15 Livšic theory
The rigidity theory of the preceding chapters concerned spectral invariants — Lyapunov exponents, entropies, dimensions. This chapter develops a different kind of rigidity, the cohomological rigidity of A. N. Livšic (1972): over a hyperbolic-type map \(T\), a Hölder observable \(\varphi \) is a coboundary — \(\varphi = u\circ T - u\) for a Hölder transfer function \(u\) — if and only if its Birkhoff sum vanishes around every periodic orbit. The forward implication is a one-line telescoping identity; the converse reconstructs \(u\) from the periodic data alone, and is the substance of the theorem. It is the exact obstruction theory for the cohomological equation \(\varphi = u\circ T - u\): the countable family of closed-orbit sums is a complete set of invariants for solvability.
The library builds the full ladder. An abstract Hölder Livšic theorem (Theorem 15.10) is proved from three regularity-free ingredients — a dense forward orbit, a Hölder observable, and a single geometric hypothesis, the exponential closing property (Definition 15.6) — following the dense-orbit construction of Katok–Hasselblatt (Introduction to the Modern Theory of Dynamical Systems, Theorem 19.2.1). That abstraction is then discharged on five concrete systems: the one-sided full shift, the two-sided (invertible) full shift, subshifts of finite type (with the golden-mean shift as headline), the doubling map of the circle, and — the culmination — the Arnold cat map, a genuine Anosov diffeomorphism. On top of the existence theorem sit the measurable rigidity tiers, which weaken the transfer function from Hölder to merely measurable: a continuous tier, a bounded-measurable tier, and the full unbounded-measurable tier (Katok–Hasselblatt 19.2.4), the last routed through the two-sided natural extension with stable/unstable essential-oscillation bounds. Finally a flow tier ports the obstruction direction to suspension flows, where it is exercised on the cat-map suspension.
Everything here is formalized sorry-free on the same substrate as the multiplicative theory and is verified by the guarded axiom audit to rest only on \(\{ \texttt{propext},\texttt{Classical.choice},\texttt{Quot.sound}\} \). Throughout, \(T:X\to X\) is a self-map of a metric space, \(\varphi :X\to \mathbb {R}\) an observable, and \(S_n\varphi \, x = \sum _{k{\lt}n}\varphi (T^k x)\) its Birkhoff sum (Mathlib’s \(\texttt{birkhoffSum}\)). We have not located a prior formalization of the Livšic theorem in a proof assistant.
15.1 The abstract Hölder Livšic theorem
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 \(u\).
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.
\(\varphi \) 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.
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 \(T\).
Induction on \(n\) through \(\texttt{birkhoffSum\_ succ}\): the successor term \(\varphi (T^k x) = u(T^{k+1}x) - u(T^k x)\) cancels against the running endpoint, leaving \(u(T^n x) - u\, x\).
A coboundary has vanishing periodic sums. Contrapositively, a single non-vanishing periodic sum defeats every coboundary: if \(T^n p = p\) and \(S_n\varphi \, p\ne 0\), then \(\varphi \) is not a coboundary of any transfer function ().
Apply the telescoping identity (Lemma 15.4) at a periodic point: \(S_n\varphi \, p = u(T^n p) - u\, p = u\, p - u\, p = 0\). The certificate is its contrapositive. Both hold for a bare \(T:X\to X\) and any \(u\); this is the downstream “no continuous section” obstruction.
The converse is the theorem. Its geometric engine is an abstracted closing property, stated in summed-bound form so that a single abstraction covers both the two-sided Anosov and the one-sided/expanding shadowing regimes.
\(T\) 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.
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,
Subtract the (vanishing) periodic Birkhoff sum of the shadowing point \(p\) from that of \(x\): \(S_n\varphi \, x = \sum _{i{\lt}n}\bigl(\varphi (T^i x) - \varphi (T^i p)\bigr)\). Bound each term by Hölder continuity, \(\left\lvert \varphi (T^i x) - \varphi (T^i p) \right\rvert \le C_\varphi \operatorname {dist}(T^i x,T^i p)^{r}\), and collapse the summed shadowing cost through the summed closing bound. The closing exponent and the Hölder exponent are the same \(r\), so the geometric bookkeeping matches term by term with no constraint on \(r\).
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.
The McShane infimal-convolution formula \(v(x) = \inf _{y\in s}\bigl(u(y) + C\operatorname {dist}(x,y)^{r}\bigr)\). For \(r\le 1\) the modulus \(t\mapsto C t^{r}\) is subadditive, so the infimum is finite, agrees with \(u\) on \(s\), and inherits the Hölder bound with the same constant.
