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Fractal geometry, dynamics and Diophantine approximations: the Markov and Lagrange spectra

We will discuss some recent results on the fractal geometry of the Markov and Lagrange spectra from Diophantine approximations, and their set difference. We will relate these results to symbolic dynamics, continued fractions and to the study of the fractal geometry of arithmetic sums of regular Cantor sets, a subject also related to the study of homoclinic bifurcations in Dynamical Systems.