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Asymptotic counting problems for periodic orbits and holonomies of rational maps and Kleinian groups
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Stylianou, Anastasios (2022) Asymptotic counting problems for periodic orbits and holonomies of rational maps and Kleinian groups. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3913380
Abstract
In this thesis we study orbit counting problems in different setups. We start by considering hyperbolic rational maps on the Riemann sphere. For such a map we study its periodic orbits and obtain an asymptotic counting result as the lengths of the orbits grow. In particular, we associate to each periodic orbit a complex number called the multiplier and we call the normalised complex number in the direction of the multiplier, the holonomy of the periodic orbit. We place restrictions on the magnitudes of the multipliers and the arguments of the holonomies of the orbits. We consider varying and potentially shrinking intervals and arcs and obtain two results which resemble a local central limit theorem for the logarithm of the absolute value of the multipliers and an equidistribution theorem for the holonomies. Using traditional ideas from probability theory and thermodynamic formalism we reduce our proofs to bounding the iterates of a certain family of transfer operators. Then we obtain our results by adapting Dolgopyat–type arguments, obtained by Oh and Winter in the situation we consider.
In the second half of this thesis we prove an analogous result to the one above in the setting of convex-cocompact hyperbolic manifolds. We consider closed geodesics in a hyperbolic manifold of arbitrary dimension and prove an asymptotic equidistribution result. We fix a Markov section for the non-wandering set of the geodesic flow on our manifold and order closed geodesics by their word length with respect to the Poincaré first return map of this section. As before, we place restrictions on the geometric length of the geodesics. Moreover, to each closed geodesic we associate a rotation element, called the holonomy, obtained by parallel transporting a frame around the closed geodesic. Once again, using symbolic dynamics and ideas from thermodynamic formalism we reduce our proofs to obtaining bounds for the iterates of a certain family of transfer operators. Adapting Dolgopyat-type estimates, established by Sarkar and Winter in a similar setup, we prove an asymptotic equidistribution result.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Riemann surfaces, Kleinian groups, Hyperbolic groups, Orbits, Holonomy groups, Hyperbolic spaces | ||||
Official Date: | June 2022 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Sharp, Richard, Dr. | ||||
Format of File: | |||||
Extent: | iv, 4 unnumbered pages, 152 pages | ||||
Language: | eng |
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