Characterizing continuity points of Lyapunov exponents for fiber-bunched smooth cocycles on surface fibers, and the shear-conjugacy refinement of the Avila–Viana invariance principle

The invariance principle of Avila–Viana (Inventiones Mathematicae 181 (2010), 115–178) states that a fiber-bunched, area-preserving smooth cocycle over a hyperbolic base whose extremal Lyapunov exponents $\lambda^\pm$ vanish has all its invariant measures disintegrated along the fibers into measures invariant under the stable and unstable holonomies, and it deduces that ergodic discontinuity points of the exponent map must fail bundle-freeness and carry invariant geometric structures: an invariant Riemannian metric on sphere or torus fibers, or an invariant continuous line field (Theorem 6.6 of the paper). This problem record collects the questions the paper left open: the characterization of the continuity points of the exponent map for smooth cocycles (Remark 2.3 of the paper, called "an important problem" there); the asserted-but-unproved refinement (Remark 6.7) that, under sufficient regularity, the line-field alternative forces torus fibers and conjugacy to a shear $(\xi,\eta) \mapsto (\xi + t(x)\eta,\eta)$; and the motivating question of when vanishing center Lyapunov exponents of partially hyperbolic diffeomorphisms can be removed by small perturbation. The later literature resolves the linear $d=2$ analogues (Backes–Brown–Butler: cocycles with invariant holonomies are continuity points in Hölder topology; Butler: irreducible locally constant $SL_2(\mathbb{R})$ cocycles are continuity points in all Hölder topologies, refuting the conjectured necessity of fiber-bunching) and important removal cases (Marin; Liang–Marin–Yang; Obata–Poletti, for symplectic systems with 2-dimensional center), while Chakraborty–Viana prove a derivative-cocycle trichotomy but state that whether rigidity holds for the invariant-line-field alternative "is not clear". The characterization in the paper's own nonlinear smooth-cocycle setting, the shear-conjugacy statement, and the general removal question are all open.

Background

A smooth cocycle over a base dynamics $f: M \to M$ is a diffeomorphism $F: E \to E$ of a fiber bundle $p: E \to M$ satisfying $p \circ F = f \circ p$, which maps each fiber $E_x = p^{-1}(x)$ diffeomorphically onto $E_{f(x)}$; a linear cocycle, given by a matrix-valued map $A: M \to GL(d,\mathbb{R})$ acting on the trivial bundle $M \times \mathbb{R}^d$, is the special case with linear fiber maps. In the source paper the base $(M,f,\mu)$ is hyperbolic (a shift, or an Anosov diffeomorphism — a diffeomorphism of a compact manifold whose tangent bundle splits into uniformly contracted and uniformly expanded invariant directions — preserving a probability $\mu$; in all cases $\mu$ has local product structure, meaning that locally $\mu$ is equivalent to the product of its conditional measures along stable and unstable manifolds) and the fibers are compact surfaces $N$ carrying an area form preserved by every fiber map (area-preserving cocycles). The extremal Lyapunov exponents $\lambda^\pm(F,\mu)$ measure the asymptotic exponential rates of contraction and expansion of curve lengths under iteration along $\mu$-typical orbits. The cocycle is fiber-bunched when the contraction and expansion of the base dynamics along its stable and unstable directions dominate the infinitesimal distortion of the fiber maps; fiber-bunching ensures the existence of invariant holonomies, that is, continuous maps $H^s_{x,y}: E_x \to E_y$ between fibers over points in a common stable manifold that commute with the cocycle, and analogously for unstable manifolds.

The invariance principle of Avila–Viana (Theorem A of the source paper) states that if a fiber-bunched area-preserving cocycle has $\lambda^\pm(F,\mu)=0$, then every $F$-invariant probability projecting to $\mu$ disintegrates along the fibers into conditional measures invariant under all stable and unstable holonomies (an su-state), and the disintegration is continuous when the cocycle is accessible, meaning any two points of the fiber bundle are joined by a finite concatenation of stable and unstable holonomy paths (an su-path); an accessibility class is an equivalence class of this relation, and the statements below are made over the saturation of such a class, that is, the union of its iterates under the cocycle. Applied to the projective extension $P(F)$ (the induced action on tangent directions of the fibers), this becomes a rigidity mechanism. A fiber-bunched cocycle is called bundle-free if $P(F)$ admits no su-states; then an ergodic $F$ has nonzero exponents that vary continuously at $F$ (Corollary 6.4 of the source paper), and accessible cocycles with fibers of genus at least 2 fall in this regime (Corollary G). Conversely, Proposition 6.3 shows that any ergodic discontinuity point fails bundle-freeness, and Theorem 6.6 then forces a topological alternative for a non-bundle-free cocycle over the saturation of an open accessibility class: either (1) the cocycle is accessible, the fiber is the sphere $S^2$ or the torus $T^2$, and there is a continuous Riemannian metric on the fibers, inducing the area form and invariant under the cocycle and its holonomies; or (2) the cocycle admits an invariant continuous line field (a continuous choice of a one-dimensional subspace of the tangent spaces of the fibers), or a pair of transverse continuous line fields. Thus discontinuity of the exponents requires very special invariant geometric structures, and classifying the cocycles that carry them is what remains for a full characterization.