Let \(T\) be continuous on a compact metric space \(X\), let \(\varphi \) be \(r\)-Hölder (\(0 {\lt} r\le 1\)), suppose \(T\) has the summed exponential closing property, and suppose some point \(x_0\) has a dense forward orbit. If \(\varphi \) has vanishing periodic sums, then \(\varphi \) is a Hölder coboundary.
The classical dense-orbit construction. Enumerate the orbit \(e_n = T^n x_0\) and set the running transfer values \(w_n = S_n\varphi \, x_0\); define \(u_0\) on the orbit by \(u_0(e_m) = w_m\). This is well-defined because two indices landing on the same orbit point give equal Birkhoff sums (the difference is a Birkhoff sum around a periodic orbit, which vanishes). The close-pair increment \(\left\lvert w_b - w_a \right\rvert \) is a Birkhoff sum over the segment \([a,b)\), so the crux estimate (Lemma 15.7) gives it a Hölder modulus on the orbit; compactness bounds \(u_0\), which upgrades the modulus to a genuine \(\texttt{HolderOnWith}\). Extend to a global Hölder \(v\) by McShane (Lemma 15.8). The on-orbit cohomology identity \(\varphi (e_m) = v(e_{m+1}) - v(e_m)\) (with \(e_{m+1} = T e_m\)) then transports to all of \(X\) by density and continuity (\(\texttt{Continuous.ext\_ on}\)), giving \(\varphi = v\circ T - v\).
Under the standing hypotheses (continuous \(T\) on a compact metric space, \(r\)-Hölder \(\varphi \) with \(0 {\lt} r\le 1\), summed exponential closing, and a dense forward orbit),
Forward: the pure telescoping obstruction (Theorem 15.5). Backward: the dense-orbit reconstruction (Theorem 15.9). This is the interface every concrete instance below discharges by supplying continuity, compactness, a closing constant, and a dense orbit.
15.2 Concrete instances
Each instance of Theorem 15.10 needs a metric substrate, continuity, compactness, an \(\texttt{ExpClosing}\) constant, and a dense forward orbit. The instances below range from the combinatorially trivial full shift to a genuine Anosov diffeomorphism.
15.2.1 The one-sided full shift
The full shift \(\sigma \) on \(\texttt{Shift}\ \alpha _0 = (\mathbb {N}\to \alpha _0)\) over a finite alphabet carries the \(\texttt{PiNat}\) ultrametric with \(\operatorname {dist}(x,y) = (1/2)^{N}\) at first disagreement index \(N\). It carries no admissibility bookkeeping — every finite word is legal — so the closing property is available unconditionally.
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.
Dropping the first coordinate moves the first-disagreement index down by one, doubling the distance at worst; compactness is a product of finite discrete spaces.
For every exponent \(\alpha {\gt} 0\) the full shift satisfies \(\texttt{ExpClosing}\, \sigma \, \alpha \, 1\, K\) with \(K = (1/2)^{\alpha }/(1 - (1/2)^{\alpha })\).
A point that almost \(n\)-returns is shadowed by the front-anchored periodization \(p_i := x_{i\bmod n}\), automatically admissible on the full shift. The \(i\)-th shifted pair agrees on the first \(n-i\) coordinates, so \(\operatorname {dist}(\sigma ^i x,\sigma ^i p)\le (1/2)^{\, n-i}\); the \(\alpha \)-powers form a geometric series summing to \(K\cdot \operatorname {dist}(x,\sigma ^n x)^{\alpha }\).
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.
Over a nonempty finite alphabet the forward orbit of the rich point is dense.
To match a target on its first \(N\) coordinates, shift the rich point to the block equal to that length-\(N\) prefix; the resulting iterate lies within \((1/2)^{N}\) of the target.
For a Hölder \(\varphi \) (exponent \(0 {\lt} r\le 1\)) on the full shift over a finite discrete alphabet,
Feed Theorem 15.10 the substrate (Lemma 15.11), the closing constant (Theorem 15.12, \(\delta = 1\)), and the dense orbit (Theorem 15.14).
The instance is non-vacuous on the binary shift. The locally constant potential \(\phi \, x = \mathbf1[x_0 = 1]\) is not a coboundary (): its period-\(1\) sum at the fixed point \(\underline1\) is \(1\ne 0\), so the obstruction certificate of Theorem 15.5 applies. By contrast \(\psi = \phi \circ \sigma - \phi \) is a Hölder coboundary () by construction, and re-deriving that fact through the hard backward direction of Theorem 15.15 exercises the dense-orbit reconstruction end-to-end on a concrete witness ().
