On the graph of the dimension function of the Lagrange and Markov spectra

TL;DR

This paper investigates the Hausdorff dimension function of the Lagrange and Markov spectra, identifying key plateaux and approximating the graph using rigorous numerical methods.

Contribution

It determines twelve nontrivial plateaux of the dimension function and uses numerical methods to approximate the graph between these plateaux.

Findings

Identified twelve nontrivial plateaux of the dimension function.

Proved the ten largest non-trivial plateaux have lengths greater than 0.005.

Provided numerical approximations of the graph between plateaux.

Abstract

We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$.