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 $\{a_n{\}}_{n\ge 1}$ is bounded. The present paper will treat the most difficult case, i.e, $\{a_n{\}}_{n\ge 1}$ is unbounded. We prove that for $V\ge 24$, $${\dim }_H{\Sigma }_{\alpha ,V}=s_{\ast }(V)\text{and}{\bar{\dim }}_B{\Sigma }_{\alpha ,V}=s^{\ast }(V),$$ where $s_{\ast }(V)$ and $s^{\ast }(V)$ 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: $s_{\ast }(V)$ and $s^{\ast }(V)$ are Lipschitz continuous on any bounded interval of $[24,\infty )$; the limits $s_{\ast }(V)\ln V$ and $s^{\ast }(V)\ln V$ exist as $V$ tend to infinity, and the limits are constants only depending on $\alpha$; $s^{\ast }(V)=1$ if and only if ${\mathrm{limsup}}_{n\to \infty }(a_1\cdots a_n{)}^{1/n}=\infty ,$ which can be compared with the fact: $s_{\ast }(V)=1$ if and only if ${\mathrm{liminf}}_{n\to \infty }(a_1\cdots a_n{)}^{1/n}=\infty$(Liu and Wen, Potential anal. 2004).

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