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

TL;DR

This paper establishes the almost sure dimensional properties of the spectrum and density of states for Sturmian Hamiltonians, providing formulas, asymptotics, and independence results that advance understanding of their spectral measures.

Contribution

It introduces a full measure set of frequencies where spectral dimensions are independent of nd satisfies Bowen and Young type formulas, improving previous results.

Findings

Hausdorff and box dimensions of spectrum coincide and are ependent of or a full measure set of frequencies.

Derived asymptotic behavior of spectral dimensions as pproaches infinity.

Proved the density of states measure is exact-dimensional with dimensions satisfying a Young type formula.

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.