On Hausdorff dimension and cusp excursions for Fuchsian groups

TL;DR

This paper explores the Hausdorff dimension and cusp excursions of Fuchsian groups, introducing strict Jarník limit sets that generalize existing concepts and connect to number theory via continued fractions.

Contribution

It introduces strict Jarník limit sets as a generalization of Jarník limit sets, revealing new weak multifractal spectra and their number-theoretical implications.

Findings

Description of strict Jarník limit sets and their properties

Connection between multifractal spectra and continued fractions

Insights into cusp excursions and Hausdorff dimension for Fuchsian groups

Abstract

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarn\'{\i}k limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This paper will describe a natural generalisation of these sets, called strict Jarn\'{\i}k limit sets, and show how these give rise to another weak multifractal spectrum. Number-theoretical interpretations of these results in terms of continued fractions will also be given.