c\) to which \(\tfrac 1n\log \left\lVert A^{(n)}(x) \right\rVert \) converges \(\mu \)-a.e. (e.g. the Furstenberg–Kesten constant) equals \(\lambda _0\).

\(\Lambda (x) := (\, \lim _n (q_n(x))_{ij}\, )_{ij}\) is the entrywise real \(\limsup \)/limit of the matrix entries of \(q_n(x)\); it is a total, measurable function of \(x\).

For \(\mu \)-a.e. \(x\), \(\Lambda (x)\) is self-adjoint and positive semidefinite.

Under the same entropy-preservation hypothesis, the amplified output modular \(it\)-cocycle equals the input one, for all \(t\in \mathbb {R}\):

If the partial-trace relative entropy is preserved, \(S(\operatorname {Tr}_B\omega \| \operatorname {Tr}_B\tau )=S(\omega \| \tau )\), then the rectangular \(-\log \) operator-Jensen gap for the Petz isometry \(W\) annihilates the output cyclic vector \(\xi =\operatorname {vec}((\operatorname {Tr}_B\omega )^{1/2})\):

For an operator \(M\) on a bipartite system \(\mathbb {C}^{n_A}\otimes \mathbb {C}^{n_B}\), the partial trace over the \(B\) factor is

an operator on \(\mathbb {C}^{n_A}\). (The left partial trace \(\operatorname{Tr}_A\) is defined symmetrically.)

Both partial traces are genuine sums of Kraus conjugations:

The left analogue is partialTraceLeft_eq_kraus.

For any state \(\rho \) and any ancilla state \(\alpha \), \(\operatorname{Tr}_E(\rho \otimes \alpha ) = \rho \).

For a finite weight family \(p : \iota \to \mathbb {R}\) and \(q \in \mathbb {R}\), the generalized partition function is

the sum over the occupied cells only. The positivity guard is load-bearing at \(q = 0\): it forces empty cells (\(p_i = 0\)) to contribute \(0\) rather than \(0^0 = 1\), so that \(Z_0\) counts the occupied cells. For \(q \ne 0\) the guard is removable (\(0^q = 0\)), and for a probability family (\(p_i \ge 0\), \(\sum _i p_i = 1\)) one has \(Z_1 = 1\).

Let \(p : \iota \to \mathbb {R}\) with \(p_a \ge 0\). For \(q \ge 1\) the partition function grows under merge, \(Z_q(p) \le Z_q(\mathrm{mergedWeights}\, f\, p)\) (partitionFunction_merge_ge, per-fibre superadditivity of \(x \mapsto x^q\)), while for \(0 \le q \le 1\) it shrinks, \(Z_q(\mathrm{mergedWeights}\, f\, p) \le Z_q(p)\) (partitionFunction_merge_le, subadditivity).

The Pauli operational partition of \(\operatorname {Matrix}_2(\mathbb {C})\) is the family of four operators \((\tfrac 12 P_i)_{i{\lt}4}\), where \(P_0 = 1\), \(P_1 = X\), \(P_2 = Y\), \(P_3 = Z\) are the Pauli matrices. Each \(P_i\) is Hermitian and unitary, so \(\sum _i (\tfrac 12 P_i)^{\dagger }(\tfrac 12 P_i) = \sum _i \tfrac 14 P_i^{2} = 4\cdot \tfrac 14\cdot 1 = 1\), an operational partition of unity of size \(4 = 2^{2}\).

At a fixed cut strictly inside a gap of width \(\ge G\), the one-step band-mass increment is the current mass damped by \(e^{-G}\) plus a tempered source term; iterating produces the per-stratum leakage envelope.

Under the same hypotheses, with \(\chi \) an unstable-Jacobian integrand (Definition 14.4), the system entropy is the literal Pesin integral

For an ergodic differentiable self-map \(T\) of \(\mathbb {R}^d\) preserving an SRB (volume-case) measure \(\mu \) (Definition 14.3) with nonsingular log-integrable derivative cocycle and nonnegative Lyapunov spectrum (hspec), and under the honest non-compactness atom-count input hgeo of the sharp Margulis–Ruelle inequality, the Kolmogorov–Sinai entropy equals the sum of the strictly positive Lyapunov exponents:

This is Pesin’s entropy formula in the absolutely continuous case (Pesin 1977; Ledrappier–Young 1985), obtained here foliation-free.

For the doubling map,

the genuine full-system Pesin equality on a compact carrier with strictly positive entropy. This is the first non-vacuous full-system instance of the volume-case Pesin formula (Theorem 14.7) in the library: the binary-expansion coding gives a genuine generating partition on a compact space, so both sides are computed rather than degenerate. Its companion Theorem 13.18 exhibits the same equality against the integrated log-Jacobian \(\int \log \left\lvert \det DT \right\rvert \, d\mu =\log 2\).

For an ergodic, absolutely continuous (\(\mu \ll \operatorname {volume}\)), differentiable, uniformly expanding self-map of \(\operatorname {EuclideanSpace}\, \mathbb {R}\, (\operatorname {Fin}d)\) with everywhere-nonsingular derivative, log-integrable derivative data, a one-sided generating injectivity partition, and integrable \(\log \rho \) and \(\log \left\lvert \det DT \right\rvert \):

Honest disclosure: this is a correct implication whose hypothesis bundle has no model on the non-compact space \(\mathbb {R}^d\) — a globally uniformly expanding map of \(\mathbb {R}^d\) admits no ergodic absolutely continuous invariant probability measure (uniform expansion forces mass to escape to infinity; for \(T = c\cdot \operatorname {id}\) the nested preimages of a ball force an atom at the fixed point). The theorem is therefore vacuously true as stated on \(\mathbb {R}^d\), and is disclosed as such; the instantiated Pesin/Rokhlin equality lives on the compact circle (Theorem 13.18).

For the doubling map and its binary partition, the partition-relative Kolmogorov–Sinai entropy equals the sum of the positive Lyapunov exponents, both being \(\log 2\): \(h_\mu (T,\alpha )=\sum \lambda ^{+}\).

