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Some spectral applications of McMullen's Hausdorff dimension algorithm


Authors: K. Gittins, N. Peyerimhoff, M. Stoiciu and D. Wirosoetisno
Journal: Conform. Geom. Dyn. 16 (2012), 184-203
MSC (2010): Primary 37F35; Secondary 37F30, 42C05, 51M10, 58J50
DOI: https://doi.org/10.1090/S1088-4173-2012-00244-5
Published electronically: July 25, 2012
MathSciNet review: 2950130
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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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Additional Information

K. Gittins
Affiliation: Mathematical Sciences, Durham University, Mountjoy Site, South Road, Durham DH1 3LE, United Kingdom
Email: katie.gittins@durham.ac.uk

N. Peyerimhoff
Affiliation: Mathematical Sciences, Durham University, Mountjoy Site, South Road, Durham DH1 3LE, United Kingdom
Email: norbert.peyerimhoff@durham.ac.uk

M. Stoiciu
Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267
Email: mstoiciu@williams.edu

D. Wirosoetisno
Affiliation: Mathematical Sciences, Durham University, Mountjoy Site, South Road, Durham DH1 3LE, United Kingdom
Email: djoko.wirosoetisno@durham.ac.uk

DOI: https://doi.org/10.1090/S1088-4173-2012-00244-5
Received by editor(s): January 13, 2012
Published electronically: July 25, 2012
Additional Notes: The first author was supported by a Nuffield Undergraduate Research Bursary.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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