Stable ergodicity of skew products of one-dimensional hyperbolic flows
Let \(A\) be an aperiodic \(k\times k\) matrix with entries from \(\{0,1\}\). Define \[ \Sigma= \{ (x_n)|_{n=-\infty}^{+\infty}\in \{1,\dots, K\}^{\mathbb{Z}}\mid A_{x_j,x_{j+1}}= 1 \text{ for }j\in \mathbb{Z}\}. \] It is clear that \(\Sigma\) is invariant under the shift \(\sigma\) defined as \((\sigma x)_j= x_{j+1}\). Let \(r\) be Hölder function in \(\Sigma\) with exponent \(\theta\in (0,1)\). Define \( \widetilde{\Sigma}_r= \{(x,t)\mid x\in \Sigma\), \(0\leq t\leq r(x)\}/\sim\) where \(\sim\) identifies \((x,r(x))\) with \((\sigma x,0)\). Then \(\widetilde{\Sigma}_r\) is a compact, one-dimensional space. The main goal of this paper is two-fold: firstly, to define and examine the relationships between various natural spaces of Hölder functions on \(\widetilde{\Sigma_r}\) and secondly, to prove the stable ergodicity of skew products within these spaces.
scientific article; zbMATH DE number 1492603