For a state \(\sigma \) and a channel \(\Lambda \), the Petz (transpose) recovery map is

realised through the continuous functional calculus (CFC.conjSqrt for the \(\sqrt{\cdot }\)-conjugations and the ring inverse for \((\Lambda \sigma )^{-1/2}\)).

Let \(\Lambda \) be any Kraus channel and \(\rho ,\sigma \) states with all four of \(\rho ,\sigma ,\Lambda \rho ,\Lambda \sigma \) positive definite. If data processing is saturated,

then the Petz recovery map reconstructs the input state,

No injectivity of the channel (or of the vectorised Petz map) is assumed; the result holds for information-losing channels such as the completely depolarising channel.

If the channel output \(\Lambda \sigma \) is positive definite, then the Petz map recovers \(\sigma \) from \(\Lambda \sigma \):

Let \(\Lambda \) and \(R\) be maps on states, each monotone under relative entropy (each satisfies the data-processing inequality against positive-definite second arguments). If \(R\) is a recovery section for the pair \(\rho ,\sigma \), i.e. \(R(\Lambda \rho )=\rho \) and \(R(\Lambda \sigma )=\sigma \) (with \(\sigma ,\Lambda \sigma \) positive definite), then the data-processing inequality is saturated:

The number of index families \(f : \operatorname {Fin} n \to \iota \) whose atom of the flat \(n\)-fold join \(\bigvee _{k{\lt}n} T^{-k}P\) (Definition 10.8) has nonzero \(\mu \)-mass. It is at most the set-nonempty atom count, yet still bounds the entropy from above (a null junk cell is not counted), giving a sharper crude Ruelle backbone.

If \(M\) is positive semidefinite then so is \(\operatorname{Tr}_B M\).

Each digit set \(T^{-n}(\operatorname {binCell}1)\) is the finite union of the generation-\((n+1)\) dyadic arcs whose last digit is \(1\), hence \(\operatorname {generateFrom}(\operatorname {dyadicArcSet})\)-measurable.

For a cell map \(c:\operatorname {Fin}d\to \operatorname {Fin}k\), the diagonal projection partition is \(\{ \operatorname {diag}\mathbf1_{c^{-1}(i)}\} _{i{\lt}k}\); it is an operational partition of unity encoding the measurable partition \(c^{-1}(\cdot )\) of \(\operatorname {Fin}d\).

The length-\(n\) qubit block index \(\mathrm{Qbits}\, n\) is defined by recursion, \(\mathrm{Qbits}\, 0 = \operatorname {Fin}1\) and \(\mathrm{Qbits}\, (n+1) = \operatorname {Fin}2\times \mathrm{Qbits}\, n\); it has cardinality \(2^{n}\), so the block algebra is \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\).

The seal’s diagonal MASA is not dynamics-invariant — \(U\cdot \operatorname {diag}(1,0)\cdot U^\dagger \) has a nonzero \((1,0)\) entry () — and the dynamics’ MASA (the eigenbasis, \(=\) commutant of the eigenprojection \(\operatorname {eigProj}\)) is not seal-invariant: the dephasing map moves \(\operatorname {eigProj}\) out of that MASA, \(E(\operatorname {eigProj})\) failing to commute with \(\operatorname {eigProj}\) ().

The norm form \(Q(p,q) = p^2 - p q - q^2\) of the cat-map matrix \(A = \bigl(\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\bigr)\) is the norm form of the ring \(\mathbb {Z}[\varphi ]\) (discriminant \(5\)). It is exactly \(A\)-invariant, \(Q(A^k n) = Q(n)\) (), and — being the norm of a nonzero element of \(\mathbb {Z}[\varphi ]\) — never vanishes on a nonzero lattice vector (), hence \(\left\lvert Q(n) \right\rvert \ge 1\) there.

The maximally coherent pure state \(|{+}\rangle \langle {+}|\) dephases to \(I/2\), a strict relative-entropy drop of \(S(I/2) - S(|{+}\rangle \langle {+}|) = \log 2 {\gt} 0\) against \(\operatorname {diagState} s\); consequently no faithful-ancilla Stinespring section recovers it.

The faithful one-parameter family \(\rho _r = \tfrac 12\! \left(\begin{smallmatrix} 1 & r \\ r & 1 \end{smallmatrix}\right)\) (\(0 {\lt} r {\lt} 1\), positive definite) dephases to \(I/2\) with a strict drop \(\log 2 - h_2\! \big((1+r)/2\big) {\gt} 0\), where \(h_2\) is Mathlib’s binary entropy (whose strict maximum at \(1/2\) supplies the strict inequality); again no faithful-ancilla Stinespring section recovers it.

For an arbitrary block \(\mathrm{blk}\) (in particular every tower level \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\)) and a faithful block state \(\beta \), no faithful-ancilla Stinespring channel on the enlarged system \(\operatorname {Matrix}_2(\mathbb {C})\otimes \operatorname {Matrix}_{\mathrm{blk}}(\mathbb {C})\) inverts the partial dephasing on \(\rho _r\otimes \beta \) and \(\operatorname {diagState} s\otimes \beta \) simultaneously.

A quantum Bernoulli shift bundles the fixed hierarchy of finite marginals of the one-sided qubit chain into one certificate:

1. the directed system — the inclusion \(\mathrm{appendQubit}\) and the shift \(\mathrm{shiftAdjoinQubit}\) commute (Theorem 17.80);

2. the inclusion- and shift-compatibility of the tracial state (Theorems 17.84 and 17.83);

3. the temporal entropy rate \(\log 2\) along the shift-window filtration (Theorem 17.89);

4. the per-stage dephasing seal at the tracial blocks, for some faithful parameters \(r,s\) (Definition 17.91, Theorem 17.92);

5. the trivial tracial modular clock at every level (Theorem 17.98).

The maps are definitionally the fixed \(\mathrm{appendQubit}\) / \(\mathrm{shiftAdjoinQubit}\), so only their laws are recorded.