15.2.2 The two-sided full shift
The invertible finale requires a bi-infinite metric. Rather than transport the \(\texttt{PiNat}\) metric through a \(\mathbb {Z}\simeq \mathbb {N}\) reindexing (which scrambles the cylinder dictionary), the library builds the \(\mathbb {Z}\)-indexed \(\theta \)-ultrametric directly, measuring first disagreement by absolute value of the integer coordinate.
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.
For \(0\le \theta {\lt}1\), \(\ \sum _{i{\lt}n}\theta ^{\min (i,\, n-i)} \le 2\, (1-\theta )^{-1}\).
The two-sided profile \(\theta ^{\min (i,\, n-i)}\) is dominated termwise by \(\theta ^{i} + \theta ^{\, n-i}\); the first arm is a geometric partial sum and the second reflects to another one, giving twice the one-sided bound.
For every \(\alpha {\gt}0\) the two-sided full shift \(\tilde\sigma \) is \(2\)-Lipschitz () and satisfies \(\texttt{ExpClosing}\, \tilde\sigma \, \alpha \, 1\, K\) with \(K = 2/(1 - (1/2)^{\alpha })\) — twice the one-sided constant.
The shadow of an almost-\(n\)-return is the central periodization \(p_j := x_{j\bmod n}\) (integer \(\bmod \)), which repeats the central block \(x_0,\dots ,x_{n-1}\). If \(x\) and \(\tilde\sigma ^n x\) first differ at \(\left\lvert \cdot \right\rvert =N\), then \(x\) is genuinely periodic on the whole window \(-N {\lt} j {\lt} n+N\), so \(p\) agrees with \(x\) there and the per-step shadowing distance follows the bilateral profile \(\theta ^{\min (i,\, n-i)}\); Lemma 15.17 sums it.
For a Hölder \(\varphi \) (\(0 {\lt} r\le 1\)) on the two-sided full shift, \(\varphi \) is a Hölder coboundary for the bilateral shift iff all its periodic Birkhoff sums vanish. The dense forward orbit is (the one-sided rich point placed on the non-negative axis, padded on the negative axis).
Instantiate Theorem 15.10 with the \(\mathbb {Z}\)-ultrametric, the doubled closing constant, and the two-sided dense orbit. Compactness comes with no metric/measure diamond because the metric reuses the product topology whose Borel structure carries the two-sided Bernoulli measure.
15.2.3 Subshifts of finite type and the golden-mean shift
A subshift of finite type (SFT) is cut out of the full shift by a \(0/1\) transition matrix \(M\). Here the closing property becomes delicate — the front-anchored periodization has a wrap transition that need not be legal — and the library isolates a clean design finding that dissolves the difficulty.
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.
If \(x\) almost \(n\)-returns within radius \(1/2\), then \(x_0 = x_n\).
\(\operatorname {dist}(x,\sigma ^n x)\le 1/2\) forces agreement on coordinate \(0\), i.e. \(x_0 = (\sigma ^n x)_0 = x_n\).
For every \(\alpha {\gt}0\) and every transition matrix \(M\), the SFT shift satisfies \(\texttt{ExpClosing}\, (\texttt{sftShiftMap}\ M)\, \alpha \, (1/2)\, K\) at the reduced closing radius \(\delta = 1/2\), with the same constant \(K = (1/2)^{\alpha }/(1-(1/2)^{\alpha })\) as the full shift — with no irreducibility hypothesis on \(M\).
The design finding: at radius \(1/2\) the periodization \(p_i := x_{i\bmod n}\) is admissible for any \(M\). By Lemma 15.21 the wrap transition \(M(x_{n-1}, x_0) = M(x_{n-1}, x_n)\) coincides with the interior transition \(M(x_{n-1}, x_{(n-1)+1})\), which is legal because \(x\) itself is admissible. No connecting word is needed. The shadow bound then transports the full-shift estimate through the subtype metric.
Irreducibility is used only to build a dense orbit. The library isolates exactly the combinatorial input the dense-orbit construction needs, disclosed honestly as strictly stronger than irreducibility.
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 \(s = 0\).
If \(M\) has a safe symbol, the SFT shift has a point with dense forward orbit.
Mirror the full-shift rich point, but first sanitize each decoded word (keep it if admissible, drop it otherwise) and pad every block with one trailing safe symbol. The pad makes every inter-block junction legal, so the whole sequence lands in the carrier; every admissible target word is retained verbatim as its own block, giving density.
