Multifractal analysis of homological growth rates for hyperbolic surfaces

TL;DR

This paper applies multifractal analysis to the homological growth rates of geodesics on hyperbolic surfaces, deriving a formula for the Hausdorff dimension of level sets using advanced dynamical systems techniques.

Contribution

It introduces a new formula linking Hausdorff dimension of growth rate level sets to a generalized Poincaré exponent, utilizing symbolic dynamics and thermodynamic formalism.

Findings

Derived a formula for Hausdorff dimension of level sets

Proved analyticity of the dimension spectrum

Connected growth rates to Poincaré exponents

Abstract

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincar\'e exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.