A concrete quantum Bernoulli shift exists. The witness assembles the already-proved theorems: the inclusion/shift commutation, the tracial inclusion- and shift-compatibility, the temporal rate \(\log 2\), the per-stage seal at the balanced parameters \(r=s=1/2\), and the trivial tracial clock.

For the constant unit roof, reading off a measurable canonical representative \(\operatorname {unitFwd} q = (\operatorname {baseIter}\lfloor s\rfloor x, \{ s\} )\) of each orbit class \(q = [x,s]\), the assignment \(t \mapsto \operatorname {cocycleZ} A\, T\, \lfloor \{ s\} + t\rfloor \, (\operatorname {unitFwd} q)_1\) is a genuine Definition 8.3 on the mapping torus. The four cocycle fields follow from Theorem 9.52 transported through the floor split \(\lfloor a + t\rfloor = \lfloor a\rfloor + \lfloor \{ a\} + t\rfloor \).

The correlation density matrix of an \(n\)-fold refinement has rank at most \(d^{2}\), uniformly in \(n\):

Let \(a:\mathbb {N}\to \mathbb {R}\) be nonnegative and uniformly bounded above by a fixed constant \(C\) (a reservoir). Then the per-step rate vanishes,

Let \(W:\mathbb {C}^q\to \mathbb {C}^p\) be an isometry (\(W^{\dagger }W=1\)) and \(X\) self-adjoint with spectrum in \((0,\infty )\). Then

Given a dynamics \(\Phi \) and an operational partition \(X=(x_i)\), the time-ordered refinement of depth \(n\) along a word \(f\in (\operatorname {Fin}k)^{\operatorname {Fin}n}\) is the operator

defined by the telescoping recursion \(\mathrm{refine}(n+1,f)=x_{f_0}\cdot \Phi \! \bigl(\mathrm{refine}(n,\mathrm{tail}\, f)\bigr)\) with \(\mathrm{refine}(0,\cdot )=1\). It records the observable measured along the first \(n\) steps of the orbit under \(\Phi \).

Let \(p, q : \mathbb {N}\to \mathbb {R}\) be antitone on \([0,d)\). If \(\# \{ p \le a\} + \# \{ q \le b\} \le d\) whenever \(a + b {\lt} 0\), and \(\sum _{j{\lt}d} q\, j = -\sum _{j{\lt}d} p\, j\), then \(q\, j = -p\, (d-1-j)\) for all \(j {\lt} d\).

The Umegaki relative entropy of \(\rho \) with respect to \(\sigma \) is \(S(\rho \| \sigma ) = \operatorname{Tr}\rho (\log \rho - \log \sigma )\). It is defined in the concrete spectral/overlap form

where \((p_k,e_k)\) are the eigenvalues/eigenvectors of \(\rho \) and \((q_m,f_m)\) those of \(\sigma \). The definition is total: with the convention \(\log 0 = 0\) the \(\sigma \)-singular columns contribute \(0\).

Tensoring both arguments with a common faithful ancilla \(\alpha \) leaves the relative entropy unchanged: for faithful \(\sigma \) and faithful \(\alpha \), \(S(\rho \otimes \alpha \, \| \, \sigma \otimes \alpha ) = S(\rho \| \sigma )\).

For any unitary \(W\), \(\; S(W\rho W^{*}\, \| \, W\sigma W^{*}) = S(\rho \| \sigma )\), where \(W\rho W^{*}\) denotes the density matrix obtained by conjugating \(\rho \).

For states \(\rho ,\sigma \) on \(n\), a faithful ancilla \(\alpha \) on \(e\), a unitary \(U\) on \(n\times e\), and faithful \(\sigma \),

For faithful (positive definite) \(\sigma \), the spectral definition agrees with the textbook trace form

where \(\log \rho ,\log \sigma \) are the Hermitian continuous functional calculus of the real logarithm.

\(D(\rho \, \| \, I/d) = \log d - S(\rho )\); the cross term collapses through the doubly-stochastic row sum to \(-\log d\cdot \operatorname {Tr}\rho = -\log d\). Unitary invariance of the von Neumann entropy () is proved alongside, via \(\operatorname {charpoly\_ mul\_ comm}\).

For faithful (positive definite) \(\sigma \), \(\; 0\le S(\rho \| \sigma )\).

\(S(\rho \| \rho ) = 0\) for every density matrix \(\rho \).

For a faithful block state \(\beta \) and the reference \(\sigma = \operatorname {diagState} s\) (\(0 {\lt} s {\lt} 1\)), partial dephasing of \(\rho _r\otimes \beta \) against \(\operatorname {diagState} s\otimes \beta \) strictly lowers the Umegaki relative entropy:

For matrices \(\rho ,\sigma \), the trace-form relative entropy is \(\mathrm{relEntropyMat}(\rho ,\sigma ) = \Re \, \operatorname{Tr}\bigl(\rho \, (\log \rho - \log \sigma )\bigr)\), the logarithms taken through the continuous functional calculus.

For a finite convex combination of faithful states — weights \(w_i\ge 0\) with \(\sum _i w_i = 1\) and positive-definite \(\rho _i,\sigma _i\) —,

For density matrices \(\rho ,\sigma \) with \(\sigma \) faithful (positive definite), \(\mathrm{relEntropyMat}(\rho ,\sigma ) = S(\rho \| \sigma )\), the Umegaki relative entropy.

For positive-definite \(\rho _1,\rho _2,\sigma _1,\sigma _2\) and \(c\in [0,1]\),

The proposition RelEntropyMonotoneUnderPartialTrace holds: the faithfulness hypothesis on \(\rho \) is unnecessary, so \(S(\operatorname{Tr}_E\rho \| \operatorname{Tr}_E\sigma )\le S(\rho \| \sigma )\) for every \(\rho \) and every faithful \(\sigma \).

