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Prime orbit theorems for closed orbits and knots in hyperbolic dynamical systems
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Waddington, Simon, 1966- (1992) Prime orbit theorems for closed orbits and knots in hyperbolic dynamical systems. PhD thesis, University of Warwick.
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WRAP_Theses_Waddington_1992.pdf - Submitted Version - Requires a PDF viewer. Download (2458Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3227933~S15
Abstract
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.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Hyperbolic groups, Numbers, Prime, Combinatorial dynamics | ||||
| Official Date: | July 1992 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Parry, William, 1934-2006| | ||||
| Sponsors: | Science and Engineering Research Council (Great Britain) | ||||
| Format of File: | |||||
| Extent: | 138 leaves | ||||
| Language: | eng |
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