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We study dynamical systems generated by skew products: $$T:[0,1)\times\mathbb{C}\to [0,1)\times\mathbb{C} \quad\quadT(x,y)=(bx\mod1,\gamma y+\phi(x))$$ 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 $\Delta\notin\mathbb{Q}$ weprove the following dichotomy for the solenoidal attractor$K^{\phi}_{b,\,\gamma}$ for $T$: Either $K^{\phi}_{b,\,\gamma}$ is a graph ofreal 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.

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