For positive-definite states \(\rho ,\sigma \) on \(n_A\times n_E\),

RelEntropyMonotoneUnderPartialTrace is the proposition that for all finite index types \(n_A, n_E\) and all states \(\rho ,\sigma \) on \(n_A\times n_E\) with \(\sigma \) faithful (positive definite), tracing out the \(E\)-factor does not increase relative entropy,

The index types are quantified over Type (universe \(0\)): finite-dimensional quantum information lives in universe \(0\), and pinning the universe lets this Prop be used as a reusable hypothesis whose subsystems unify with any consumer’s.

The positive linear functional \(M\mapsto \langle \mathrm{vec}\, 1,\, M\, \mathrm{vec}\, 1\rangle \) recovers the trace-form relative entropy from the perspective:

The Rényi (generalized) dimension of \(p\) at scale \(\varepsilon \) is

At \(q = 1\) the general formula is the indeterminate \(0/0\), so the L’Hôpital value — the information dimension — is supplied directly as a separate branch. (By the Mathlib convention \(\log 0 = 0\), the \(q = 1\) numerator needs no positivity guard.)

Let \(p\) be a probability weight family (\(p_i \ge 0\), \(\sum _i p_i = 1\), at least one \(p_i {\gt} 0\)) and \(0 {\lt} \varepsilon {\lt} 1\). Then \(q \mapsto D_q\) is non-increasing (Antitone) on all of \(\mathbb {R}\) — including across the information-dimension branch point \(q = 1\).

For a measure-preserving flow \(\varphi \) with invariant probability measure \(\mu \) (Definition 8.1), a partition \(P\), and \(\varepsilon , q\), the Rényi dimension of the flow’s invariant measure is \(D_q(\varphi , P, \varepsilon ) = D_q(\mu , P, \varepsilon )\). The flow is an explicit (unused) argument whose type documents that \(\mu \) is flow-invariant; the multifractal API consumes any invariant probability measure.

For a measure-preserving flow \(\varphi \) of a probability measure \(\mu \), a partition \(P\), and \(0 {\lt} \varepsilon {\lt} 1\), the flow Rényi dimension \(q \mapsto D_q(\varphi , P, \varepsilon )\) is non-increasing in \(q\).

Let \(\alpha _0\) consist of exactly two symbols \(i \ne j\), let \(\nu \) charge both with positive but different masses (\(\nu \{ i\} \ne \nu \{ j\} \) after toReal), and let \(0 {\lt} \varepsilon {\lt} 1\). Then the Rényi dimension of the suspension flow’s invariant measure on the witness partition takes different values at two exponents:

The exhibited exponents are the explicit \(q_1 = 0\), \(q_2 = 1\) (renyiDimFlow_bernSuspension_zero_ne_one): concretely \(D_0 = \log 2 / (-\log \varepsilon )\) (both cells occupied) while \(D_1 = H(\nu )/(-\log \varepsilon )\) (the information dimension), and these differ precisely because the bias forces the strict inequality \(H(\nu ) {\lt} \log 2\). The witness is therefore non-vacuous: the \(q\)-dependence is driven by the genuine bias of \(\nu \), not satisfied trivially.

For a measure \(\mu \) on \(\alpha \), a finite measurable partition \(P\) of \(\mu \) indexed by \(\iota \), and \(\varepsilon , q \in \mathbb {R}\), the Rényi dimension of \(\mu \) at partition scale \(\varepsilon \) is the abstract \(D_q\) of the cell-mass family \(i \mapsto \mu (P_i)\) (real-valued via toReal). The companion definitions partitionFunctionMeasure and massExponentMeasure specialize \(Z_q\) and \(\tau \) in the same way.

For a probability measure \(\mu \), a finite measurable partition \(P\), and a scale \(0 {\lt} \varepsilon {\lt} 1\), the Rényi dimension \(q \mapsto D_q(\mu , P, \varepsilon )\) is non-increasing in \(q\).

For a probability measure \(\mu \), a partition \(P\), and any \(\varepsilon \),

where \(H(P) = \sum _i \operatorname {negMulLog}\bigl(\mu (P_i)\bigr) = -\sum _i \mu (P_i)\log \mu (P_i)\) is the Shannon entropy of the partition. For \(0 {\lt} \varepsilon {\lt} 1\) this is the familiar \(D_1 = H(P)/\log (1/\varepsilon )\).

The Rényi entropy of order \(q\) of a family \(p : \iota \to \mathbb {R}\) is

reusing the generalized partition function \(Z_q\) (Definition 12.1) verbatim as the power sum \(\sum _{p_i {\gt} 0} p_i^{\, q}\).

For every \(f : \iota \to \kappa \), family \(p \ge 0\), and order \(0 \le q\), \(q \ne 1\), coarse-graining does not increase Rényi entropy:

Moreover (renyiEntropy_merge_lt) the inequality is strict whenever the merge glues two atoms of the support, for \(q \in (0,1) \cup (1,\infty )\).

The preimage of a length-\(n\) cylinder of \(\mathrm{Shift}\, B\) under \(\mathrm{blockCode}\, \varphi \) is the finite disjoint union of the length-\(n\) cylinders over the fibre of \(\mathrm{wordMap}\, \varphi \, n\), so by additivity the pushed cylinder masses are exactly the merged weights of the originals (cylinderMass_map_blockCode). The static DPI (Theorem 12.29) then gives the per-length bound \(H_q^{(n)}(\mathrm{blockCode}\, \varphi _*\mu ) \le H_q^{(n)}(\mu )\) for every \(0 \le q\), \(q \ne 1\).

Pushing a Bernoulli measure forward along a one-block code is again Bernoulli, with the mapped single-symbol law \(\mathrm{Measure.map}\, \varphi \, \nu \) (map_blockCode_bern). Combining the exact rate (Theorem 12.34) with the strict static DPI, a one-block code that glues two atoms of positive \(\nu \)-mass strictly lowers the Rényi rate for every \(q \in (0,1) \cup (1,\infty )\) (renyiRateSup_map_blockCode_bern_lt, renyiRateInf_map_blockCode_bern_lt). A concrete \(A = \operatorname {Fin}3 \to B = \operatorname {Fin}2\) with the uniform law and an atom-gluing symbol map certifies the strict drop non-vacuously at \(q = 2\).

