Diophantine approximations, Markov and Lagrange spectra and dynamical Cantor sets

TL;DR

This paper surveys the interplay between Diophantine approximations, Markov and Lagrange spectra, and dynamical Cantor sets, highlighting recent advances in low-dimensional dynamics and number theory.

Contribution

It provides an expanded overview of the connections between number theory and dynamical systems, focusing on Cantor sets and spectral properties.

Findings

Enhanced understanding of the structure of Markov and Lagrange spectra.

Insights into the dynamics of Cantor sets in low-dimensional systems.

Connections established between number theory and dynamical systems.

Abstract

This text is a slightly expanded version of a survey article on certain aspects of low dimensional dynamics and number theory written after a kind invitation by the editors of the Notices of the American Mathematical Society.