For a Hölder \(\varphi \) (\(0 {\lt} r\le 1\)) on the SFT and a dense forward orbit under the SFT shift, \(\varphi \) is a Hölder coboundary iff all its periodic Birkhoff sums vanish.
Instantiate Theorem 15.10 with the subtype substrate and the unconditional \(\delta = 1/2\) closing (Theorem 15.22); the dense orbit is the sole hypothesis, the only place a combinatorial assumption on \(M\) enters.
For the golden-mean shift (: forbid the block \(11\), Lind–Marcus §1.2), a Hölder \(\varphi \) (\(0 {\lt} r\le 1\)) is a Hölder coboundary iff all its periodic Birkhoff sums vanish — with no external hypotheses.
The symbol \(0\) is safe (it forbids nothing), so Theorem 15.24 supplies the dense orbit and Theorem 15.25 closes the equivalence. The shift is a proper subshift — the all-ones sequence contains the forbidden block, — so the instance is not the full shift in disguise.
15.2.4 The doubling map
The doubling map \(y\mapsto 2y\) on the circle is the canonical smooth expanding endomorphism. Its dense orbit comes from a generic ergodicity lemma, and its closing is an explicit rounding construction.
On a second-countable space with an open-positive probability measure, an ergodic map has a point with dense forward orbit.
For each basic open \(o\) the visiting set \(U_o = \bigcup _n T^{-n}o\) is almost invariant, hence null or conull by ergodicity; since \(o\subseteq U_o\) has positive measure it is conull. Intersecting over the countable basis leaves a conull set of points whose orbit visits every basic open set, i.e. a dense orbit; nonemptiness follows from the probability normalization. This generic lemma is Mathlib-upstreamable and is reused for the cat map.
For every \(r{\gt}0\) the doubling map satisfies \(\texttt{ExpClosing}\ r\ 1\ (2^{r}/(2^{r}-1))\).
Because \(T^n = (2^n)\cdot \) is a group endomorphism, the periodic shadow of \(x = \bar a\) is obtained by rounding: with \(M = 2^n - 1\) and \(k = \operatorname {round}(Ma)\), the point \(p = \bar a - \overline{(Ma-k)/M}\) is genuinely \(n\)-periodic (\(M\cdot p = 0\)) and lies within \(\left\lVert x - 2^n x \right\rVert /M\) of \(x\) (). The \(i\)-th orbit points satisfy \(\operatorname {dist}(2^i x, 2^i p)\le 2^i\left\lVert x-p \right\rVert \), and the \(r\)-powers sum to a geometric series bounded via \(2^n/(2^n-1)\le 2\).
For a Hölder \(\varphi \) (\(0 {\lt} r\le 1\)) on the circle, \(\varphi \) is a Hölder coboundary for the doubling map iff all its periodic Birkhoff sums vanish.
The doubling map is continuous, the circle compact, the closing is Theorem 15.28, and the dense orbit is Theorem 15.27 applied to the ergodicity of the doubling map for Haar measure (). Non-vacuity: the constant \(1\) is not a coboundary (), its sum at the fixed point \(0\) being \(1\ne 0\).
15.2.5 The Arnold cat map
The culmination is Livšic on a genuine Anosov diffeomorphism: the hyperbolic toral automorphism \(\texttt{catTorus}:\mathbb {T}^2\to \mathbb {T}^2\) induced by \(\bigl(\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\bigr)\). Here the Anosov closing lemma admits an exact, non-iterative solution.
\(\texttt{catProj}:\mathbb {R}^2\to \mathbb {T}^2\) 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 \(\mathbb {R}^2\).
For every exponent \(0 {\lt} \alpha \le 1\), \(\texttt{catTorus}\) has the summed exponential closing property with radius \(1\) and an explicit constant \(2\, C_{\mathrm{sh}}^{\alpha }/(1-\theta ^{\alpha })\), where \(\theta {\lt}1\) is the contraction ratio (the small eigenvalue).
Lift an almost-return to \(d = \texttt{cat}_{\mathbb {R}}^n v - v\), reduce it to its nearest-integer representative \(e\), solve \(w = \texttt{catShadowSol}\ n\ e\), and project \(p = \texttt{catProj}(v - w)\). Then \(p\) is exactly \(n\)-periodic because \(d - e\) is an integer vector (invisible to the projection), and its forward orbit two-sidedly geometrically shadows that of \(x\): the shadow displacement at step \(i\) is \(\left\lVert \texttt{cat}_{\mathbb {R}}^i w \right\rVert \), whose eigenbasis decomposition gives the two-sided \(\theta ^{\min (i,\, n-i)}\) profile with summed \(\alpha \)-power bounded by the return-gap constant.
