A Dichotomy for the dimension of SRB measure

We study dynamical systems generated by skew products: $T: [0,1)\times\mathbb{R}\to [0,1)\times\mathbb{R} \quad\quad T(x,y)=(bx\mod1,γy+ϕ(x))$ where integer $b\ge2$, $0<γ<1$ and $ϕ$ is a real analytic $\mathbb{Z}$-periodic function. We prove the following dichotomy for the SRB measure $ω$ for $T$: Either the support of $ω$ is a graph of real analytic function, or the dimension of $ω$ is equal to $\min\{2,1+\frac{\log b}{\log1/γ}\}$. Furthermore, given $b$ and $ϕ$, the former alternative only happens for finitely many $γ$ unless $ϕ$ is constant.

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