Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups

TL;DR

This paper demonstrates that the convergence rate of the cusp winding spectrum to the Hausdorff dimension in generalized Schottky groups is polynomial, revealing new multifractal phenomena influenced by the limit set's dimension.

Contribution

It introduces a polynomial convergence rate result and links multifractal analysis differences to the Hausdorff dimension of limit sets in generalized Schottky groups.

Findings

Convergence rate of cusp winding spectrum is polynomial.

Differences in Hausdorff dimension affect multifractal analysis.

Provides new characterization of geodesic flow and limit sets.

Abstract

We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which differences in the Hausdorff dimension of the limit set generated by a Markov system cause essentially different results on multifractal analysis. This paper also provides a new characterization of the geodesic flow on the Poinca\'re disc model of two-dimensional hyperbolic space and the limit set of a generalized Schottky group. To prove our main theorem we use thermodynamic formalism on a countable Markov shift, gamma function, and zeta function.