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

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.

57 pages, 4 figures. Error in the previous versions fixed adding the extra Assumption 2.1 to the main Theorem 1.2: we assume that there exists a finite index free subgroup whose Ford domain is an ideal polygon. Added Appendix A to show a class of non-uniform lattices satisfying Assumption 2.1 (all these examples give a punctured sphere as quotient space). Reintegrated proof of Theorem 4.1