A Dichotomy for the dimension of SRB measure

Abstract

We study dynamical systems generated by skew products: $$T:[0,1)\times \mathbb{R}\to [0,1)\times \mathbb{R}T(x,y)=(bx\mathrm{mod}1,\gamma y+\varphi (x))$$ where integer $b\ge 2$, $0<\gamma <1$ and $\varphi$ is a real analytic $\mathbb{Z}$-periodic function. We prove the following dichotomy for the SRB measure $\omega$ for $T$: Either the support of $\omega$ is a graph of real analytic function, or the dimension of $\omega$ is equal to $\min \{2,1+\frac{\log b}{\log 1/\gamma }\}$. Furthermore, given $b$ and $\varphi$, the former alternative only happens for finitely many $\gamma$ unless $\varphi$ is constant.