A multifractal analysis for cuspidal windings on hyperbolic surfaces

TL;DR

This paper studies the multifractal structure of limit sets of Fuchsian groups, linking Hausdorff dimension to a free energy function, and extends multifractal formalism to infinite graph systems.

Contribution

It provides a complete determination of the multifractal spectrum for cusp windings on hyperbolic surfaces and generalizes multifractal formalism to infinite graph directed Markov systems.

Findings

Hausdorff dimension matches Legendre transform of free energy

Multifractal spectrum explicitly characterized

Extended formalism to infinite systems

Abstract

In this paper we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp winding number. We will completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalise previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.