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A topological version of Furstenberg-Kesten theorem

Aihua FanMeng Wu
Dec 2022
摘要
Let $A(x): =(A_{i, j}(x))$ be a continuous function defined on some subshiftof $\Omega:= \{0,1, \cdots, m-1\}^\mathbb{N}$, taking $d\times d$ non-negativematrices as values and let $\nu$ be an ergodic $\sigma$-invariant measure onthe 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 entryfunction $A_{i,j}(\cdot)$ is either identically zero or bounded from below by apositive number which is independent of $i$ and $j$, it is proved that for any$\nu$-generic point $\omega\in \Omega$, the limit defining the Lyapunovexponent $\lim_{n\to \infty} n^{-1} \log \|A(\omega) A(\sigma\omega)\cdotsA(\sigma^{n-1}\omega)\|$ exists.
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