The paper itself states (Remark 2.3) that it is "an important problem to characterize those cocycles which are continuity points for the Lyapunov exponents", noting that the results of Bochi and of Bochi–Viana show that continuity depends quite subtly on the topology on the cocycle space: for $C^0$ linear cocycles over aperiodic bases, the Bochi–Mañé rigidity phenomenon (proved by Bochi following Mañé's program) leaves only the uniformly hyperbolic cocycles and the zero-exponent cocycles as continuity points. It also asserts, without proof, the refinement (Remark 6.7) that under sufficient regularity alternative (2) of Theorem 6.6 strengthens to: the fiber is $T^2$ and the cocycle is conjugate to a shear $(\xi,\eta) \mapsto (\xi + t(x)\eta,\eta)$ on the fibers — a fact the authors explicitly say is "neither proved nor used in this paper". The introduction motivates the theory by the question of when vanishing center Lyapunov exponents of partially hyperbolic diffeomorphisms can be removed by small perturbations: a partially hyperbolic diffeomorphism is one whose tangent bundle splits into uniformly contracted stable and uniformly expanded unstable directions with an intermediate center bundle, and its center Lyapunov exponents are the asymptotic exponential rates of tangent vectors in the center bundle. For symplectic pseudo-Anosov toral automorphisms with 2-dimensional center (a linear toral automorphism, with exactly two eigenvalues on the unit circle, that is ergodic — no eigenvalue is a root of unity — and whose characteristic polynomial is irreducible over the integers and not a polynomial in $t^n$ for any $n \geq 2$), the paper proves a rigidity alternative (Theorem I): every nearby symplectic diffeomorphism is either non-uniformly hyperbolic (has all Lyapunov exponents nonzero) or is conjugate to the automorphism by a volume-preserving homeomorphism.

The subsequent literature developed the linear-cocycle analogue intensively. Bocker-Neto–Viana proved $L^\infty$-continuity for locally constant $SL(2,\mathbb{C})$ cocycles over Bernoulli shifts. Backes–Brown–Butler proved the case $d=2$ of Viana's conjecture that $\theta$-Hölder $GL(d)$-cocycles with invariant holonomies (in particular, fiber-bunched ones) are continuity points in the Hölder topology; Freijo–Marin extended this to a non-uniform fiber-bunching condition; Backes–Poletti–Sánchez proved the corresponding openness of nonzero exponents over accessible, center-bunched partially hyperbolic bases. Butler constructed discontinuity points arbitrarily close to the fiber-bunched locus and later proved that irreducible locally constant $SL_2(\mathbb{R})$ cocycles are continuity points in every $C^\alpha$ topology, $\alpha>0$, giving negative answers to the conjecture in Viana's book Lectures on Lyapunov Exponents (Conjecture 10.14) that failure of fiber-bunching implies discontinuity and to the corresponding problem in Viana's survey, and posing a new conjecture (Butler's Conjecture 2.8). Viana–Yang treated the $C^0$ topology over non-invertible bases, where Bochi–Mañé fails. Viana's survey states the cases $d \geq 3$ as open (Conjectures 4.3 and 4.7). The invariance principle itself was extended to partially hyperbolic bases by Avila–Santamaria–Viana (Astérisque 358 (2013)), with a rough-regularity theory for holonomies, but that volume contains no proof of Remark 6.7 and no characterization of continuity points; the recent survey of Marin–Poletti restates the mechanism (discontinuity implies an su-invariant disintegration, their Proposition 6.3, stated for Hölder cocycles over Anosov and partially hyperbolic bases) without a characterization. In the parallel derivative-cocycle setting (the tangent dynamics of fibred partially hyperbolic maps), Chakraborty–Viana proved a trichotomy — center exponents distinct with continuity, or invariant line fields on $T^2$-fibers, or invariant conformal structures on $S^2$- or $T^2$-fibers — together with conjugacy to affine extensions in the conformal case, an explicit example (their Example 3) of an accessible symplectic system with an invariant line field of exactly the shear form, and the statement that whether a rigidity statement holds for the line-field alternative "is not clear". The removal program for zero center exponents has partial results for symplectic partially hyperbolic diffeomorphisms with 2-dimensional center bundle (Marin; Liang–Marin–Yang; Obata–Poletti), with related random-perturbation and rigidity results by Berger–Carrasco, Blumenthal–Xue–Young, Brown–Rodriguez Hertz, Tahzibi–Yang, and Saghin–Yang.