For a probability measure \(\mu \) on \(\mathrm{Shift}\, A\), the length-\(n\) cylinder mass of a word \(w : \operatorname {Fin}n \to A\) is \(\mathrm{cylinderMass}\, \mu \, n\, w = \mu (\text{cylinder}\, w)\), the length-\(n\) Rényi entropy is \(H_q^{(n)}(\mu ) = \mathrm{renyiEntropySeq}\, \mu \, q\, n\) — the Rényi entropy (Definition 12.27) of the family of cylinder masses — and the upper and lower Rényi rates are its normalized limit superior and inferior, \(\mathrm{renyiRateSup}\, \mu \, q = \limsup _n H_q^{(n)}(\mu )/n\) and \(\mathrm{renyiRateInf}\, \mu \, q = \liminf _n H_q^{(n)}(\mu )/n\).

For the i.i.d. (Bernoulli) measure \(\mathrm{bern}\, \nu \) on \(\mathrm{Shift}\, A\) with single-symbol law \(\nu \), and any \(q \ne 1\), both rates equal the static single-symbol Rényi entropy of \(\nu \):

realized as an honest \(\limsup = \liminf \) limit (renyiRateInf_bern).

For every probability measure \(\mu \) on \(\mathrm{Shift}\, A\), symbol map \(\varphi : A \to B\), and order \(0 \le q\), \(q \ne 1\), the Rényi rate does not increase under the pushforward along \(\mathrm{blockCode}\, \varphi \):

and likewise for \(\mathrm{renyiRateInf}\) (renyiRateInf_map_blockCode_le).

On the representative, the doubling map acts by \(x\mapsto 2x\) on the left half and \(x\mapsto 2x-1\) on the right, uniformly \(\operatorname {rep}(Ty)=2\operatorname {rep}(y)-d_0(y)\).

The representative splits into its first \(N\) digits plus a rescaled tail: \(\operatorname {rep}(y)=\operatorname {binPartialSum}N(y)+\operatorname {rep}(T^N y)\, 2^{-N}\).

For a measurable family \(V : X \to \mathrm{Submodule}\, \mathbb {R}\, (\mathbb {R}^d)\), set \(\mathrm{restGen}\, A\, V\, x = A(x)\cdot P_{V(x)}\) (with \(P_K\) the orthogonal projector onto \(K\)), \(\mathrm{sFloor}\, A\, T\, n\, x = \prod _{j{\lt}n}\left\lVert (A(T^{[j]}x))^{-1} \right\rVert ^{-1}\), and

Kingman applied to \(\mathrm{restLog}\) over \(T\) yields a constant \(\chi _V\); the orbit-reversed cocycle \(h_n(x) = \mathrm{restLog}_n(T^{-n}x)\) is subadditive over \(T^{-1}\) with the same integral means, hence converges a.e. to the same \(\chi _V\).

For the forward level \(V_i\), the restricted Kingman constant equals the \(i\)-th exponent: \(\chi _{V_i} = \mathrm{expEnum}\, \mathrm{lam0}\, d\, i\).

At each ambient flag level, the dimension captured by the subbundle is bounded by the ambient dimension: \(\dim \big(W(x) \cap V_i(x)\big) \le \dim V_i(x)\), so the restricted multiplicities are a sub-multiset of the ambient ones.

For ergodic \(T\), invertible \(A\), and an invariant subbundle \(W\), collapsing the constant-dimension levels of \(i \mapsto W \cap V_i\) yields a genuine strict Oseledets filtration inside \(W\): a strictly antitone \(\lambda '\) and a measurable family \(V'\) with, \(\mu \)-a.e., \(V'_0(x) = W(x)\), \(V'_{k'}(x) = 0\), strictly descending and \(A\)-equivariant, with exact growth rate \(\lambda '_i\) per stratum and all levels \(\le W(x)\).

For a single-qubit state \(\rho \), the product state \(\rho ^{\otimes n}\) on the length-\(n\) block is defined by \(\rho ^{\otimes 0} = I/1\) (the unique state on \(\operatorname {Fin}1\), of zero entropy) and \(\rho ^{\otimes (n+1)} = \rho \otimes \rho ^{\otimes n}\).

The \(n\)-fold product of the maximally mixed single-qubit state is the maximally mixed state on the whole length-\(n\) block, \(\mathrm{rhoPow}\, \mathrm{mm}\, n=\mathrm{mm}\): the tracial state is a fixed point of the tower construction.

Writing \(\tau _m(y)=\operatorname{tr}\bigl((\mathrm{rhoPow}\, \rho \, m)\cdot y\bigr)\), pairing the enlarged state against an observable \(\mathrm{shiftAdjoinQubit}\, x=1\otimes x\) that acts trivially on the fresh qubit returns the previous-block pairing:

The family \((\tau _n)\) thus fits together into one shift-invariant state of the one-sided qubit chain.

For the doubling map and the binary partition,

the genuinely instantiated Pesin/Rokhlin equality on a real expanding system (in contrast to the vacuous-on-\(\mathbb {R}^d\) assembly Theorem 13.12). The integrand is the honest log-Jacobian: the generator \((2)\) is proved to be the Fréchet derivative of the doubling map’s linear lift \(x\mapsto 2x\) to the universal cover (), and the covering projection \(\mathbb {R}\to \mathbb {T}\) intertwines that lift with the doubling map () — so \((2)\) genuinely is \(DT\), not an arbitrary constant of the right determinant.

Let \(e\) be an ergodic measure-preserving automorphism of a standard Borel probability space with non-atomic \(\mu \). For every height \(N \ge 1\) and every \(\varepsilon {\gt} 0\) there is a measurable base \(B\) whose first \(N\) iterates \(B, eB, \dots , e^{N-1}B\) are pairwise disjoint and whose union covers all but \(\varepsilon \) of the space:

(Ergodicity plus non-atomicity supplies the aperiodicity the classical statement assumes.)

