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

arXiv2212.11371MaRDI QIDQ6421298

P. Vytnova, Carlos Matheus, Carlos Gustavo Moreira

Publication date: 21 December 2022

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 $>0.005$.

Has companion code repository: https://github.com/polevita/dimension_function

Mathematics Subject Classification ID

Continued fractions (11A55) Thermodynamic formalism, variational principles, equilibrium states for dynamical systems (37D35) Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.) (37M25) Markov and Lagrange spectra and generalizations (11J06)

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