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Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum

Carlos Gustavo Moreira, Christian Camilo Silva Villamil

Abstract

We prove that for any $η$ that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets $k^{-1}((-\infty,η])$ and $k^{-1}(η)$, which are the sets of irrational numbers with best constant of Diophantine approximation bounded by $η$ and exactly $η$ respectively, have the same Hausdorff dimension. We also show that, as $η$ varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.

Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum

Abstract

We prove that for any that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets and , which are the sets of irrational numbers with best constant of Diophantine approximation bounded by and exactly respectively, have the same Hausdorff dimension. We also show that, as varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.
Paper Structure (10 sections, 15 theorems, 149 equations)

This paper contains 10 sections, 15 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Continued fractions and regular Cantor sets
  4. Results on Dynamical Markov and Lagrange spectra
  5. The horseshoe $\Lambda(N)$
  6. Proof of the main theorem
  7. Connection of subhorseshoes
  8. Dimension estimates
  9. Putting unstable Cantor sets into $k^{-1}(\eta)$
  10. Hyperbolic sets of finite type

Key Result

Theorem 1.1

Define $T:=int(L)=int(M)$. For any $\eta\in \overline{T}$, $D(\eta)=HD(k^{-1}(\eta))$ i.e. Even more, In particular, $D|_{X}$ is strictly increasing, where $X$ is $T$ or any interval contained in $\overline{T}$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 15 more