A topological version of Furstenberg-Kesten theorem

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 ${\lim }_{n\to \infty }{n}^{-1}\log \Vert A(\omega )A(\sigma \omega )\cdots A({\sigma }^{n-1}\omega )\Vert$ exists.

24 pages