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arxiv: 2604.13401 · v1 · submitted 2026-04-15 · 🧮 math.DS

Periodic data rigidity for cocycles and hyperbolic automorphisms

Boris Kalinin , Victoria Sadovskaya This is my paper

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classification 🧮 math.DS
keywords periodic data rigidityhyperbolic automorphismsAnosov diffeomorphismslinear cocyclesHolder cohomologytopological conjugacystable holonomiesweak irreducibility
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The pith

Conjugate periodic data implies Holder cohomology for cocycles over hyperbolic systems, making topological conjugacies smooth for weakly irreducible automorphisms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that two Holder continuous linear cocycles over a hyperbolic dynamical system are Holder cohomologous when they have conjugate periodic data, provided conditions hold such as the target cocycle having narrow spectrum periodic data with the conjugacy Holder at a periodic point, or the cocycle being constant and diagonalizable over the complex numbers with Lyapunov spaces of dimension at most two. It further proves that any topological conjugacy between a weakly irreducible hyperbolic automorphism L of the d-torus and an Anosov diffeomorphism f must be smooth whenever the derivative cocycles L and Df are conjugate. These cohomology and smoothness results together yield global periodic data rigidity theorems for the automorphisms. A reader would care because the results reduce questions of smoothness and classification to verifiable data from periodic orbits rather than the entire dynamics.

Core claim

For Holder continuous cocycles A and B over a hyperbolic system with conjugate periodic data, the cocycles are Holder cohomologous under the conditions that the periodic data of B has narrow spectrum and the conjugacy C(p) is Holder continuous at a periodic point, or that B is constant and diagonalizable over C with Lyapunov spaces at most two-dimensional or C(p) bounded. A topological conjugacy between a weakly irreducible hyperbolic automorphism L and an Anosov diffeomorphism f of the d-torus is smooth if the cocycles L and Df are conjugate. This implies global periodic data rigidity for such automorphisms, and the argument also yields differentiability of stable holonomies in low-regular,

What carries the argument

The periodic data conjugacy C(p) between cocycles, which transfers local linear information at periodic points to a global Holder continuous cohomology map, together with the differentiability of stable holonomies established in low regularity.

Load-bearing premise

The periodic data of the target cocycle either has narrow spectrum with Holder continuous conjugacy at one periodic point, or the cocycle is constant and diagonalizable over the complex numbers with Lyapunov spaces of dimension at most two, plus the automorphism being weakly irreducible and all cocycles being Holder continuous.

What would settle it

A concrete pair of Holder continuous cocycles over a hyperbolic system where the periodic data are conjugate via a map that is Holder at one periodic point, yet no Holder continuous cohomology exists between them, or a topological conjugacy between a weakly irreducible hyperbolic automorphism and an Anosov diffeomorphism that remains non-differentiable despite conjugate derivative cocycles.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies Holder cohomology of linear cocycles over hyperbolic systems when periodic data are conjugate, establishing Holder cohomology under conditions including narrow spectrum of the periodic data for one cocycle together with Holder continuity of the conjugacy at a periodic point, constancy of one cocycle with measurable cohomology, or constancy plus diagonalizability over C with Lyapunov space dimension at most two or bounded conjugacy data. It proves that a topological conjugacy between a weakly irreducible hyperbolic automorphism L and an Anosov diffeomorphism f on the d-torus is smooth whenever the derivative cocycles are conjugate. Combining this with the cohomology results yields global periodic data rigidity theorems for weakly irreducible hyperbolic automorphisms. The argument also includes a result on differentiability of stable holonomies in a low-regularity setting.

Significance. If the proofs are correct, the work advances rigidity theory in hyperbolic dynamics by linking periodic data conjugacy to smooth conjugacy and global rigidity for automorphisms, under explicitly stated hypotheses. The technical contribution on differentiability of stable holonomies in low regularity is a potentially useful tool for future work on cocycle regularity and Anosov conjugacies. The results are conditional on natural assumptions such as weak irreducibility and Holder continuity, which aligns with the literature on Livsic-type theorems and cocycle cohomology.

major comments (2)
  1. [Main rigidity theorem (after the conjugacy smoothness result)] The global periodic data rigidity theorem for weakly irreducible hyperbolic automorphisms (stated after the smoothness result for derivative cocycles) relies on the new differentiability of stable holonomies; however, the precise range of Holder exponents for which this differentiability holds is not made fully explicit in the main statements, which could limit verification of applicability to the rigidity conclusion.
  2. [Cohomology theorems for constant cocycles] In the cohomology results for cocycles with conjugate periodic data, the case where B is constant and diagonalizable over C requires either Lyapunov spaces of dimension at most two or C(p) bounded; the manuscript should clarify whether these alternatives are sharp or if counterexamples exist when both fail, as this directly affects the scope of the Holder cohomology claim.
minor comments (2)
  1. [Introduction and statements of main theorems] Notation for the periodic data conjugacy C(p) and the narrow spectrum condition should be defined at first use with a reference to the precise definition of spectrum in the hyperbolic setting.
  2. [Introduction] The abstract mentions 'global periodic data rigidity results' but the precise statement of what 'global' means (e.g., for all periodic points or all orbits) would benefit from an explicit sentence in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address each major comment below and will incorporate the requested clarifications into the revised version.

read point-by-point responses

  1. Referee: The global periodic data rigidity theorem for weakly irreducible hyperbolic automorphisms (stated after the smoothness result for derivative cocycles) relies on the new differentiability of stable holonomies; however, the precise range of Holder exponents for which this differentiability holds is not made fully explicit in the main statements, which could limit verification of applicability to the rigidity conclusion.

    Authors: We agree that explicitness improves readability. The differentiability of stable holonomies is proved in Section 4 for all Holder exponents strictly less than the minimum of the hyperbolicity constants of the underlying system and the regularity of the cocycle. In the revised manuscript we will restate the main rigidity theorem (Theorem 1.3) with this precise range of exponents included in the hypothesis, so that applicability is immediately verifiable. revision: yes

  2. Referee: In the cohomology results for cocycles with conjugate periodic data, the case where B is constant and diagonalizable over C requires either Lyapunov spaces of dimension at most two or C(p) bounded; the manuscript should clarify whether these alternatives are sharp or if counterexamples exist when both fail, as this directly affects the scope of the Holder cohomology claim.

    Authors: The two alternatives are presented as sufficient conditions under which the measurable cohomology implies Holder cohomology. The manuscript does not claim sharpness. We are not aware of counterexamples when both conditions fail simultaneously, and proving or disproving sharpness lies outside the scope of the present work. In the revision we will add a short remark after the statement of the relevant theorem noting that it remains open whether the result continues to hold without either assumption. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes theorems on Holder cohomology for linear cocycles over hyperbolic systems and smoothness of conjugacies under explicit conditional hypotheses (narrow spectrum of periodic data, Holder continuity of C(p) at periodic points, weak irreducibility, and Holder continuity of the cocycles). These results are derived from standard dynamical systems techniques including new differentiability of stable holonomies in low regularity, without any reduction of the central claims to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained against the stated assumptions and external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

No free parameters, invented entities, or non-standard axioms are visible in the abstract. The results rest on standard domain assumptions of hyperbolic dynamics.

axioms (2)
  • domain assumption The underlying dynamical system is hyperbolic (Anosov or similar) with Holder continuous cocycles.
    Invoked throughout the abstract for all cohomology and conjugacy statements.
  • domain assumption Weak irreducibility of the hyperbolic automorphism.
    Required for the global periodic data rigidity conclusion.

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