The $C^1$ endpoint of the global Livšic theorem for diffeomorphism-valued cocycles over hyperbolic systems

The global Livšic theorem for cocycles taking values in the group of diffeomorphisms of a closed manifold asks whether trivial periodic data forces a cocycle over a hyperbolic base to be a coboundary. Avila, Kocsard and Liu (GAFA, 2018) proved this theorem, without any localization hypothesis and in arbitrary fiber dimension, for Hölder cocycles valued in $\mathrm{Diff}^{1+\beta}(N)$ with $\beta>0$; their proof runs through fiber-wise Pesin theory, which consumes the $\beta$-Hölder modulus of the fiber derivatives and therefore stops strictly above $C^1$. The paper's own remarks leave two undeveloped directions: whether the theorem survives at the sharp endpoint of merely $C^1$ fiber regularity (the regularity at which the predecessor characterization theorem of Kocsard and Potrie is stated), and whether the constructions extend from product skew products to a-priori nontrivial fiber bundles — an extension the authors themselves assess as routine but leave unwritten. Kocsard and Potrie explicitly conjectured the arbitrary-dimension statement and posed the $C^{\alpha,1}$ bundle-map version as the remaining step. No proof or refutation of the $C^1$ endpoint is known from the later literature (Sadovskaya, Hurtado, Damjanović–Xu, Backes–Kocsard, Cantarino–Freijo, and recent periodic-data rigidity work), nor is any published fiber-bundle extension. The essential open core (part (A)) is: decide whether a Hölder cocycle $A:M\to\mathrm{Diff}^1(N)$ over a hyperbolic homeomorphism, with $\dim N\ge 2$, satisfying the periodic orbit condition is necessarily a Hölder coboundary (equivalently, whether identity periodic data forces vanishing of all fiber Lyapunov exponents). A secondary direction (part (B)) concerns the same $C^{1+\beta}$ theorem on a-priori nontrivial fiber bundles; it is not part of the open core.