Problem Statement

Let $f: M \to M$ be a hyperbolic base transformation preserving a probability $\mu$ with local product structure (a shift or an Anosov diffeomorphism in the source paper), and let $F: E \to E$ be a fiber-bunched, area-preserving smooth cocycle over $f$ whose fibers are compact orientable surfaces. Consider the exponent map $G \mapsto \lambda^\pm(G,\mu)$ on the space of such cocycles, endowed with the topologies defined in the source paper: the Lipschitz topology on the space $B_{vol}(f)$ of area-preserving fiber-bunched Lipschitz cocycles with uniformly $C^1$-bounded fiber derivatives (Section 2.5 of the paper) and, when $f$ is a $C^r$ Anosov diffeomorphism and the bundle is trivial, the uniform $C^r$ topology on the space $B^r_{vol}(f)$ of area-preserving fiber-bunched $C^r$ cocycles (Section 2.6 of the paper).

(P1) Characterize the continuity points of the exponent map in this smooth-cocycle setting. Known: an ergodic fiber-bunched cocycle that is bundle-free is a continuity point with $\lambda^- < 0 < \lambda^+$, and accessible cocycles with fibers of genus at least 2 fall in this regime (Corollary G); conversely, every ergodic discontinuity point fails bundle-freeness and has $\lambda^- < 0 < \lambda^+$ (Theorem F, Proposition 6.3, and the semicontinuity of $\lambda^\pm(\cdot,\mu)$ recorded in Section 6.1 of the source paper), and, over the saturation of an open accessibility class, admits the invariant structures of Theorem 6.6 of the source paper (invariant Riemannian metric with fiber $S^2$ or $T^2$, or invariant continuous line field or transverse pair); moreover, by area preservation together with these semicontinuity properties, every cocycle with $\lambda^\pm = 0$ is already a continuity point of the exponent map. The open target is to determine exactly which cocycles carrying these invariant structures — on the nonzero-exponent, non-bundle-free side — are continuity points, and for which of the topologies defined in the source paper.

(P2) Prove or refute the shear-conjugacy assertion of Remark 6.7 of the source paper: if the cocycle is sufficiently regular, alternative (2) of Theorem 6.6 (an invariant continuous line field, or transverse pair, over the saturation of an open accessibility class) strengthens considerably — the fiber is $N = T^2$ and the cocycle is conjugate to a shear $(\xi,\eta) \mapsto (\xi + t(x)\eta,\eta)$ on the fibers. Since the source fixes neither the regularity hypothesis on the cocycle nor the regularity of the conjugacy, this part of the target includes determining the regularity class of the cocycle (ideally the weakest one) under which the shear-conjugacy holds, together with the regularity of the conjugacy it delivers; the assertion itself is stated without proof ("neither proved nor used in this paper"), no proof has appeared in the subsequent literature, and the analogous problem in the derivative-cocycle setting is stated by Chakraborty–Viana as not known.

(P3) The motivating application is the pair of removal questions of the source paper's introduction, for partially hyperbolic diffeomorphisms: when is it possible to remove vanishing center Lyapunov exponents by a small perturbation of the map, and, supposing the center Lyapunov exponents do vanish stably, what can be said about the dynamical behavior? The source's question fixes no differentiability class for the perturbation, so the differentiability class of the perturbation under which the vanishing exponents can be removed is itself undetermined by the source and belongs to the question, as does the dynamical behavior in the stably-vanishing case. Known context: the source paper settles the neighborhood of symplectic pseudo-Anosov toral automorphisms with 2-dimensional center (Theorem I, stated in the $C^\infty$ symplectic category and remaining true for $C^k$ with $k \geq 22$), and the later literature settles the symplectic, accessible, 2-dimensional-center cases (Marin; Liang–Marin–Yang; Obata–Poletti).

(P1), (P2), and (P3) are the open questions posed by the source, and all three are required parts of the target: the target is resolved only when each part is settled on its own terms — for (P1) the classification across the topologies defined in the source paper, for (P2) the shear-conjugacy with the regularity class of the cocycle fixed, and for (P3) a resolution of the removal questions as stated — and settling one part does not resolve the others.

References

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