Periodic data rigidity for cocycles and hyperbolic automorphisms

Abstract: We study cohomology of Holder continuous linear cocycles over a hyperbolic dynamical system and regularity of conjugacy between Anosov systems. For cocycles $A$ and $B$ with conjugate periodic data, we establish Holder cohomology under various conditions: the periodic data of $B$ has narrow spectrum and the periodic data conjugacy $C(p)$ is Holder continuous at a periodic point; $B$ is constant and the cocycles are measurably cohomologous; $B$ is constant and diagonalizable over $\mathbb C$ and either its Lyapunov spaces are at most two-dimensional or $C(p)$ is in a bounded set. We also prove that a topological conjugacy between a weakly irreducible hyperbolic automorphism $L$ and an Anosov diffeomorphism $f$ of $\mathbb T^d$ is smooth if their derivative cocycles $L$ and $Df$ are conjugate. Using this and our results on cohomology of cocycles we obtain global periodic data rigidity results for weakly irreducible hyperbolic automorphisms. In the argument we also establish differentiability of stable holonomies in low regularity setting.

• The paper establishes new periodic data rigidity theorems for Hӧlder cocycles, ensuring a unique global conjugacy under narrow spectral conditions.
• It employs techniques like dominated splittings and non-stationary linearization to deduce smoothness criteria for conjugacies between Anosov diffeomorphisms and hyperbolic automorphisms.
• The work provides counterexamples that illustrate the necessity of spectral constraints, thereby advancing the classification of smooth dynamical systems on tori.

Rigidity of Periodic Data for Cocycles over Hyperbolic Systems and Applications

Introduction

This work provides a comprehensive investigation into the rigidity phenomena for H\"older continuous linear cocycles over hyperbolic dynamical systems, addressing the relationship between periodic data conjugacies and full cohomological conjugacy. The central questions concern when conjugacy of periodic data for two cocycles implies the existence of a globally continuous conjugacy and the implications of these results on the regularity (smoothness) of conjugacies between Anosov diffeomorphisms and hyperbolic toral automorphisms. The authors supply new general periodic data rigidity results, including optimality statements and applications to the global rigidity of Anosov automorphisms, relying on a blend of techniques from smooth dynamics, cocycle theory, and geometric group theory.

Cohomology and Periodic Data Conjugacy

The paper studies linear cocycles $A$ and $B$ over a hyperbolic base $f\colon X\to X$, focusing on the cohomology equation

$A_x=C(fx) \circ B_x \circ C(x)^{-1},$

where $C\colon X\to GL(d,\mathbb R)$ is called a conjugacy or transfer map.

The classical Livšic theorem settles the scalar (Abelian) case, but for non-commutative (matrix-valued) cocycles, the situation is more subtle. Continuous conjugacy clearly forces conjugacy of periodic data, that is, for each periodic point $p=f^n p$, there exists $C(p)$ such that $A_p^n = C(p) B_p^n C(p)^{-1}$, but the converse may fail, even in smooth or low-dimensional settings. This is evidenced by constructed counterexamples in dimension two and above, demonstrating that additional regularity or spectral constraints are required for periodic data to determine cohomology.

Main Theorems on Cohomological Rigidity

The authors establish new periodic data rigidity theorems for H\"older cocycles:

• Theorem 1: If $B$ has a $\delta$-narrow periodic data spectrum (generalizing almost-constancy) and the periodic data conjugacies $B$0 are $B$1-H\"older continuous near a periodic point, then a unique global H\"older conjugacy $B$2 exists.
• Theorem 2: If $B$3 is constant and $B$4 is measurably cohomologous to $B$5, then the measurable conjugacy coincides almost everywhere with a H\"older conjugacy.
• Theorem 3: If $B$6 is constant, diagonalizable, and either all its Lyapunov spaces are at most two-dimensional, or $B$7 is uniformly bounded, then a H\"older conjugacy exists. Without diagonalizability, this may fail, and the provided examples show the necessity of these assumptions.

Key to the proofs is the recent dominated splitting criterion from periodic data established in "Dominated splitting from constant periodic data and global rigidity of Anosov automorphisms" [DG24]. By leveraging the $B$8-narrowness of periodic data and the geometric structure induced by splitting, the approach generalizes earlier Livšic-type results and shows optimality up to the precise boundary of diagonalizability and fiber dimension.

