Fast approximation of Lyapunov exponents: Beyond the locally constant case

TL;DR

This paper introduces a fast method for approximating the maximal Lyapunov exponent of certain cocycles over Gibbs states, achieving high accuracy even with complex dependencies, by leveraging periodic points and decay conditions.

Contribution

It provides a novel approximation technique for Lyapunov exponents in systems with infinite coordinate dependencies and exponential decay of variations.

Findings

Approximation accuracy of O(n^{-kn}) using periodic points.

Applicable to cocycles with exponential decay in variations.

Method extends beyond locally constant cases.

Abstract

We study the problem of estimating the maximal Lyapunov exponent of dominated cocycles. In particular we are concerned with cocycles over Gibbs states on shifts of finite type for which both the function defining the cocycle and the potential defining the Gibbs state may depend on infinitely many coordinates but are still very regular. We show that when the $n$th variation of both the cocycle and the potential is $O(e^{-cn^{2}})$ for some $c>h_{\text{top}}$ then using periodic points of period less then $n$ the Lyapunov exponent can be approximated to an accuracy $O(n^{-kn})$ for some explicit $k>0$.