Limiting modular symbols and their fractal geometry

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an `almost complete' multifractal description of the higher--dimensional level sets arising from Manin--Marcolli's limiting modular symbols.Comment: 20 pages, 1 figur

A note on the algebraic growth rate of Poincar\'e series for Kleinian groups

In this note we employ infinite ergodic theory to derive estimates for the algebraic growth rate of the Poincar\'e series for a Kleinian group at its critical exponent of convergence.Comment: 8 page

Phase transitions for suspension flows

This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.Comment: Example 5.2 expanded. Typos corrected. Section 6.1 superced the note "Thermodynamic formalism for the positive geodesic flow on the modular surface" arXiv:1009.462

Jarník and Julia; a Diophantine analysis for parabolic rational maps

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarn'ik in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a `weak multifractal analysis' of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan's famous dictionary translating between the theory of Kleinian groups and the theory of rational maps. 1 Statement of main results In this paper we derive a Diophantine analysis for Julia sets J(T ) of parabolic rational maps T : C ! C . We generalise two classical number theoretical theorems of Dirichlet and Jarn'ik to the theory of iterations of rational maps. We then show that these results embed in the concept of conformal measures, where they admit a `weak multifractal analysis' of the dimH (J(T )) -conformal measure which is naturally associated with the..