Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transitivity and an abelian Livsic theorem for covers

Mark Pollicott, Richard Sharp

TL;DR

This paper extends the abelian Livšic theorem to the setting where the lifted flow on a regular cover is transitive, showing that zero-periodic-integral data on Frobenius-trivial orbits suffices to express the cocycle as a coboundary plus a closed 1-form. The authors adapt the argument to subshifts of finite type, then transfer it to transitive Anosov flows, providing a sharp criterion that requires no extra covering-group conditions. They apply the result to marked length spectrum rigidity, establishing rigidity from data on non-universal covers, and extend the methodology to cocycles valued in connected Lie groups under a distortion hypothesis, revealing a central-obstruction decomposition. Collectively, the work broadens the reach of Livšic-type rigidity to covers and non-abelian contexts, with concrete geometric implications.

Abstract

We show that the abelian Livšic theorem recently obtained by A. Gogolev and F. Rodriguez Hertz for null-homologous periodic orbits of homologically full Anosov flows continues to hold when restricted to periodic orbits which are trivial with respect to any regular cover for which the lifted flow is transitive.

Transitivity and an abelian Livsic theorem for covers

TL;DR

This paper extends the abelian Livšic theorem to the setting where the lifted flow on a regular cover is transitive, showing that zero-periodic-integral data on Frobenius-trivial orbits suffices to express the cocycle as a coboundary plus a closed 1-form. The authors adapt the argument to subshifts of finite type, then transfer it to transitive Anosov flows, providing a sharp criterion that requires no extra covering-group conditions. They apply the result to marked length spectrum rigidity, establishing rigidity from data on non-universal covers, and extend the methodology to cocycles valued in connected Lie groups under a distortion hypothesis, revealing a central-obstruction decomposition. Collectively, the work broadens the reach of Livšic-type rigidity to covers and non-abelian contexts, with concrete geometric implications.

Abstract

We show that the abelian Livšic theorem recently obtained by A. Gogolev and F. Rodriguez Hertz for null-homologous periodic orbits of homologically full Anosov flows continues to hold when restricted to periodic orbits which are trivial with respect to any regular cover for which the lifted flow is transitive.

Paper Structure

This paper contains 5 sections, 8 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Subshifts of finite type
  3. Anosov flows
  4. Application to marked length spectrum rigidity
  5. Skew products and non-abelian cocycles

Key Result

Theorem 1.1

Let $X^t : M \to M$ be a transitive Anosov flow and let $\widehat{M}$ be a regular cover of $M$, with covering group $\Gamma$, such that the lifted flow $\widehat{X}^t : \widehat{M} \to \widehat{M}$ is transitive. If a Hölder continuous function $f : M \to \mathbb R$ satisfies then for some Hölder continuous $u : M \to \mathbb R$ which is continuously differentiable along flow lines and some smo

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Remark 3.2: Coudene, coudene Theorem 3
  • Lemma 3.3
  • proof
  • ...and 9 more