The low and high routes are essential: a naive minimum of only the direct and two wrap routes provably fails the triangle inequality (when \(T\) strongly contracts a pair, the cheap path climbs both fibres to the seam and traverses at cost \(d(Tx, Ty)\) without crossing it). Even the full five-route minimum \(\operatorname {routeDist}\) is only bi-Lipschitz to, not equal to, the chain-infimum distance, so it is a gauge, not a metric — a documented counterexample rules out the triangle inequality. The genuine metric is built by embedding instead.

For a measure-preserving \(T\) and an integrable \(f : X \to \mathbb {R}\),

For a measure-preserving \(T\) and a measurable integrable \(g\), and every \(N : \mathbb {N}\),

The capacity-enlargement step is the embedding \(\operatorname {Matrix}_{2^{n}}(\mathbb {C})\hookrightarrow \operatorname {Matrix}_{2^{n+1}}(\mathbb {C})\) adjoining one fresh qubit on the left, \(A\mapsto 1\otimes A\). It is unital, multiplicative, and \(*\)-preserving — a unital \(*\)-homomorphism realising the tower inclusion — and it is injective, since \(1\otimes A\) determines \(A\) through the entry identity \((1\otimes A)_{(i,a),(i,b)} = A_{a,b}\).

The capacity-enlargement shift \(\mathrm{shiftAdjoinQubit}\) (\(A\mapsto 1\otimes A\)) and the far-end inclusion \(\mathrm{appendQubit}\) (\(x\mapsto x\otimes 1\)) commute:

Hence the family of finite levels is a genuine directed system carrying the shift as an endomorphism.

The \(k\)-fold shift \(A_n\hookrightarrow A_{n+k}\) iterates the capacity-enlargement embedding \(\mathrm{shiftAdjoinQubit}\) (\(A\mapsto 1\otimes A\)) \(k\) times, by the recursion \(\mathrm{shiftIter}\, 0=\mathrm{id}\) and \(\mathrm{shiftIter}\, (k+1)=\mathrm{shiftAdjoinQubit}\circ \mathrm{shiftIter}\, k\). It is a unital \(*\)-homomorphism and is injective.

The fixed state \(\mathrm{rhoPow}\, \rho \), read through \(k\) shift-iterates of a level-\(n\) observable \(x\), returns the level-\(n\) marginal:

This is the shift-invariance of the whole hierarchy — the temporal stationarity of the quantum Bernoulli shift (“the past slides out”): a level-\(n\) observable, however far the shift carries its window, is still evaluated by the level-\(n\) state.

Under the reflection \(\mathrm{mu0}\, j = -\mathrm{lam0}\, (d-1-j)\), the backward count of distinct exponents equals the forward one, and the index \(\mathrm{sidx}\, i = \mathrm{cast}\, (\mathrm{Fin.rev}\, i)\) satisfies \(\mathrm{expEnum}\, \mathrm{mu0}\, d\, (\mathrm{sidx}\, i) = -\mathrm{expEnum}\, \mathrm{lam0}\, d\, i\).

The singularity spectrum of \(p\) at scale \(\varepsilon \) is the Legendre transform of the mass exponent,

It is an infimum: since \(\tau \) is concave (Theorem 12.6), the supremum of \(q\alpha - \tau (q)\) would be \(+\infty \).

Write \(k=\mathrm{specCard}(A,T,x)\) for the number of distinct exponents (the cardinality of the spectrum). The exponent list \(\mathrm{specList}\colon \mathrm{Fin}\, k\to \mathbb {R}\) enumerates the spectrum in strictly descending order, \(\lambda _0{\gt}\lambda _1{\gt}\cdots {\gt}\lambda _{k-1}\), obtained from the order embedding of the finset composed with index reversal. It is strictly antitone, every \(\mathrm{specList}(i)\) lies in the spectrum, and every spectrum value is \(\mathrm{specList}(i)\) for a unique \(i\).

For \(\mu \)-a.e. \(x\), every index \(i\) and every vector \(v\) in the \(i\)-th stratum \(\mathrm{vflag}\, i.\mathrm{castSucc}\setminus \mathrm{vflag}\, i.\mathrm{succ}\) one has

For every \(n\), \(x\): \(\mathrm{sprod}\, A\, T\, d\, n\, x = \left\lvert \det A^{(n)}(x) \right\rvert \).

A map \(T\) preserves an SRB (volume-case) measure \(\mu \) if \(\mu \) is absolutely continuous with respect to ambient Lebesgue volume, \(\mu \ll \operatorname {volume}\). This is the volume-case specialization of the Ledrappier–Strelcyn–Young SRB condition. It is a genuine, non-vacuous, dischargeable hypothesis: \(\operatorname {volume}\ll \operatorname {volume}\) witnesses inhabitation, while a Dirac mass is not \(\ll \operatorname {volume}\). Three earlier framings on issue #10 were all defective and are recorded so the rescope is auditable: an \(\texttt{acConditionalUnstable}:\texttt{True}\) field (vacuous, dischargeable by \(\langle \texttt{trivial}\rangle \)); an existential \(\exists \pi \, \eta ,\ \operatorname {singularPart}(\operatorname {condDistrib}\dots )=0\) (also vacuous — satisfied by every \(\mu \), since an existential over an arbitrary measurable factor self-trivializes); and an \(\texttt{opaque}\) marker (honest but unconsumable, so the reverse inequality stayed a documented \(\texttt{BLOCKED}\) leaf). The field \(\mu \ll \operatorname {volume}\) is the honest fix that is both genuine and consumable — the reverse inequality is proved from it.

Let \(\mathrm{stepVal}\, \lambda ^0\, D\) be the step function \(e^{\lambda ^0_{D-1}} + \sum _{k=1}^{D-1}(e^{\lambda ^0_{k-1}}-e^{\lambda ^0_k})\, \mathbf1_{(c_k,\infty )}\) with thresholds \(c_k = e^{(\lambda ^0_k+\lambda ^0_{k-1})/2}\) strictly inside the \(k\)-th gap. If \(\lambda ^0\) is antitone on \([0,D)\) and \(j{\lt}D\), then \(\mathrm{stepVal}\, \lambda ^0\, D\, (e^{\lambda ^0_j}) = e^{\lambda ^0_j}\).