Background

Hyperbolic base systems. A hyperbolic homeomorphism is a bi-Lipschitz transitive homeomorphism $f$ of a compact metric space $M$ that admits local stable and unstable sets, $W^s_\varepsilon(x)$ and $W^u_\varepsilon(x)$, on which forward (respectively backward) iterates contract at uniform exponential rates, and such that for sufficiently close points the local stable set of one and the local unstable set of the other intersect in a single point depending continuously on the pair (local product structure). Transitive Anosov diffeomorphisms of compact manifolds (diffeomorphisms whose tangent space splits into uniformly expanding and contracting invariant directions) and mixing two-sided subshifts of finite type (closed shift-invariant sets of bi-infinite sequences over a finite alphabet determined by finitely many forbidden blocks) are the standard examples. Cocycles and the cohomological equation. For a topological group $G$, a cocycle over $f$ is a continuous map $A:M\to G$; it generates the skew product $F(x,t)=(f(x),A(x)t)$. The cohomological equation asks for a transfer map $u:M\to G$ with $A(x)=u(f(x))\,u(x)^{-1}$ for all $x\in M$; when a Hölder solution exists, $A$ is called a Hölder coboundary. Since a coboundary has trivial return behavior on periodic orbits, the periodic orbit condition (POC), $A(f^{n-1}p)\circ\cdots\circ A(p)=\mathrm{id}$ for every $p\in\mathrm{Fix}(f^n)$ and every $n\ge 1$, is necessary. Livšic's classical theorem for $G=\mathbb{R}$ states that for Hölder real-valued cocycles over hyperbolic bases, POC is also sufficient and the transfer map is Hölder (Livšic, 1972). Non-abelian groups. For groups without a complete bi-invariant distance the classical proof breaks down: one cannot control the distortion produced by right translations. The first results for $G=\mathrm{Diff}^r(N)$, the group of $C^r$-diffeomorphisms of a closed manifold $N$ (Niţică–Török, 1995; de la Llave–Windsor, 2010), required localization hypotheses, meaning the cocycle is assumed uniformly close to the identity cocycle, together with higher regularity of the fiber maps. Kalinin (2011) settled the linear global problem: every Hölder $GL(d,\mathbb{R})$-valued cocycle over a hyperbolic base satisfying POC is a Hölder coboundary, proved through Lyapunov-exponent criteria with no localization. Diffeomorphism-valued cocycles and the Kocsard–Potrie characterization. Kocsard–Potrie (2016) obtained the first global results for $G=\mathrm{Diff}^1(N)$. For circle fibers, POC implies the existence of an $\alpha$-Hölder transfer $u:M\to\mathrm{Diff}^1(\mathbb{R}/\mathbb{Z})$ (their Theorem A; the compact-interval case is covered by the same argument, their Remark 1.1). In arbitrary fiber dimension, working in the category of $C^{\alpha,1}$ fiber bundles, meaning bundles whose local trivializations are Hölder in the base and $C^1$ in the fiber, they proved (Theorem 3.1) that for a bundle map covering $f$ whose return map on the fiber over each periodic point is the identity, the following are equivalent: all fiber Lyapunov exponents vanish; the map is $\alpha$-dominated; the map is a $C^{\alpha,1}$-coboundary, in which case the bundle is a-posteriori trivial, that is, the conclusion itself supplies a global trivialization. Here the fiber Lyapunov exponents are those of the derivative cocycle $(x,t)\mapsto D_tA_x$ acting on tangent spaces along the fibers, and $\alpha$-domination (Kocsard–Potrie, Section 2.8) is a uniform norm/co-norm comparison of the fiber derivative against the base rates, with no invariant splitting assumed a priori: writing $F$ for the bundle map and $\partial_{\mathrm{fib}}F^{\ell}(\zeta)$ for the derivative of $F^{\ell}$ along the fiber at $\zeta$, there exists $\ell\ge 1$ such that $\|\partial_{\mathrm{fib}}F^{\ell}(\zeta)\|\le\frac{1}{2}\big(\nu_u^{(\ell)}(\pi(\zeta))\big)^{\alpha}$ and $\|(\partial_{\mathrm{fib}}F^{\ell}(\zeta))^{-1}\|^{-1}\ge 2\big(\nu_s^{(\ell)}(\pi(\zeta))\big)^{\alpha}$ for every $\zeta$, where $\nu_u,\nu_s$ are the expansion and contraction rate functions in the hyperbolicity data of the base and $\nu_u^{(\ell)},\nu_s^{(\ell)}$ their products along $\ell$-step orbit segments. Kocsard–Potrie explicitly conjectured that POC forces vanishing of fibered Lyapunov exponents in any dimension, and posed as Question 5.1 whether a $C^{\alpha,1}$ bundle map with identity periodic data is $\alpha$-dominated, describing this as the only remaining step toward the Livšic theorem for diffeomorphism cocycles in any dimension. The Avila–Kocsard–Liu theorem and its limitation. Avila–Kocsard–Liu (2018) proved the conjecture in the product (trivial-bundle) category at fiber regularity $C^{1+\beta}$ with $\beta>0$: a Hölder cocycle $A:M\to\mathrm{Diff}^{1+\beta}(N)$ satisfies POC if and only if it is a Hölder coboundary (Theorem 1.1), and the higher-regularity statement for $C^r$, $r>1$, follows via the smooth-dependence bootstrap of de la Llave–Windsor (2011). The proof reduces (Theorem 2.5) to showing that POC forces vanishing of all fiber Lyapunov exponents and then invokes the Kocsard–Potrie characterization. The mechanism is fiber-wise Pesin theory, that is, a Lyapunov block form with a measurable linearization of the derivative cocycle along typical orbits in the spirit of Pesin charts for non-uniformly hyperbolic systems, together with fake invariant sets, which are partitions of the fibered phase space mimicking local stable and unstable laminations without assuming hyperbolicity of the skew product, glued by a fiber closing lemma, which promotes orbit segments that return close to their starting point into exact periodic orbits of the base using the Anosov closing lemma. Both inputs consume the $\beta$-Hölder modulus of the fiber derivatives: the finite-time linearization estimate is of order $O(r^\beta)$ and degenerates at exactly $C^1$. The authors state in Section 2.5 that they must work in the $C^{1+\beta}$ topology instead of $C^1$ because the proof involves techniques from Pesin theory, and that although their techniques are local or semi-local and could have been run in the fiber bundle category, they present everything on product spaces to avoid technicalities, leaving the a-priori nontrivial bundle version unwritten.

