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Title: Prime orbit theorems for closed orbits and knots in hyperbolic dynamical systems
Author: Waddington, Simon
ISNI:       0000 0001 3549 6649
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1992
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This thesis consists of four chapters, each with its own notation and references. Chapters 1, 2 and 3 are independent pieces of research. Chapter 0 is an introduction which sets out the definitions and results needed in the main part of the thesis. In Chapter 1, we derive asymptotic formulae for the number of closed orbits of a toral automorphism which is ergodic, but not necessarily hyperbolic. Previously, such formulae were known only in the hyperbolic case. The proof uses an analogy with the Prime Number Theorem. We also give a new proof of the uniform distribution of periodic points. In Chapter 2, we derive various asymptotic formulae for the numbers of closed orbits in the Lorenz and Smale horseshoe templates with given knot invariants, (specifically braid index and genus). We indicate how these estimates can be applied to more complicated flows by giving a bound for the genus of knotted periodic orbits in the ' figure of eight template'. In Chapter 3, we prove a dynamical version of the Chebotarev density theorem for group extensions of geodesic flows on compact manifolds of variable negative curvature. Specifically, the group is taken to be the weak direct sum of a finite abelian group. We outline an application to twisted orbits.
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