For a Hölder \(\varphi \) (\(0 {\lt} r\le 1\)) on \(\mathbb {T}^2\), \(\varphi \) is a Hölder coboundary for the Anosov diffeomorphism \(\texttt{catTorus}\) iff all its periodic Birkhoff sums vanish.
\(\texttt{catTorus}\) is continuous, \(\mathbb {T}^2\) compact, the closing is Theorem 15.32, and the dense orbit is Theorem 15.27 applied to the genuine ergodicity of the cat map for Haar measure. Non-vacuity: the constant \(1\) is not a coboundary (), its sum at the fixed point \(0\) being \(1\ne 0\).
15.3 Measurable rigidity tiers
The existence theorem produces a Hölder transfer function. The measurable rigidity programme asks the converse robustness question: if the cohomological equation is solved only almost everywhere by a merely measurable \(u\), must \(\varphi \) already have vanishing periodic sums (hence be a genuine Hölder coboundary)? This splits into three tiers by the regularity assumed of \(u\).
\(\varphi \) 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.
15.3.1 Continuous and bounded tiers
If \(\varphi \) is continuous and equals the coboundary of a continuous \(u\) only \(\mu \)-a.e., for a fully supported \(\mu \) (\(\texttt{IsOpenPosMeasure}\)), then \(\varphi \) has vanishing periodic sums.
Full support forces a.e.-equal continuous functions to be equal, so the a.e. equation upgrades to an everywhere coboundary and Theorem 15.5 finishes. No ergodicity or hyperbolicity is used. On the full shift, a fully supported Bernoulli law is open-positive, giving the corollary .
Let \(T\) preserve a probability measure \(\mu \), let \(p\) be \(n\)-periodic, and let \(\varphi = u\circ T - u\) \(\mu \)-a.e. with \(u\) bounded (\(\left\lvert u \right\rvert \le M\)). If each cylinder \(D_m\) around \(p\) has positive measure and shadows the Birkhoff sum of \(p\) up to a uniform additive constant \(B\), then \(S_n\varphi \, p = 0\).
Around the periodic orbit \(S_{nm}\varphi \, p = m\cdot S_n\varphi \, p\). A.e. telescoping gives \(S_{nm}\varphi \, x = u(T^{nm}x) - u\, x\), bounded by \(2M\) uniformly in \(m\). Picking a witness in the positive-measure \(D_m\) yields \(\left\lvert m\cdot S_n\varphi \, p \right\rvert \le B + 2M\), a bound independent of \(m\); if \(S_n\varphi \, p\ne 0\) the left side is unbounded, a contradiction. The uniformity of \(B\) and \(2M\) in \(m\) is exactly where unboundedness of \(u\) would break the argument.
Instantiated on the full shift with a fully supported Bernoulli law — cylinders are charged, and a Hölder observable shadows along a cylinder with a depth-independent constant — the bounded core gives , which promotes to the Hölder-coboundary conclusion through Theorem 15.15.
15.3.2 The full measurable tier via the natural extension
For a genuinely unbounded measurable \(u\) the uniform endpoint control of the bounded tier breaks, and the theorem becomes the classical Livšic regularity theorem (Katok–Hasselblatt 19.2.4). The library discharges it through the two-sided natural extension, and does so with two disclosed proof innovations: there is no Hölder-version construction of the transfer function and no Birkhoff ergodic theorem — the oscillation bounds are obtained by reverse Fatou.
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.
The factor map is \(1\)-Lipschitz for the respective \(\theta \)-ultrametrics and pushes the two-sided Bernoulli measure to the one-sided one; pulling the a.e. identity back along it and rewriting through the intertwining transports the coboundary equation.
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).
The past (\(j{\lt}0\)) and future (\(j\ge 0\)) coordinate blocks are independent under the i.i.d. Bernoulli measure, so the joining map is measure preserving; this is the substrate for the Fubini glue below.
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), .
Points sharing a future have forward orbits that converge, so the telescoped Birkhoff sums of \(\varphi \) along them differ by a convergent geometric tail; Lusin regularity plus reverse Fatou on the return set (in place of a Birkhoff density of returns) turns this into an a.e. essential bound on \(u\)’s oscillation. The unstable bound is the \(\tilde\sigma ^{-1}\) mirror, reversing the cocycle to \(\psi = -\varphi \circ \tilde\sigma ^{-1}\).
