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

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 $> 0.005$.