Livsic theorem for matrix cocycles

TL;DR

This paper proves a version of Livsic's theorem for matrix cocycles over hyperbolic systems, establishing conditions under which a cocycle is cohomologous to the identity based on its periodic data.

Contribution

It extends Livsic's theorem to $GL(m,\mathbb R)$ cocycles, providing new insights into the relation between periodic data, Lyapunov exponents, and growth estimates.

Findings

Proves Livsic's theorem for arbitrary $GL(m,\mathbb R)$ cocycles.

Establishes conditions for cocycles to be cohomologous to the identity.

Develops new results linking periodic data, Lyapunov exponents, and growth estimates.

Abstract

We prove the Liv\v{s}ic Theorem for arbitrary $GL(m,\mathbb R)$ cocycles. We consider a hyperbolic dynamical system $f : X \to X$ and a H\"older continuous function $A: X \to GL(m,\mathbb R)$. We show that if $A$ has trivial periodic data, i.e. $A(f^{n-1} p) ... A(fp) A(p) = Id$ for each periodic point $p=f^n p$, then there exists a H\"older continuous function $C: X \to GL(m,\mathbb R)$ satisfying $A (x) = C(f x) C(x) ^{-1}$ for all $x \in X$. The main new ingredients in the proof are results of independent interest on relations between the periodic data, Lyapunov exponents, and uniform estimates on growth of products along orbits for an arbitrary H\"older function $A$.