A merely measurable (possibly unbounded) a.e. solution \(u\) of the cohomological equation over the two-sided Bernoulli measure is essentially bounded.
The Fubini glue: over the \(\text{past}\otimes \text{future}\) product a typical base point \((a_0,b_0)\) connects to a.e. \((a,b)\) in two holonomy moves, \((a,b)\xrightarrow {\text{unstable}}(a,b_0)\xrightarrow {\text{stable}}(a_0,b_0)\), so \(\left\lvert u \right\rvert \le \left\lvert u(a_0,b_0) \right\rvert + C_s + C_u\) almost everywhere by the two oscillation bounds of Lemma 15.39.
Over a fully supported two-sided Bernoulli measure, an \(r\)-Hölder \(\varphi \) that is the a.e. coboundary of a merely measurable (possibly unbounded) \(u\) has vanishing periodic sums.
With the essential bound \(M\) from Theorem 15.40, the clamp \(v = \max (-M',\min (M',u))\) is a bounded-everywhere measurable function agreeing with \(u\) a.e., hence itself an a.e. transfer function (the shift being measure preserving); the already-shipped bounded tier (Theorem 15.36, two-sided form) discharges vanishing periodic sums. No Hölder-version construction, no ergodic theorem.
Over a fully supported one-sided Bernoulli measure, an \(r\)-Hölder \(\varphi \) that is the a.e. coboundary of a merely measurable transfer function has vanishing periodic sums.
Transport \(\varphi ,u\) up the natural-extension factor (Lemma 15.37); apply the two-sided headline (Theorem 15.41) to \(\varphi \circ \texttt{toShift}\); then descend through a periodic lift — a one-sided \(n\)-periodic point lifts to the bi-infinite \(n\)-periodization, whose lifted Birkhoff sum equals the one-sided sum, so vanishing descends.
Over a fully supported Bernoulli measure, an \(r\)-Hölder \(\varphi \) (\(0 {\lt} r\le 1\)) admits some measurable a.e. transfer function iff all its periodic Birkhoff sums vanish.
Forward: the one-sided measurable headline (Theorem 15.42). Backward: vanishing periodic sums feed the substantive backward direction of Theorem 15.15 to produce a genuine Hölder transfer function, which — being Hölder — is continuous, hence measurable and an exact (so a.e.) coboundary. Composed with the uniqueness of the a.e. solution modulo constants (ergodicity of the Bernoulli shift), the measurable solution is a.e. the Hölder one up to an additive constant ().
15.4 The flow tier
The final layer ports the Livšic obstruction to continuous-time flows. For a one-parameter flow \(\Phi :\mathbb {R}\to Q\to Q\) and observable \(F:Q\to \mathbb {R}\), the cohomological equation is read integrally.
\(F\) 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.
If \(\Phi _P q = q\) and the closed-orbit integral \(\int _0^P F(\Phi _s q)\, ds\ne 0\), then \(F\) is not a flow coboundary of any transfer function.
The fundamental-theorem-of-calculus telescoping around a closed orbit: a transfer function would force the orbit integral to equal \(u(\Phi _P q) - u\, q = u\, q - u\, q = 0\). Needs only a bare flow.
This obstruction is landed on the concrete suspension (mapping-torus) flow \(\zeta _t\) of a base automorphism \(T\) under a roof \(\tau \), via the cross-section \(x\mapsto [x,0]\).
The induced base observable of a flow observable \(F\) is the one-lap roof integral \(\texttt{inducedBaseCocycle}\ F\ x = \int _0^{\tau x} F([x,s])\, ds\).
If \(F\) is a flow coboundary of the suspension flow with transfer function \(u\), then its induced base observable is a discrete coboundary for the base map \(T\), with transfer function \(u\circ (x\mapsto [x,0])\).
Evaluate the flow-coboundary equation at the section point \([x,0]\) over one lap \(t = \tau x\): flowing that long carries \([x,0]\) to \([Tx,0]\), collapsing the left side to \(u([Tx,0]) - u([x,0])\), while the descent \(\zeta _s[x,0] = [x,s]\) identifies the lap integrand with the induced observable.
If \(p\) is an \(n\)-periodic base point and the periodic Birkhoff sum of the induced base observable does not vanish, then \(F\) is not a flow coboundary of the suspension flow.
Combine Theorem 15.47 with the discrete obstruction certificate of Theorem 15.5 applied to the induced base observable.
The integral of \(F\) around one full closed flow orbit above an \(n\)-periodic base point equals the base Birkhoff sum of the induced base observable, identifying the tier-1 and (flow-native) tier-2 obstruction quantities.