Given the partial-trace DPI (Definition 16.39), every Stinespring-dilated channel \(\Lambda \rho = \operatorname{Tr}_E\bigl(U(\rho \otimes \alpha )U^{*}\bigr)\) — adjoin a faithful ancilla \(\alpha \), conjugate by a unitary dilation \(U\), then trace out the ancilla — is relative-entropy monotone on faithful states: \(S(\Lambda \rho \| \Lambda \sigma )\le S(\rho \| \sigma )\).

For a measure-preserving \(T\) and a finite measurable partition \(\xi \), the strict-future \(\sigma \)-algebra generated by the pullbacks of the iterated joins lies below the comap of \(T\):

On the diagonal corner the correlation-entropy sequence \(n\mapsto S\bigl(\mathrm{corrMatrix}\, (\mathrm{adPerm}\, \sigma )\, \rho _\mu \, (\mathrm{projPartition}\, c)\, n\bigr)\) is subadditive, \(S(m+n)\le S(m) + S(n)\).

For an ultrametric growth function \(g\) and threshold \(t\in \mathbb {R}\), the sublevel set

is a submodule of \(E\): it contains \(0\); closure under addition is the non-Archimedean inequality together with \(\max (g v,g w)\le t\); closure under scaling is scaling-invariance. These submodules are monotone in \(t\) (sublevel_mono).

The block inclusions satisfy the Kraus completeness / trace-preservation relation on the full \(A\otimes B\) space,

The left analogue is sum_conjTranspose_mul_krausInclusionLeft.

Any exponent sequence \(\mathrm{lam0}\) realizing the a.e. singular-value limits satisfies \(\sum _{j{\lt}d}\mathrm{lam0}\, j = \int _X \log \left\lvert \det A(x) \right\rvert \, d\mu \).

For a backward exponent sequence \(\mathrm{mu0}\), \(\sum _{j{\lt}d}\mathrm{mu0}\, j = -\sum _{j{\lt}d}\mathrm{lam0}\, j\).

Let \(g\) be an ultrametric growth function, \(s\) a nonempty finite index set, and \(v\colon s\to E\) a family of nonzero vectors with \(g\circ v\) injective on \(s\). Then \(\sum _{i\in s} v_i \neq 0\) and \(g\bigl(\sum _{i\in s} v_i\bigr) = \sup _{i\in s} g(v_i)\).

For a probability vector \(p\) supported on a nonempty finite set \(s\) (\(p_i\ge 0\) and \(\sum _{i\in s}p_i = 1\)), the entropy sum is bounded by the log-cardinality,

with equality at the uniform distribution \(p\equiv 1/\left\lvert s \right\rvert \).

The refinement of an operational partition of unity is again an operational partition of unity: for every \(n\),

The sum of all Lyapunov exponents equals the integral of \(\log \left\lvert \det \right\rvert \) of the generator:

For \(2 {\lt} s\) the family \(n\mapsto \langle n\rangle ^{-s}\) is summable over \(\mathbb {Z}^2\), and the sum over any set of frequencies with \(\langle n\rangle \ge R\) is at most \(C_s\, R^{-(s-2)/2}\) (), with \(C_s\) the total sum of the faster-decaying family \(\langle n\rangle ^{-(s+2)/2}\). This is exactly the shape the mixing argument consumes to bound a geometric tail \(\sum _{\langle b\rangle {\gt}\lambda _+^{k/2}} K\langle b\rangle ^{-s} \le C\, \theta ^{k}\).

Writing \(k_+ = \# \{ i : 0 {\lt} \mathrm{exponents}_i\} \), the positive-exponent sum equals the partial sum \(\Gamma _{k_+} = \sum _{i{\lt}k_+}\mathrm{exponents}_i\).

For an ergodic, log-integrable, differentiable uniformly expanding self-map \(T\), the sum of the strictly positive Lyapunov exponents — the Pesin / Margulis–Ruelle right-hand side — equals the integrated volume distortion — the Rokhlin right-hand side:

This is the honest foliation-free instance of the identity between the two right-hand sides; it is not a claim of the full Pesin entropy formula (no entropy appears in the statement).

For a uniformly expanding map the positive-part exponent sum is the full exponent sum: \(\sum \lambda ^{+} = \sum _i\lambda _i\) (with multiplicity).

If every Lyapunov exponent of the derivative cocycle is nonnegative, \(0\le \lambda _i\) for all \(i\), then the positive-part exponent sum equals the full exponent sum, \(\sum _i\lambda _i^{+}=\sum _i\lambda _i\).

Let \(T\) be ergodic on \(\mathbb {R}^d\) with a nonsingular, log-integrable derivative cocycle, differentiable, preserving an SRB (volume-case) measure (\(\mu \ll \operatorname {volume}\), Definition 14.3), equipped with a nonempty injectivity partition \(\xi \) and integrable \(\log \rho ,\ \log \left\lvert \det DT \right\rvert \). If the Lyapunov spectrum is nonnegative (\(0\le \lambda _i\) for all \(i\), the hypothesis hspec), then

the reverse of the sharp Margulis–Ruelle bound (Pesin 1977; Mañé 1981; Ledrappier–Young 1985), discharged here via Rokhlin’s inequality.

The sum \(\mathrm{sumPosExp}\) of the strictly positive exponents is nonnegative, vanishes iff all exponents are \(\le 0\), and is strictly positive iff some exponent is positive (and dually \(\mathrm{sumNegExp} \le 0\) with the mirror characterizations).

For measurable automorphisms \(T : X\simeq X\), \(S : Y\simeq Y\) with unit roofs and a base factor map \(\pi : X\to Y\) (measurable, \(\pi \circ T = S\circ \pi \)), the fibrewise raw map \((x,s)\mapsto (\pi x, s)\) on \(X\times \mathbb {R}\) descends through the two orbit quotients to \(\operatorname {suspensionFactorMap} : \widehat X\to \widehat Y\), \([x,s]\mapsto [\pi x, s]\).

