Transfer operators and dimension of bad sets for nonuniform Fuchsian lattices
Abstract
The set of real numbers which are badly approximable by rationals admits a 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 nonuniform 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. We relate geometric good approximations to a notion of bounded partial quotients for the BowenSeries expansion. This gives a family of Cantor sets and associated quasicompact transfer operators, with simple and positive maximal eigenvalue. Perturbative analysis of spectra applies.
 Publication:

arXiv eprints
 Pub Date:
 December 2018
 arXiv:
 arXiv:1812.11921
 Bibcode:
 2018arXiv181211921M
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Number Theory
 EPrint:
 Minor changes with respect to version 2. 59 pages, 4 figures