  We study dynamical systems generated by skew products: $$T:
[0,1)\times\mathbb{R}\to [0,1)\times\mathbb{R} \quad\quad
T(x,y)=(bx\mod1,\gamma y+\phi(x))$$ where integer $b\ge2$, $0<\gamma<1$ 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 $\min\{2,1+\frac{\log b}{\log1/\gamma}\}$. Furthermore, given $b$ and
$\phi$, the former alternative only happens for finitely many $\gamma$ unless
$\phi$ is constant.
