Some spectral applications of McMullen's Hausdorff dimension algorithm

2011/12/05 by Katie Gittins, K. Gittins, N. Peyerimhoff +8
Mathematics · #37F35 (Primary) 37F30 #42C05 #51M10 #58J50 (Secondary) #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Spectral Theory (math.SP) #math.DS #math.SP #msc:37F30 #msc:37F35 #msc:42C05 #msc:51M10 #msc:58J50

paper · pdf · doi:10.48550/arxiv.1112.1020

24 pages, 10 figures; ancillary files uploaded

openalex publication_date 2011/12/05 · arxiv created 2011/12/07 · arxiv updated 2011/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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