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

We study dynamical systems generated by skew products: \(T: [0,1)\times \mathbb {C}\rightarrow [0,1)\times \mathbb {C} \quad \quad T(x,y)=(bx\mod 1,\gamma y+\phi (x))\) $$T:[0,1)\times \mathbb{C}\to [0,1)\times \mathbb{C}T(x,y)=(bxmod1,\gamma y+\varphi (x))$$ where integer \(b\ge 2\) $b\ge 2$ , \(\gamma \in \mathbb {C}\) $\gamma \in \mathbb{C}$ are such that \(0<|\gamma |<1\) $0<|\gamma |<1$ , and \(\phi \) $\varphi$ is a real analytic \(\mathbb {Z}\) $\mathbb{Z}$ -periodic function. Let \(\Delta \in [0,1) \) $\mathrm{\Delta }\in [0,1)$ be such that \(\gamma =|\gamma |e^{2\pi i\Delta }\) $\gamma =|\gamma |e^{2\pi i\mathrm{\Delta }}$ . For the case \(\Delta \notin \mathbb {Q}\) $\mathrm{\Delta }\notin \mathbb{Q}$ we prove the following dichotomy for the solenoidal attractor \(K^{\phi }_{b,\,\gamma }\) $K_{b,\gamma }^{\varphi }$ for T: Either \(K^{\phi }_{b,\,\gamma }\) $K_{b,\gamma }^{\varphi }$ is the graph of a real analytic function, or the Hausdorff dimension of \(K^{\phi }_{b,\,\gamma }\) $K_{b,\gamma }^{\varphi }$ is equal to \(\min \{3,1+\frac{\log b}{\log 1/|\gamma |}\}\) $min\{3,1+\frac{logb}{log1/|\gamma |}\}$ . Furthermore, given b and \(\phi \) $\varphi$ , the former alternative only happens for countably many \(\gamma \) $\gamma$ unless \(\phi \) $\varphi$ is constant.