Problem Statement

Throughout, $f$ is a hyperbolic homeomorphism of a compact metric space, $N$ is a closed smooth manifold, and $\mathrm{Diff}^r(N)$ denotes the group of $C^r$-diffeomorphisms of $N$. (A) Sharp endpoint, the essential open core. Let $A:M\to\mathrm{Diff}^1(N)$ be an $\alpha$-Hölder cocycle, that is, Hölder with respect to the $C^1$ topology on $\mathrm{Diff}^1(N)$, satisfying the periodic orbit condition

$$A(f^{n-1}p)\circ A(f^{n-2}p)\circ\cdots\circ A(p)=\mathrm{id}_N,\quad\forall p\in\mathrm{Fix}(f^n),\ \forall n\ge 1.$$

Decide whether $A$ must be an $\alpha$-Hölder coboundary, that is, whether there exists an $\alpha$-Hölder map $u:M\to\mathrm{Diff}^1(N)$ with $A(x)=u(f(x))\circ u(x)^{-1}$ for all $x\in M$. The open endpoint is $\dim N\ge 2$; the case of one-dimensional fibers is settled by Kocsard–Potrie (their Theorem A for the circle, their Remark 1.1 for the compact interval). A negative answer is an explicit counterexample: a hyperbolic base, a closed manifold $N$ with $\dim N\ge 2$, and an $\alpha$-Hölder $\mathrm{Diff}^1(N)$-valued cocycle satisfying the periodic orbit condition that is not a Hölder coboundary. Writing $A^0_x:=\mathrm{id}_N$, $A^n_x:=A(f^{n-1}x)\circ\cdots\circ A(x)$ for $n\ge 1$, and $A^n_x:=A(f^{n}x)^{-1}\circ\cdots\circ A(f^{-1}x)^{-1}$ for $n<0$ (the convention of the source paper, Section 2.3) for the composition of the fiber maps along the orbit segment from $x$, and $D_tA^n_x$ for the derivative at $t\in N$ of this composition (the derivative cocycle of the background), the characterization theorem of Kocsard–Potrie (Theorem 3.1), which is valid at regularity $C^{\alpha,1}$, implies that for a fixed cocycle satisfying the periodic orbit condition each of the following is equivalent to the coboundary conclusion, so that deciding part (A) is equivalent to deciding whether the periodic orbit condition implies either one: (i) $\lim_{n\to\pm\infty}\frac{1}{n}\log\|D_tA^n_x\|=0$ for every $(x,t)\in M\times N$, the vanishing of all fiber Lyapunov exponents of the derivative cocycle; (ii) the cocycle is $\alpha$-dominated (this is Kocsard–Potrie Question 5.1). (B) Category generalization, a secondary direction. Extend the Avila–Kocsard–Liu theorem, at its existing $C^{1+\beta}$ regularity, from product skew products on $M\times N$ to a-priori nontrivial $C^{\alpha,1+\beta}$ fiber bundles $\pi:E\to M$ with fiber $N$ (local trivializations that are $\alpha$-Hölder in the base and $C^{1+\beta}$ in the fiber): prove that a Hölder bundle map $F:E\to E$ covering $f$ (that is, $\pi\circ F=f\circ\pi$) whose return map on the fiber over every periodic point of the base is the identity is a $C^{\alpha,1+\beta}$-coboundary, so that in particular the bundle is a-posteriori trivial. The natural strongest form combines (A) and (B) at regularity $C^{\alpha,1}$ in the bundle category. This direction is recorded because the source authors assess it as following from their local and semi-local techniques but leave it unwritten; it is not part of the open core. Part (A) is the essential scientific question: it is the surviving endpoint of the Kocsard–Potrie conjecture. Part (B) is the same $C^{1+\beta}$ theorem stated in its natural category, which the source authors describe as following from their local and semi-local techniques but do not develop; it is a secondary direction, independent of part (A).

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