Title: Dynamical Lagrange and Markov spectra

The Lagrange and Markov spectra are subsets of the real line with interesting fractal properties that appear naturally in the study of Diophantine approximations. Using the symbolic-dynamic characterization discovered by Perron, these objects give a rich connection between Number theory and Dynamical systems. In one part of the talk, I will explain the Dynamical Lagrange and Markov spectra, which are generalizations of the classical objects that have a fruitful relation with hyperbolic dynamics. In the other part, I will specialize to the classical case, where I will explain recent joint work with Carlos Gustavo Moreira, Rodolfo Gutiérrez-Romo, Sergio Romaña and Luke Jeffreys, were we obtained an asymptotic expansion of the dimension function of the Markov and Lagrange spectra near 3 as well as results about monotonicity of the local dimension.