论文标题
furstenberg-kesten定理的拓扑版本
A topological version of Furstenberg-Kesten theorem
论文作者
论文摘要
令$ a（x）：=（a_ {i，j}（x））$是$ω：= \ {0,1，\ cdots，m-1 \}^\ mathbb {n} $定义的连续函数其中$σ$是移位图。在$ a（x）a（σx）\ cdots a（σ^{\ ell-1} x）$的条件下，在支持$ν$的某个点$ x $和某些Integer $ \ ell \ ge 1 $的某个点$ x $中是一个正矩阵事实证明，对于任何$ν$ - 生成点$ω\ inω$，定义lyapunov指数的极限 $ \ lim_ {n \ to \ infty} n^{ - 1} \ log \ | a（ω）a（σω）
Let $A(x): =(A_{i, j}(x))$ be a continuous function defined on some subshift of $Ω:= \{0,1, \cdots, m-1\}^\mathbb{N}$, taking $d\times d$ non-negative matrices as values and let $ν$ be an ergodic $σ$-invariant measure on the subshift where $σ$ is the shift map. Under the condition that $ A(x)A(σx)\cdots A(σ^{\ell-1} x)$ is a positive matrix for some point $ x$ in the support of $ν$ 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 $ν$-generic point $ω\in Ω$, the limit defining the Lyapunov exponent $\lim_{n\to \infty} n^{-1} \log \|A(ω) A(σω)\cdots A(σ^{n-1}ω)\|$ exists.
