A Liv\v{s}ic theorem for matrix cocycles over non-uniformly hyperbolic systems

TL;DR

This paper establishes a Livšic-type theorem for matrix cocycles over non-uniformly hyperbolic systems, showing that under certain conditions, cocycles are cohomologous to the identity via a measurable or Hölder continuous transfer map.

Contribution

It extends Livšic's theorem to matrix-valued cocycles over non-uniformly hyperbolic systems, including regularity of the transfer map when the measure has local product structure.

Findings

Existence of a measurable transfer map P satisfying the cohomological equation.

Hölder continuity of P on large measure sets under local product structure.

Generalization of Livšic's theorem to non-uniformly hyperbolic systems with matrix cocycles.

Abstract

We prove a Liv\v{s}ic-type theorem for H\"older continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever $(f,\mu)$ is a non-uniformly hyperbolic system and $A:M \to GL(d,\mathbb{R}) $ is an $\alpha$-H\"{o}lder continuous map satisfying $ A(f^{n-1}(p))\ldots A(p)=\text{Id}$ for every $p\in \text{Fix}(f^n)$ and $n\in \mathbb{N}$, there exists a measurable map $P:M\to GL(d,\mathbb{R})$ satisfying $A(x)=P(f(x))P(x)^{-1}$ for $\mu$-almost every $x\in M$. Moreover, we prove that whenever the measure $\mu$ has local product structure the transfer map $P$ is $\alpha$-H\"{o}lder continuous in sets with arbitrary large measure.