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Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$

Christian Camilo Silva Villamil, Carlos Gustavo Moreira

TL;DR

The paper investigates the fine structure of the classical Lagrange and Markov spectra near the critical value $3$ through a dynamical-systems framework. It constructs a decreasing sequence $\{a_r\}$ approaching $3$ and associated sets $\mathcal{B}_r$ within $(a_{r+1},a_r)\cap \mathcal{L}'$ that isolate a dominant part of $\mathcal{L}$ on which the spectral dimension behaves regularly, with $HD(((a_{r+1},a_r)\cap \mathcal{L})\setminus\mathcal{B}_r)<HD(\mathcal{B}_r)$ and a strictly increasing $D(t)$ on $\mathcal{B}_r$. It also provides a quantitative upper bound for the dimension of $\mathcal{M}\setminus\mathcal{L}$ near $3$, showing that $HD((\mathcal{M}\setminus\mathcal{L})\cap(-\infty,3+\rho))$ decays like $\frac{\log(\lvert\log\rho\rvert)-\log(\log(\lvert\log\rho\rvert))+C}{\lvert\log\rho\rvert}$ for small $\rho>0$. The methodology fuses Farey theory, renormalization of words, and the geometry of subhorseshoes to control the distribution of Lagrange/Markov values and to quantify the local fractal dimensions, thereby linking Diophantine properties to hyperbolic dynamics on surfaces. The results advance the understanding of how dynamical representations capture the intricate, near-threshold behavior of these spectra with explicit dimension bounds and monotonicity phenomena.

Abstract

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence $\{a_r\}_{r\in \mathbb{N}}$ converging to $3$, such that for any $r$ one can find a subset $\mathcal{B}_r\subset (a_{r+1},a_r)\cap \mathcal{L}^{'}$ with the property that the Hausdorff dimension of $((a_{r+1},a_r)\cap \mathcal{L})\setminus \mathcal{B}_r$ is less than the Hausdorff dimension of $\mathcal{B}_r$ and for $t\in \mathcal{B}_r$ the sets of irrational numbers with Lagrange value bounded by $t$ and exactly $t$ respectively, have the same Hausdorff dimension. We also show that, as $t$ varies in $\mathcal{B}_r$, this Hausdorff dimension is a strictly increasing function. Finally, in relation to $\mathcal{M}\setminus \mathcal{L}$, we find $C>0$ such that we can bound from above the Hausdorff dimension of $(\mathcal{M}\setminus \mathcal{L})\cap (-\infty,3+ρ)$ by $\frac{\log (\abs{\log ρ})-\log (\log(\abs{\log ρ}))+C}{\abs{\log ρ}}$ if $ρ>0$ is small.

Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$

TL;DR

The paper investigates the fine structure of the classical Lagrange and Markov spectra near the critical value through a dynamical-systems framework. It constructs a decreasing sequence approaching and associated sets within that isolate a dominant part of on which the spectral dimension behaves regularly, with and a strictly increasing on . It also provides a quantitative upper bound for the dimension of near , showing that decays like for small . The methodology fuses Farey theory, renormalization of words, and the geometry of subhorseshoes to control the distribution of Lagrange/Markov values and to quantify the local fractal dimensions, thereby linking Diophantine properties to hyperbolic dynamics on surfaces. The results advance the understanding of how dynamical representations capture the intricate, near-threshold behavior of these spectra with explicit dimension bounds and monotonicity phenomena.

Abstract

We study the sets and near , where and are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence converging to , such that for any one can find a subset with the property that the Hausdorff dimension of is less than the Hausdorff dimension of and for the sets of irrational numbers with Lagrange value bounded by and exactly respectively, have the same Hausdorff dimension. We also show that, as varies in , this Hausdorff dimension is a strictly increasing function. Finally, in relation to , we find such that we can bound from above the Hausdorff dimension of by if is small.
Paper Structure (26 sections, 47 theorems, 262 equations)

This paper contains 26 sections, 47 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. The Lagrange spectrum
  3. The Markov spectrum
  4. Structure of the Lagrange and Markov spectra
  5. Dynamical spectra
  6. Preliminares
  7. Sets of finite type and connection of subhorseshoes
  8. The horseshoe $\Lambda(2)$
  9. Cuts, alphabets and renormalization of words
  10. Sequences in $\Sigma(3)$
  11. Cuts and intervals determined by words
  12. Weakly renormalizable words
  13. Farey sequences and subhorseshoes
  14. Some results about cuts and finite words
  15. Dimension of subhorseshoes that do not connect with $\psi_b$
  16. ...and 11 more sections

Key Result

Theorem 1.1

There exists a decreasing sequence $\{a_r\}_{r\in \mathbb{N}}$ with $a_1<t_1$, $d(a_{r+1})<d(a_r)$ and $\lim \limits_{n\rightarrow \infty}a_r=3$, such that, given $r\in \mathbb{N}$ we can find a subset $\mathcal{B}_r\subset (a_{r+1},a_r)\cap \mathcal{L}$ with the following properties:

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 2.8
  • ...and 77 more