A topological version of Furstenberg-Kesten theorem

TL;DR

This paper establishes the existence of Lyapunov exponents for a class of matrix cocycles over subshifts, extending Furstenberg-Kesten type results to a topological setting with specific positivity conditions.

Contribution

It introduces a topological framework for Furstenberg-Kesten theorems, proving Lyapunov exponent existence under new positivity and boundedness conditions for matrix cocycles.

Findings

Lyapunov exponent exists for generic points under given conditions

Positivity of matrix products at some point ensures exponent existence

Extends classical results to a topological and subshift setting

Abstract

Let $A(x): =(A_{i, j}(x))$ be a continuous function defined on some subshift of $\Omega:= \{0,1, \cdots, m-1\}^\mathbb{N}$, taking $d\times d$ non-negative matrices as values and let $\nu$ be an ergodic $\sigma$-invariant measure on the subshift where $\sigma$ is the shift map. Under the condition that $ A(x)A(\sigma x)\cdots A(\sigma^{\ell-1} x)$ is a positive matrix for some point $ x$ in the support of $\nu$ and some integer $\ell\ge 1$ and that every entry function $A_{i,j}(\cdot)$ is either identically zero or bounded from below by a positive number which is independent of $i$ and $j$, it is proved that for any $\nu$-generic point $\omega\in \Omega$, the limit defining the Lyapunov exponent $\lim_{n\to \infty} n^{-1} \log \|A(\omega) A(\sigma\omega)\cdots A(\sigma^{n-1}\omega)\|$ exists.