Dimension of bad sets for non-uniform Fuchsian lattices

Luca Marchese (Paris 13)

Monday 14th January 16:00-17:00 Maths 311B

Abstract

The set of real numbers which are badly approximable by rationals admits a filtration by sets Bad(epsilon), whose dimension converges to 1 as epsilon goes to zero. D. Hensley computed the asymptotic for the dimension up to the first order in epsilon, via an analogous estimate for the 
set of real numbers whose continued fraction has all entries uniformly bounded. We generalize this setting considering diophantine approximations by any non-uniform lattice in PSL(2,R). In particular we give a definition of epsilon-badly approximable points which naturally generalizes the case of rationals. Then we use the thermodynamic method of Ruelle and Bowen to compute the dimension of the set of such points up to the first order in epsilon. Our estimates of spectral radii of transfer operators follow Hensley's scheme, but we use Banach spaces of piecewise Lipschitz functions.

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