Transfer operators and dimension of bad sets for non-uniform Fuchsian lattices

TL;DR

This paper studies the dimension of sets of points badly approximable by parabolic fixed points in non-uniform Fuchsian lattices, using transfer operators and thermodynamic methods to obtain first-order asymptotics.

Contribution

It extends the analysis of badly approximable sets to non-uniform Fuchsian lattices using transfer operators and thermodynamic formalism, providing new asymptotic dimension estimates.

Findings

Computed the dimension of epsilon-badly approximable points up to first order in epsilon.

Established a connection between geometric good approximations and bounded partial quotients in Bowen-Series expansions.

Developed a family of Cantor sets with associated transfer operators having positive maximal eigenvalues.

Abstract

The set of real numbers which are badly approximable by rationals admits an exhaustion 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 consider diophantine approximations by parabolic fixed points of any non-uniform lattice in PSL(2,R) and a geometric notion of $\epsilon$-badly approximable points. We compute the dimension of the set of such points up to the first order in $\epsilon$, via the thermodynamic method of Ruelle and Bowen. Geometric good approximations are related to a notion of bounded partial quotients for the Bowen-Series expansion. This gives a family of Cantor sets and associated quasi-compact transfer operators, with simple and positive maximal eigenvalue. Perturbative analysis of spectra applies. Our techniques only apply to non-uniform lattices admitting a finite index free subgroup satisfying a specific property.