A Dichotomy for the dimension of solenoidal attractors on high dimensional space

Abstract: 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.