Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jie Cao, Yanhui Qu

TL;DR

This work establishes that for a full-measure set of irrational frequencies, the spectrum of Sturmian Hamiltonians has a single, frequency-independent fractal dimension D(λ) in the large-coupling regime, with Hausdorff and box dimensions coinciding and governed by a Bowen-type formula via a relativized pressure. It further proves that the density of states is exact-dimensional with a dimension d(λ) given by a Young-type relation involving the entropy of a Gibbs measure on a global symbolic space and the geometric potential; both D(λ) and d(λ) are Lipschitz in λ and admit precise asymptotics as λ → ∞. The arguments rely on an intricate thermodynamic formalism for countable Markov shifts, a global coding that couples continued-fraction dynamics with spectral bands, and two geometric lemmas (gap and tail) that transfer fiber-measure dimensions to the spectrum and DOS. Collectively, these results extend and refine the large-coupling picture for Sturmian Hamiltonians, answering longstanding questions about the almost-sure fractal structure and linking spectral data to dynamical invariants. The methods yield a robust, λ-uniform description of typical-frequency fractal properties, with implications for transport and spectral theory in quasicrystal models.

Abstract

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

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

TL;DR

This work establishes that for a full-measure set of irrational frequencies, the spectrum of Sturmian Hamiltonians has a single, frequency-independent fractal dimension D(λ) in the large-coupling regime, with Hausdorff and box dimensions coinciding and governed by a Bowen-type formula via a relativized pressure. It further proves that the density of states is exact-dimensional with a dimension d(λ) given by a Young-type relation involving the entropy of a Gibbs measure on a global symbolic space and the geometric potential; both D(λ) and d(λ) are Lipschitz in λ and admit precise asymptotics as λ → ∞. The arguments rely on an intricate thermodynamic formalism for countable Markov shifts, a global coding that couples continued-fraction dynamics with spectral bands, and two geometric lemmas (gap and tail) that transfer fiber-measure dimensions to the spectrum and DOS. Collectively, these results extend and refine the large-coupling picture for Sturmian Hamiltonians, answering longstanding questions about the almost-sure fractal structure and linking spectral data to dynamical invariants. The methods yield a robust, λ-uniform description of typical-frequency fractal properties, with implications for transport and spectral theory in quasicrystal models.

Abstract

In this paper, we find a full Lebesgue measure set of frequencies such that for any , the Hausdorff and box dimensions of the spectrum of the Sturmian Hamiltonian coincide and are independent of . Denote the common value by , we show that satisfies a Bowen type formula, and is locally Lipschitz. We obtain the exact asymptotic behavior of as tends to This considerably improves the result of Damanik and Gorodetski (Comm. Math. Phys. 337, 2015). We also show that for any , the density of states measure of is exact-dimensional; its Hausdorff and packing dimensions coincide and are independent of . Denote the common value by , we show that satisfies a Young type formula, and is Lipschitz. We obtain the exact asymptotic behavior of as tends to During the course of study, we also answer several questions in the same paper of Damanik and Gorodetski.
Paper Structure (48 sections, 54 theorems, 389 equations, 3 figures)

This paper contains 48 sections, 54 theorems, 389 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and previous results
  3. Main results
  4. Sketch of the main ideas
  5. Further remarks
  6. Preliminaries
  7. Recall on the theory of continued fraction
  8. The covering structure of the spectrum.
  9. The symbolic space and the coding of $\Sigma_{a,\lambda}$
  10. The symbolic space $\Omega^a$
  11. The coding map $\pi^a_\lambda$ and basic sets
  12. Useful facts for Sturmian Hamiltonians
  13. Almost sure dimensional properties of the spectrum
  14. Pressure functions and generalized Bowen's formulas
  15. Relativized pressure function and dimension formula
  16. ...and 33 more sections

Key Result

Theorem 1.1

There exist a subset $\hat{{\mathbb I}}\subset [0,1]\setminus {\mathbb Q}$ of full Lebesgue measure and a function $D:[24,\infty)\to (0,1)$ such that the following hold: (1) For any $(\alpha,\lambda)\in\hat{{\mathbb I}}\times [24,\infty)$, the spectrum $\Sigma_{\alpha,\lambda}$ satisfies (2) $D(\lambda)$ satisfies a Bowen's type formula: $D(\lambda)$ is the unique zero of the relativized pressure

Figures (3)

  • Figure 1: The covering structure
  • Figure 2: weak-Lipschitz of $\pi_{a,\lambda}^{-1}$
  • Figure 3: The case $q=2$, ${\mathbf t}_w={\mathbf 2}$ and $h_w'|_{B_w}>0$

Theorems & Definitions (109)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4: YKLS
  • Proposition 2.5
  • proof
  • ...and 99 more