A Dichotomy for the Dimension of Solenoidal Attractors on High Dimensional Space

We study dynamical systems generated by skew products:

$$T: [0,1)\times \mathbb {C}\rightarrow [0,1)\times \mathbb {C} \quad \quad T(x,y)=(bx\mod 1,\gamma y+\phi (x))$$

where integer \(b\ge 2\), \(\gamma \in \mathbb {C}\) are such that \(0<|\gamma |<1\), and \(\phi \) is a real analytic \(\mathbb {Z}\)-periodic function. Let \(\Delta \in [0,1) \) be 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 the graph of a real analytic function, or the Hausdorff dimension of \(K^{\phi }_{b,\,\gamma }\) is equal to \(\min \{3,1+\frac{\log b}{\log 1/|\gamma |}\}\). Furthermore, given b and \(\phi \), the former alternative only happens for countably many \(\gamma \) unless \(\phi \) is constant.