A topological version of Furstenberg-Kesten theorem (2212.12890v1)

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.