The fractal dimensions of the spectrum of Sturm Hamiltonian

Abstract

Let $\alpha\in(0,1)$ be irrational and $[0;a_1,a_2,\cdots]$ be the continued fraction expansion of $\alpha$. Let $H_{\alpha,V}$ be the Sturm Hamiltonian with frequency $\alpha$ and coupling $V$, $\Sigma_{\alpha,V}$ be the spectrum of $H_{\alpha,V}$. The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when $\{a_n\}_{n\ge1}$ is bounded. The present paper will treat the most difficult case, i.e, $\{a_n\}_{n\ge1}$ is unbounded. We prove that for $V\ge24$, $$ \dim_H\ \Sigma_{\alpha,V}=s_*(V)\ \ \ \text{and}\ \ \ \bar{\dim}_B\ \Sigma_{\alpha,V}=s^*(V), $$ where $s_*(V)$ and $s^*(V)$ are lower and upper pre-dimensions respectively. By this result, we determine the fractal dimensions of the spectrums for all Sturm Hamiltonians. We also show the following results: $s_*(V)$ and $s^*(V)$ are Lipschitz continuous on any bounded interval of $[24,\infty)$; the limits $s_*(V)\ln V$ and $s^*(V)\ln V$ exist as $V$ tend to infinity, and the limits are constants only depending on $\alpha$; $s^\ast(V)=1$ if and only if $\limsup_{n\to\infty}(a_1\cdots a_n)^{1/n}=\infty,$ which can be compared with the fact: $s_\ast(V)=1$ if and only if $\liminf_{n\to\infty}(a_1\cdots a_n)^{1/n}=\infty$(Liu and Wen, Potential anal. 2004).