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Abelian Livsic theorems for Anosov flows

Richard Sharp

TL;DR

This work provides two concise proofs of the abelian Livšic theorem for transitive Anosov flows, showing extensions to null-homologous orbits of positive density and to amenable covers. The methods combine weighted equidistribution for null-homologous periodic orbits with asymptotic orbit counting in a thermodynamic formalism framework to identify f as a flow coboundary plus a fixed cohomological term. Consequently, Hölder observables with zero period integrals on the relevant orbit sets are cohomologous to a closed-1-form along the flow, up to a coboundary. The results broaden the abelian Livšic toolkit to general abelian and amenable covers, yielding new positive-density variants and strengthening cohomological rigidity results for Anosov flows.

Abstract

We give two short proofs of the abelian Livsič theorem of Gogolev and Rodriguez Hertz. We show that these proofs may be extended to give new abelian Livsic theorems for positive density sets of null-homologous orbits and for amenable covers.

Abelian Livsic theorems for Anosov flows

TL;DR

This work provides two concise proofs of the abelian Livšic theorem for transitive Anosov flows, showing extensions to null-homologous orbits of positive density and to amenable covers. The methods combine weighted equidistribution for null-homologous periodic orbits with asymptotic orbit counting in a thermodynamic formalism framework to identify f as a flow coboundary plus a fixed cohomological term. Consequently, Hölder observables with zero period integrals on the relevant orbit sets are cohomologous to a closed-1-form along the flow, up to a coboundary. The results broaden the abelian Livšic toolkit to general abelian and amenable covers, yielding new positive-density variants and strengthening cohomological rigidity results for Anosov flows.

Abstract

We give two short proofs of the abelian Livsič theorem of Gogolev and Rodriguez Hertz. We show that these proofs may be extended to give new abelian Livsic theorems for positive density sets of null-homologous orbits and for amenable covers.

Paper Structure

This paper contains 5 sections, 11 theorems, 66 equations.

Table of Contents

  1. introduction
  2. Anosov flows and cohomology
  3. Two proofs of Theorem \ref{['abelian_livsic_theorem']}
  4. General abelian covers
  5. Amenable covers

Key Result

Theorem 1.1

Let $X^t : M \to M$ be a homologically full transitive Anosov flow. If $f : M \to \mathbb R$ is Hölder continuous and satisfies then for some smooth closed $1$-form $\omega$ and some Hölder continuous $u : M \to \mathbb R$ which is continuously differentiable along flow lines.

Theorems & Definitions (25)

  • Theorem 1.1: Gogolev and Rodriguez Hertz GRH
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 15 more