The lap decomposition: cut the closed orbit at the successive cross-section returns (times \(S_k\tau \, p\)); each lap contributes exactly \(\texttt{inducedBaseCocycle}\ F\ (T^k p)\) by a change of variables, and summing the adjacent laps (under per-lap interval integrability) reassembles the base Birkhoff sum. The flow-native tier-2 obstruction follows by closing a base periodic orbit into a flow-periodic one of period \(S_n\tau \, p\).
The constant observable \(1\) is not a flow coboundary of the suspension flow over the Arnold cat map under the unit roof \(\tau \equiv 1\).
Take the fixed point \(0\) of the cat map (base period \(1\)): the induced base observable summed around it is the single lap integral \(\int _0^{\tau (0)} 1\, ds = \tau (0) = 1\ne 0\), so the tier-1 packaged obstruction (Theorem 15.48) defeats every transfer function. This exhibits a concrete \(F\) with a genuine nonzero closed-orbit integral, certifying the flow obstruction is non-vacuous.
15.4.1 The tier-III converse for constant-roof suspension flows
The obstruction direction above is only half the flow Livšic story. The converse — vanishing of all closed-orbit integrals implies \(F\) is a flow coboundary — is delivered here for constant-roof suspension flows (\(\tau \equiv c\)), and delivered exactly: because IsFlowCoboundary is a regularity-free cohomological predicate (Definition 15.44, no metric on the suspension), the fundamental-domain seam is glued by an identity, not a Hölder estimate. The one input required is that the base map discharge its own discrete Livšic converse for the induced base observable; that is exactly the interface every concrete base in Theorem 15.10 already supplies.
Fix a constant roof \(\tau \equiv c\), a base transfer function \(u_0\) cobounding the induced base observable, and build the fundamental-domain transfer candidate \(\texttt{uCover}(x,s) = u_0(x) + \int _0^{s} F([x,\sigma ])\, d\sigma \) () lap by lap on the strip. Across one fundamental-domain seam — the identification \([T x, s] = [x, s + c]\) — the candidate is unchanged, because the base jump \(u_0(T x) - u_0(x)\) is precisely the lap integral \(\int _0^{c} F([x,\sigma ])\, d\sigma \) that the fibre integral accumulates over one roof height.
The base cohomological equation \(\texttt{inducedBaseCocycle}\, F\, x = u_0(T x) - u_0(x)\) supplies the value of the seam jump; with a constant roof the one-lap integral \(\int _0^{\tau x}F([x,\sigma ]) \, d\sigma = \int _0^{c}F([x,\sigma ])\, d\sigma \) is exactly this jump, so the two descriptions of \(\texttt{uCover}\) across the seam agree by interval-integral additivity. No estimate; a pure identity. Its upgrade to full \(\mathbb {Z}\)-invariance under the suspension action is .
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
Forward is the general obstruction: a flow coboundary induces a base coboundary (Theorem 15.47), which has vanishing periodic sums (Theorem 15.5). Backward: vanishing periodic sums feed the base converse to produce a base transfer \(u_0\); Lemma 15.51 then makes the fundamental-domain candidate seam-invariant, so it descends through the suspension quotient (, a \(\texttt{Quotient.lift}\)) to a genuine flow transfer function verifying the continuous-time coboundary equation ().
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.
Chain Theorem 15.52 through the lap-decomposition bridge (Theorem 15.49), which identifies each closed-orbit flow integral with the base Birkhoff sum of the induced observable.
This equivalence lands on the Arnold cat-map suspension flow, where the discrete base converse is the already-proved Hölder Livšic theorem for the cat map (Theorem 15.33).
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.
Instantiate Theorem 15.52 at \(c = 1\), discharging the base converse with the Hölder Livšic theorem for the cat map (Theorem 15.33, \(\texttt{livsic\_ catTorus}\)).
The equivalence has content on both sides. The obstruction side is Theorem 15.50: the constant observable \(1\) is not a flow coboundary. The coboundary side is a concrete zero-mean fibre profile.
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.
The one-lap integral \(\int _0^{1}\sin (2\pi s)\, ds = 0\) makes \(\texttt{inducedBaseCocycle}\, F\) identically zero, so its periodic Birkhoff sums vanish trivially; the backward direction of Theorem 15.54 supplies the flow transfer function. Together with the obstruction witness \(F = 1\) (Theorem 15.50), this certifies the tier-III equivalence is non-vacuous on both sides — some observables cobound, others do not.
