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Title: The Ruelle operator, zeta functions and the asymptotic distribution of closed orbits
Author: Pollicott, Mark
ISNI:       0000 0001 2439 6212
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1981
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This thesis is composed of three independent chapters and an appendix. Each chapter has its own introduction, references and notation. In chapter One a new proof of a theorem of Ruelle about real Perron-Frobenius type operators is given. t This theorem is then extended to complex Perron-Frobenius type operators in analogy with Wielandt's theorem for matrices. Finally two questions raised by Ruelle and Bowen concerning analyticity properties of seta functions for flows are answered. In Chapter Two we improve a result of Ruello on the domain of analyticity of the zeta function for an Axiom A flow. The method used requires results on complex Perron-Frobenius operators derived in the first chapter. These results are reproduced with alternative proofs. Finally, asymptotic estimates for numbers of closed orbits are deduced by analogy with the prime number theorem. This extends a result of Margulis. In the first section of Chapter Three we give a relationship between periodic points and certain equilibrium states for subshifto of finite type. We next study geodesic flows on surfaces of constant negative curvature. We compare the zeta functions of a geodesic flow and a certain suspension flow. These results are then used to recover asymptotic estimates by Margulis and Bowen on the distribution of closed geodesics. Finally new results are given in two special cases. The Appendix is an outline of Bowen's symbolic dynamics for Axiom A flows. This material is purely expository.
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