We study dynamical systems generated by skew products: $$T: [0,1)\times\mathbb{ C}\to [0,1)\times\mathbb{ C} \quad\quad T(x,y)=(bx\mod1,\gamma y+\phi(x))$$ where integer $b\ge2$, $\gamma\in\mathbb{ C}$ such that $0<|\gamma|<1$, and $\phi$ is a real analytic $\mathbb{ Z}$-periodic function. Let $\Delta\in[0,1) $ such that $\gamma=|\gamma|e^{ 2\pi i\Delta}$. For the case $\Delta\notin\mathbb{ Q}$ we prove the following dichotomy for the solenoidal attractor $K^{ \phi}_{ b,\,\gamma}$ for $T$: Either $K^{ \phi}_{ b,\,\gamma}$ is a graph of real analytic function, or the Hausdorff dimension of $K^{ \phi}_{ b,\,\gamma}$ is equal to $\min\{ 3,1+\frac{ \log b}{ \log1/|\gamma|}\}$. Furthermore, given $b$ and $\phi$, the former alternative only happens for countable many $\gamma$ unless $\phi$ is constant.
Categories: math.DS