15.4.2 Hölder-regularity flow-Livšic
The tier-III equivalence above is regularity-free: IsFlowCoboundary (Definition 15.44) asks nothing of the transfer function \(u\), so the seam is glued by a bare identity. Issue #63 upgrades it to the classical-strength Livšic theorem, where a Hölder flow observable is required to admit a Hölder flow transfer function. Regularity is now a genuine constraint, measured against the Bowen–Walters embedding metric \(\operatorname {embDist}\) (Definition 9.59, built in Chapter 9). This closes the former Hölder-regularity front.
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.
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.
The one-lap integral \(\int _0^{1} F[x, \sigma ]\, d\sigma \) inherits the embedding-Hölder modulus of \(F\); the base distance \(d(x, y)\) is recovered from the embedding metric through the two Kuratowski test bundles \(\operatorname {muFun}/\operatorname {nuFun}\) (a \(2\times 2\) elimination) and, in the seam-wrap regime, through \(\operatorname {dist\_ map\_ le\_ embDist\_ wrap}\), while Lipschitz continuity of \(T\) controls the \(d(Tx, Ty)\) term.
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.
Following the exact-seam philosophy of Lemma 15.51: on the fundamental strip the fibre integral is controlled by the embedding-Hölder modulus of \(F\), the base value by the discrete Hölder transfer \(u_0\), and the seam is bridged by two algebraically equivalent (from-below and from-above) representations of \(\operatorname {uCover}\), matched by the base cohomological identity \(\operatorname {inducedBaseCocycle} F\, x = u_0(Tx) - u_0(x)\). The base-distance recoveries reuse the two Kuratowski bundles and the seam-wrap primitive.
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.
Forward: a Hölder flow coboundary is a flow coboundary, which induces a base coboundary (Theorem 15.47) with vanishing periodic sums (Theorem 15.5). Backward: vanishing periodic sums feed the base Hölder-Livšic converse (Theorem 15.9, applied to the Hölder induced observable Lemma 15.57) to produce a Hölder base transfer \(u_0\); the cross-seam gluing Theorem 15.58 makes the fundamental-domain candidate \(\operatorname {embDist}\)-Hölder and seam-invariant, so it descends to a genuine Hölder flow transfer function.
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.
Repeat the constant-roof argument on the normalized fibre coordinate. The metric compares points at normalized heights \(u = s/\tau x\) while the transfer integrates raw heights; the change of variables \(\sigma = \tau x\cdot w\) puts both fibre integrands at a common normalized height and lets the variable-roof horizontal-length bound control the fibre difference. The general-roof seam identity (, seam at height \(\tau x\)) glues the candidate; Lipschitz continuity of \(\tau \) is essential — the fibre-comparison factor \(\left\lvert \tau x - \tau y \right\rvert \) stays at exponent \(1\), so the assembled transfer keeps exponent \(r\); a merely \(r\)-Hölder roof would degrade it to \(r^2\) (disclosed, not delivered).
This lands on the Arnold cat-map suspension flow — the classical-strength Livšic theorem for a genuine Anosov flow presentation, the library’s first Hölder-regularity flow-Livšic result — by discharging the base hypotheses with the cat map’s diameter bound, Lipschitz constant, ergodic dense orbit and Anosov closing (Theorem 15.32).
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.
Instantiate Theorem 15.59 at the cat map: the diameter bound \(\operatorname {diam}\mathbb {T}^2\le 1\) (), the \(3\)-Lipschitz constant (), continuity, the Anosov summed closing (Theorem 15.32) and the ergodic dense orbit (Theorem 15.27) discharge every base hypothesis.
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.
Coboundary side: \(\int _0^1\sin (2\pi s)\, ds = 0\) gives vanishing periodic sums, so the backward direction of Theorem 15.61 supplies the Hölder flow transfer. Obstruction side: \(\operatorname {IsHolderFlowCoboundary.isFlowCoboundary}\) reduces to Theorem 15.50, where the period-\(1\) fixed point charges the single lap integral \(1\ne 0\).
The remaining wall.
The Hölder-regularity converse above is delivered for a Lipschitz roof; what is not delivered is the flow-Livšic theorem for a roof that is only \(r\)-Hölder, where the fibre-comparison change of variables produces a roof-difference factor \(\left\lvert \tau x - \tau y \right\rvert \) that a Hölder bound would degrade to exponent \(r^2\) — a genuine exponent-loss obstruction, disclosed rather than elided. The classical Bowen–Walters hypothesis that roofs be Lipschitz (Barreira–Radu–Wolf §2.1) is thus exactly the boundary of the current theory.