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Title: Asymptotic formulae for closed orbits of hyperbolic flows
Author: Sharp, Richard John
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1990
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This thesis consists of three chapters and an appendix each with its own notation and references. Chapter 0 is an introduction which sets out the definitions and results needed in the main part of the thesis. In Chapter 1 we prove an analogue of Mertens' theorem of prime number theory for the closed orbits of an Axiom A flow (restricted to a non-trivial basic set). A similar result is established for the geodesic flow on a non-compact, finite area surface of constant negative curvature. Applying this result to the modular suface yields some asymptotic formulae concerning quadratic forms. In Chapter 2 we give a new proof of an asymptotic formula for the number of closed orbits of a weak-mixing Axiom A flow subject to certain constaints due to S. Lalley. We extend this result to cover the case of finite group extensions and, for transitive Anosov flows, give an application to homology. We also discuss asymptotics for closed orbits in a fixed homology class, extending a result of A. Katsuda and T. Sunada. The appendix, which is included for completeness, is an account of the proof of a technical result of A. Katsuda and T. Sunada which is used extensively in Chapter 2.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics