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The spectral characteristics of the Sturm Hamiltonian with eventually periodic type

Jie Cao, Zhenyu Yu

Abstract

In this paper we consider the spectral characteristics of the Sturm Hamiltonian with eventually periodic type frequencies under large coupling and establish strict inequalities between the optimal Hölder exponent of the density of states measure, the dimension of the density of states measure, the dimension of the spectrum, and the transport exponent by analyzing the thermodynamic pressure function. Also, we provide the large coupling asymptotic properties of the four spectral characteristics.

The spectral characteristics of the Sturm Hamiltonian with eventually periodic type

Abstract

In this paper we consider the spectral characteristics of the Sturm Hamiltonian with eventually periodic type frequencies under large coupling and establish strict inequalities between the optimal Hölder exponent of the density of states measure, the dimension of the density of states measure, the dimension of the spectrum, and the transport exponent by analyzing the thermodynamic pressure function. Also, we provide the large coupling asymptotic properties of the four spectral characteristics.
Paper Structure (26 sections, 22 theorems, 160 equations, 1 figure)

This paper contains 26 sections, 22 theorems, 160 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and previous results
  3. Main results
  4. Ideas of the proof
  5. Recall on Thermodynamical Formalism and Multifractal Analysis
  6. Thermodynamical Formalism
  7. Analysis of the pressure function
  8. Multifractal Analysis
  9. Structure of the Spectrum of Sturm Hamiltonian
  10. The covering structure of the spectrum
  11. The symbolic space and the coding of $\Sigma_{\alpha,\lambda}$
  12. The symbolic space $\Omega^{\alpha}$
  13. The coding map $\pi^{\alpha}_\lambda$
  14. Useful results for Sturm Hamiltonian
  15. Sturm Hamiltonian and thermodynamical formalism
  16. ...and 11 more sections

Key Result

Theorem 1.1

Assume that $\lambda>20$, $\mathbf{a}\in{\mathbb N}^k$ and $\alpha\in\mathcal{EP}(\mathbf{a})$. Then we have

Figures (1)

  • Figure 1: Pressure function and spectral characteristics of the Strumian Hamiltonian

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1: Ba2006Mu2006
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 30 more