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Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra

Davi Lima, Carlos Gustavo Moreira, Christian Camilo Silva Villamil

TL;DR

The paper studies the continuity and equality properties of the Lagrange and Markov dynamical spectra for conservative generic Markov and Lagrange spectra on hyperbolic horseshoes. By constructing stable and unstable Cantor-set projections and employing Markov partitions, the authors prove that, for generic perturbations, the dimension functions d_u(t) and d_s(t) are continuous and that L(t)=M(t)=min{1,HD(Λ_t)} with HD(Λ_t)=2d_u(t). Central to the approach are combinatorial critical-windows arguments and the approximation of Λ_t by complete subshifts whose Cantor-set dimensions closely match the target values. The results extend Cerqueira-Matheus-Moreira to arbitrary HD horseshoes and reveal robust fractal structure in dynamical spectra under conservative perturbations, with implications for the geometry of dynamical billiards and Diophantine-type spectra in smooth dynamics. The analysis hinges on bounded distortion, dynamically defined Cantor sets, and a careful dimension-theory treatment of subhorseshoes and their projections.

Abstract

Let $\varphi_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a transitive horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $\varphi$ of $\varphi_0$, let $L_{\varphi, f}$ and $M_{\varphi, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $Λ(\varphi)$ of the horseshoe $Λ_0$ and $f$. We show that for generic choices of $\varphi$ and $f$, the Hausdorff dimension of the sets $L_{\varphi, f}\cap (-\infty, t)$ and $M_{\varphi, f}\cap (-\infty, t)$ are equal and determine a continuous function as $t\in \mathbb{R}$ varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension.

Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra

TL;DR

The paper studies the continuity and equality properties of the Lagrange and Markov dynamical spectra for conservative generic Markov and Lagrange spectra on hyperbolic horseshoes. By constructing stable and unstable Cantor-set projections and employing Markov partitions, the authors prove that, for generic perturbations, the dimension functions d_u(t) and d_s(t) are continuous and that L(t)=M(t)=min{1,HD(Λ_t)} with HD(Λ_t)=2d_u(t). Central to the approach are combinatorial critical-windows arguments and the approximation of Λ_t by complete subshifts whose Cantor-set dimensions closely match the target values. The results extend Cerqueira-Matheus-Moreira to arbitrary HD horseshoes and reveal robust fractal structure in dynamical spectra under conservative perturbations, with implications for the geometry of dynamical billiards and Diophantine-type spectra in smooth dynamics. The analysis hinges on bounded distortion, dynamically defined Cantor sets, and a careful dimension-theory treatment of subhorseshoes and their projections.

Abstract

Let be a smooth conservative diffeomorphism of a compact surface and let be a transitive horseshoe of . Given a smooth real function defined in and a small smooth conservative perturbation of , let and be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation of the horseshoe and . We show that for generic choices of and , the Hausdorff dimension of the sets and are equal and determine a continuous function as varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension.
Paper Structure (8 sections, 13 theorems, 135 equations, 2 figures)

This paper contains 8 sections, 13 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical spectra
  3. Dynamical spectra
  4. Acknowledgements
  5. Statement of the results
  6. Preliminary results
  7. Critical windows and combinatorial lemmas
  8. Proofs of the Theorems 1 and 2

Key Result

Corollary 1

Let $\varphi_0\in \operatorname{Diff}^{2}_{\omega}(S)$ with a transitive horseshoe $\Lambda_0$ with $HD(\Lambda_0)>1$ and $\mathcal{V}$ a $C^{2}$-sufficiently small neighbourhood of $\varphi_0$ in $\operatorname{Diff}^{2}_{\omega}(S)$ such that $\Lambda_0$ admits a continuation $\Lambda$ for every $ but for all $\delta>0$. Moreover, one has

Figures (2)

  • Figure 1: Markov partition and projections.
  • Figure 2: Letters on the left of $\alpha$ determine part of the letters on the right.

Theorems & Definitions (35)

  • Remark 1.1
  • Remark 2.1
  • Corollary 1: Theorem D of LM
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Proposition 1
  • Remark 4.1
  • ...and 25 more