A Dichotomy for the dimension of SRB measure

We study dynamical systems generated by skew products: $$T:[0,1)\times \mathbb{R}\to [0,1)\times \mathbb{R}T(x,y)=(bx\mathrm{m}\mathrm{o}\mathrm{d}1,\gamma y+\phi (x))$$ where integer $b\ge 2$, $0<\gamma <1$ and $\phi$ 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 $\min \{2,1+\frac{\log b}{\log 1\mathrm{/}\gamma }\}$. Furthermore, given $b$ and $\phi$, the former alternative only happens for finitely many $\gamma$ unless $\phi$ is constant.