Almost sure dimensional properties for the spectrum and the density of states of Sturmian Hamiltonians

Abstract

In this paper, we find a full Lebesgue measure set of frequencies $\check \II\subset [0,1]\setminus \Q$ such that for any $(\alpha,\lambda)\in \check \II\times [24,\infty)$, the Hausdorff and box dimensions of the spectrum of the Sturmian Hamiltonian $H_{\alpha,\lambda,\theta}$ coincide and are independent of $\alpha$. Denote the common value by $D(\lambda)$, we show that $D(\lambda)$ satisfies a Bowen type formula, and is locally Lipschitz. We obtain the exact asymptotic behavior of $D(\lambda)$ as $\lambda$ tends to $ \infty.$ This considerably improves the result of Damanik and Gorodetski (Comm. Math. Phys. 337, 2015). We also show that for any $(\alpha,\lambda)\in \check \II\times [24,\infty)$, the density of states measure of $H_{\alpha,\lambda,\theta}$ is exact-dimensional; its Hausdorff and packing dimensions coincide and are independent of $\alpha$. Denote the common value by $d(\lambda)$, we show that $d(\lambda)$ satisfies a Young type formula, and is Lipschitz. We obtain the exact asymptotic behavior of $d(\lambda)$ as $\lambda$ tends to $ \infty.$ During the course of study, we also answer several questions in the same paper of Damanik and Gorodetski.