For a compact metric base \(X\) with \(\operatorname {diam} X\le 1\) and a homeomorphism \(T\), the \(\operatorname {embDist}\) metric induces exactly the quotient topology (, via \(\texttt{MetricSpace.ofDistTopology}\)), under which the suspension space is compact, complete and separable — hence Polish.

If \(T\) is measure-preserving and \(g\in L^1(\mu )\), then for \(\mu \)-a.e. \(x\), \(\tfrac 1n\, g(T^n x)\to 0\).

If \(\sum _i \mathrm{exponents}_i {\lt} 0\) then \(\mu \)-a.e. \(\left\lvert \det A^{(n)}(x) \right\rvert \to 0\).

\(\operatorname {binPartialSum}N(y)\to \operatorname {rep}(y)\) as \(N\to \infty \).

For a finite measure, a measure-preserving \(T\) and an integrable \(g\), the Birkhoff averages converge \(\mu \)-a.e. to the conditional expectation of \(g\) onto the invariant \(\sigma \)-algebra:

For an ergodic \(T\) on a probability space and integrable \(g\), the Birkhoff averages converge \(\mu \)-a.e. to the space average \(\int g\, d\mu \).

The per-step entropy rate converges to the single-qubit entropy,

as an eventually-constant sequence (equal to \(S(\rho )\) for every \(n\ge 1\)).

Let \(U_k v = v\circ \operatorname {catTorus}^{[k]}\) be the Koopman operator (an \(L^2\)-isometry, ) and \(\Phi _k(u,v) = \langle u, U_k v\rangle \) the correlation (). For every \(u,v\in L^2(\mathbb {T}^2)\),

the orthogonal projection onto the constants.

On the diagonal corner the averaged entropies converge to the per-partition CNT rate,

The a priori infimum is thus a true limit on the abelian corner.

For every operational partition, the entropy rate converges to the partition entropy, \(S(\mathrm{corrMatrix}\, n)/n\to h_\Phi (\rho ,X)\); the \(\inf \)-definition captures a true limit here because \(0\le S(\mathrm{corrMatrix}\, n)/n\le \log (d^{2})/n\to 0\).

If \(P^{c}_n(x)\to P\) with \(Pv\neq 0\), then \(\tfrac 1n\log \left\lVert P^{c}_n(x)\, v \right\rVert \to 0\).

For a finite measure, a measure-preserving \(T\), an integrable subadditive cocycle \(g\) whose normalized integrals are bounded below, there is a \(T\)-invariant integrable \(G\) with

For an ergodic \(T\) on a probability space, an integrable subadditive cocycle with normalized integrals bounded below, there is a constant \(c\) with \(n^{-1}\, g\, n\, x \to c\) for \(\mu \)-a.e. \(x\).

Under the hypotheses of the ergodic Kingman theorem, there is a constant \(c\) with

The averaged iterated-join entropies converge: \(\frac1n H\bigl(\bigvee _{k{\lt}n}T^{-k}\alpha \bigr)\to h(T,\alpha )\) as \(n\to \infty \).

Fix a flow \( \varphi \), a flow cocycle \( A \) with uniform forward/inverse one-step controls \( g, g' \) on \( [0,1] \), a point \( x \), and a nonzero vector \( v \). Suppose the integer fluctuation error vanishes and the integer-time average converges, \( n^{-1}\log \left\lVert A(n,x)v \right\rVert \to L \). Then the continuous-time average converges to the same limit:

On a probability space, with \(A\) invertible and \(0 {\lt} k\), \(\mu \)-a.e. the operator-norm growth rate of the cocycle converges to the top exponent:

For \(k \le d\) and \(\mu \)-a.e. \(x\), the operator-norm growth of the compound cocycle converges to \(\Gamma _k\): \(\tfrac 1n\log \left\lVert C_k(A^{(n)}(x)) \right\rVert \to \Gamma _k\), the \(k\)-dimensional volume growth rate.

For each sorted index \(i\) and \(\mu \)-a.e. \(x\), the singular-value exponents of the inverse-matrix cocycle are the negated, reversed exponents of \(A\):

For \(\mu \)-a.e. \(x\), \(q_n(x)\to \Lambda (x)\) in the matrix metric, and \(\Lambda \) is measurable.

For subspaces \(K, L\) of \(\mathbb {R}^d\), with \(P_K, P_L\) the orthogonal projectors,

For \(\mu \)-a.e. \(x\) the approximants \(q_n(x)\) converge in the matrix metric to a single matrix \(\Lambda (x)\).

For ergodic \(T\) and a possibly-singular measurable generator with only \(\log ^+\left\lVert A \right\rVert \in L^1\) (no invertibility, no inverse integrability), the normalized positive-part log-norms \(\tfrac 1n\log ^+\left\lVert A^{(n)}(x) \right\rVert \) converge \(\mu \)-a.e. to a constant \(\lambda _1^+\).

The per-window rate \(\mathrm{windowEntropy}\, \mathrm{mm}\, k / k\) is the constant \(\log 2\) for every \(k\ge 1\), hence converges to \(\log 2\):

This is the site-window entropy rate of the quantum Bernoulli shift at the tracial state.

The irrational roof \(r := \sqrt2\) yields an ergodic time-\(1\) map of the cat suspension.

For the Arnold cat map with Haar \(\operatorname {volume}\) on \(\mathbb {T}^2\) and any positive irrational roof \(r\), the time-\(1\) map of the constant-roof suspension flow is ergodic for the invariant suspension probability measure.

A measurable \(g : \mathbb {T}^2 \to \mathbb {C}\) with \(g(\operatorname {catTorus} x) = l\cdot g(x)\) for all \(x\), where \(\left\lVert l \right\rVert = 1\) and \(l \ne 1\), vanishes almost everywhere.

For any s-finite measure \(\mu \) on a standard Borel base, a measurable subspace graph makes \(x\mapsto \mathrm{orthProjMatrix}(V x)\) AEMeasurable.