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