Some spectral applications of McMullen's Hausdorff dimension algorithm

TL;DR

This paper applies McMullen's Hausdorff dimension algorithm to analyze the limit sets of reflection groups in hyperbolic geometry, revealing minima in the parameter space and exploring associated orthogonal polynomials.

Contribution

It extends McMullen's algorithm to compute limit measures and their moments, providing new insights into the structure of these fractal sets and related polynomials.

Findings

Identified four minima in the parameter space of reflection groups.

Extended the algorithm to compute limit measures and moments.

Numerical observations on polynomial zeros and coefficients.

Abstract

Using McMullen's Hausdorff dimension algorithm, we study numerically the dimension of the limit set of groups generated by reflections along three geodesics on the hyperbolic plane. Varying these geodesics, we found four minima in the two-dimensional parameter space, leading to a rigorous result why this must be so. Extending the algorithm to compute the limit measure and its moments, we study orthogonal polynomials on the unit circle associated with this measure. Several numerical observations on certain coefficients related to these moments and on the zeros of the polynomials are discussed.