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

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