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Seminar in Mathematical physics and related subjects: “Diophantine approximations, dimension and thermodynamic formalism for Fuchsian groups”

Thursday, May 2, 2019, 14:30 - Room 1BC45 - Luca Marchese

ARGOMENTI: Seminars

Luca Marchese (Parigi 13)
“Diophantine approximations, dimension and thermodynamic formalism for Fuchsian groups”
Thursday, May 2, 2019, 14:30 - 15:30
Room 1BC45

Abstract
In classical diophantine approximations it is natural to consider the set "Bad" of those real numbers which are badly approximable by rationals: it is a set of zero Lebesgue measure and full dimension. Finer metric properties have been investigated in depth, both for the classical case and for several generalizations, which arise from the relation between diophantine approximations and the dynamics on homogeneous spaces (or other moduli spaces). The set Bad admits a natural exhaustion by sub-sets Bad(c), in terms a positive parameter c>0, and the dimension of Bad(c) converges to 1 as c goes to 0. D. Hensley computed the asymptotic for the dimension up to the first order in c, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded.
I will prove a generalization of Hensley's asymptotic formula in the context of Fuchsian groups, considering the set off points in the boundary of the hyperbolic space which are badly approximable by the orbits of a non-uniform lattice G in PSL(2,R), and an exhaustion of such set by subsets Bad(G,c), in terms of a parameter c>0. Bowen and Series introduced a "boundary expansion" which enables to approximate any set Bad(G,c) by a dynamically defined Cantor set, whose dimension can be estimated with great precision by thermodynamic techniques introduced by Ruelle and Bowen. A perturbative analysis of the spectral radius of the transfer operator gives the dimension of Bad(G,c) up to the first order in c.