The H\"older cohomology is ultimately reduced to a fiber-bunched setting for the cocycle restrictions to Lyapunov blocks, where earlier results guarantee rigidity under periodic data conjugacy (cf. [S15], [S17], [KSW23]). The proofs integrate advanced notions such as non-stationary linearization, dominated splittings, and holonomy regularity.

Applications to Rigidity of Hyperbolic Automorphisms

The second part of the paper applies the new cohomological rigidity theorems to smooth dynamics on tori and Anosov diffeomorphisms. For an Anosov $B$9 and the induced hyperbolic automorphism $f\colon X\to X$0, a topological conjugacy $f\colon X\to X$1 exists. The question of when $f\colon X\to X$2 is actually smooth is a central theme in rigidity theory.

Classically, periodic data conjugacy of the derivatives $f\colon X\to X$3 and $f\colon X\to X$4 is a necessary condition for smoothness. The authors prove that for weakly irreducible $f\colon X\to X$5 (a generalization of irreducibility involving density of Lyapunov foliation leaves), conjugacy of derivative cocycles is also sufficient:

• Theorem 4: If $f\colon X\to X$6 and $f\colon X\to X$7 are conjugate via a continuous (or measurable and sufficiently close) map, then $f\colon X\to X$8 and $f\colon X\to X$9 are $A_x=C(fx) \circ B_x \circ C(x)^{-1},$0 conjugate. If $A_x=C(fx) \circ B_x \circ C(x)^{-1},$1 is $A_x=C(fx) \circ B_x \circ C(x)^{-1},$2, the conjugacy is $A_x=C(fx) \circ B_x \circ C(x)^{-1},$3.

This is significant: for generic "weakly irreducible" automorphisms, smooth conjugacy reduces to periodic data considerations for derivatives (subject to dichotomies like diagonalizability and boundedness). The proof innovates by bootstrapping regularity along Lyapunov foliations via non-stationary linearization, regularity of cocycle holonomies (addressing cases where Jordan blocks are present), and group-theoretic properties of isometry actions.

A practical upshot is a global periodic data rigidity characterization (Corollary) for Anosov diffeomorphisms of tori, with only minimal necessary algebraic assumptions:

• If $A_x=C(fx) \circ B_x \circ C(x)^{-1},$4 is diagonalizable and has distinct modulus eigenvalues in small blocks, or conjugacy maps $A_x=C(fx) \circ B_x \circ C(x)^{-1},$5 are uniformly bounded or H\"older at a point, then $A_x=C(fx) \circ B_x \circ C(x)^{-1},$6 is smoothly conjugate to $A_x=C(fx) \circ B_x \circ C(x)^{-1},$7.

Technical Contributions and Optimality

The work provides a set of optimal and highly general conditions for when periodic data determines cocycle cohomology and the smooth classification of Anosov dynamical systems on tori. It systematically addresses obstructions arising from reducibility, non-diagonalizability, and degeneracy of Lyapunov splitting, supplying both positive and negative examples (constructed counterexamples for necessity claims).

Technically, the arguments develop new methods to deduce global regularity properties from periodic data, in part by establishing the differentiability of cocycle holonomies (even in low-regularity or non-conformal cases), and adapting techniques for dealing with Jordan blocks—going substantially beyond settings restricted to conformal or fully irreducible automorphisms.

Implications and Future Perspectives

Practically, these results strengthen the toolkit for classifying Anosov systems up to smooth equivalence: for large classes of cocycles and diffeomorphisms, periodic orbit information suffices to determine global, sometimes even smooth, classification. This is directly relevant for rigidity programs in smooth dynamics, geometric group actions, and mathematical physics, and opens the door to leveraging periodic data methods in non-uniformly hyperbolic and partially hyperbolic contexts.

Theoretically, the thorough delineation of rigidity boundaries via periodic data, and the precise role of spectral and geometric constraints (such as diagonalizability and boundedness), further clarify the interplay between local and global dynamical behavior. The developed machinery, especially around holonomy regularity and nonstationary normal forms, is likely to find broader application in the analysis of more general cocycle and group action problems.

Conclusion

This paper provides a complete and technically deep analysis of periodic data rigidity for linear cocycles over hyperbolic systems, achieving optimal H\"older and smooth cohomological rigidity results. The corresponding applications establish powerful new global rigidity theorems for Anosov dynamics on tori under natural algebraic and geometric conditions. The developed techniques and results represent a foundational advancement in the periodic data approach to rigidity phenomena in smooth dynamics and have substantial implications for future work in classification and cohomological problems for group actions and cocycles.