Dependencies

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 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.e. \(x\),

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. \(x\).

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

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. \(x\).

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

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.

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

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

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

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:

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

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:

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 \(T^{-1}\).

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., \(\mathrm{finrank}\, V_i(x) = m_i\).

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 \(\sum _i m_i = d\).

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 \(\Lambda (x)\).

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 \(\mathrm{topExponent}\).

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 \( A(t,\cdot ) \).

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 \(x\in X\) 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 \(\mathtt{toEuclideanCLM}\).

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 \(V_x \sqcap U_x = \bot \).

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 \(f\), \(\int \sum _{k{\lt}n} f(T^{k}x)\, d\mu = n\int f\, d\mu \).

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

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 \(0 {\lt} k\).

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 \(T^{[\sum _{j{\lt}i}\ell _j]}x\).

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.

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 \(\mathbb {R}\).

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,

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 \(\mathrm{lyapunovSpectrum}(x) = \mathrm{distinctExp}\, \lambda ^0\, d\).

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.

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 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.

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,

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 \(M\).

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

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 \(\mathrm{IsOseledetsFiltration}\ \mu \ T\ A\ k\ \lambda \ V\).

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.

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.

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 \(j {\lt} d\).

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

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

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 \(\mathrm{expEnum}\, \mathrm{mu0}\, d\, (\mathrm{sidx}\, i) = -\mathrm{expEnum}\, \mathrm{lam0}\, d\, i\).

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 \(i\).

Let \(v\neq 0\) lie in the slow subspace. Given the slow restricted-norm exponent bound \(\limsup _n \tfrac 1n\log r_n\le \lambda _i\) and the restriction estimate \(\left\lVert A^{(n)}(x)v \right\rVert \le r_n\left\lVert v \right\rVert \) (valid for \(v\) slow), one has

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

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 \(\mathrm{stepVal}\, \lambda ^0\, D\, (e^{\lambda ^0_j}) = e^{\lambda ^0_j}\).

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).

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 \(g\bigl(\sum _{i\in s} v_i\bigr) = \sup _{i\in s} g(v_i)\).

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

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

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).

If \(T\) is measure-preserving and \(g\ge 0\) with \(g\in L^1(\mu )\), 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. \(\left\lvert \det A^{(n)}(x) \right\rvert \to 0\).

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 \(\int g\, d\mu \).

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

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 \(\lambda _1^+\).

For a flow cocycle \( A \) over \( \varphi \), every \( n \in \mathbb {N}\) and every \( x \),

i.e. at integer times the continuous-time cocycle equals the discrete iterated cocycle generated by its time-one map \( A(1,\cdot ) \) over the time-one dynamics \( \varphi (1) \).

\(\mu \)-a.e. the largest exponent of the inverse cocycle is \(-\, \mathrm{exponents}_{d-1}\), the negative of the smallest forward exponent.

\(\mathrm{TopGapMassEnvelope}\, A\, T\, \lambda ^0\, x\) asserts the uniform geometric leakage of fast-band mass across each genuine gap: for every cut tolerance \(\delta \) there is a constant \(C\) controlling, eventually and uniformly, the band mass that crosses the top gap of each stratum.

For \(\mu \)-a.e. \(x\), the top-gap fast-band-mass envelope \(\mathrm{TopGapMassEnvelope}\, A\, T\, \lambda ^0\, x\) holds.

On a probability space, any two Oseledets filtration data \((k, \lambda , V)\) and \((k_2, \lambda _2, V_2)\) for the same cocycle agree: \(k = k_2\), the exponents coincide under the index cast, and \(\mu \)-a.e. the flags agree level by level.

The limsup flag at \(x\) is the family \(\mathrm{vflag}(A,T,x)\colon \mathrm{Fin}\, (k+1)\to \mathrm{Submodule}\) with

With the descending enumeration, level \(j\) is the sublevel set at \(\lambda _j\), so the flag decreases from the whole space down to \(\bot \).

Under the standing hypotheses, for a.e. \(x\) the spectrum is invariant, \(\mathrm{lyapunovSpectrum}(A,T,x)=\mathrm{lyapunovSpectrum}(A,T,Tx)\) (lyapunovSpectrum_equivariant_ae), and the action of \(A x\) maps each flag level (each sublevel subspace) at \(x\) onto the corresponding level at \(Tx\):

On the good set, \(\mathrm{vflag}(A,T,x)(i{+}1) \subsetneq \mathrm{vflag}(A,T,x)(i)\) for each interior index \(i\).

On the good set, \(\mathrm{vflag}(A,T,x)(0)=\top \); and unconditionally \(\mathrm{vflag}(A,T,x)(\mathrm{last})=\bot \) (vflag_last).

Under the spectral upper bound and the reverse inclusion, for \(\mu \)-a.e. \(x\) and every \(t\),

On the ultrametric-growth good set, every vector \(v\) of the \(\Lambda \)-slow band \(\mathrm{vslow}(e^t)\) with \(\limsup _n\tfrac 1n\log \left\lVert A^{(n)}v \right\rVert \le t\) lies in the growth sublevel \(\{ v:\overline\lambda (x,v)\le t\} \).