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Cherry fields and the rotation numbers of one parameter families of maps of the circle
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Boyd, Colin Alexander (1984) Cherry fields and the rotation numbers of one parameter families of maps of the circle. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b1445507~S15
Abstract
This thesis is concerned with a class of flows on the 2-torus and certain properties of maps of the circle. The introduction explai the historical background to the work. In the first chapter Cherry fields are defined, and a class of natural paths introduced. It is shown that for these paths, the set of parameters for which the path takes an unstable field is a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds, and this is used to show that the natural paths through them are stable paths. The second chapter is concerned with rotation intervals for endomorphisms of the circle and the individual rotation intervals of different points on the circle. A shift space is constructed for each endomorphism and this is used to give new proofs of some known results and to give new information about how individual rotation intervals are distributed around the circle. The third chapter generalises a Theorem from Chapter 1 concerning rotation numbers, to one about rotation intervals.
Regarding the method of investigation, aside from the traditional methods of Pure Mathematics, some experimentation on examples was done by computer enabling the author to deduce the likely outcome of some problems. These experiments are not referred to elsewhere in the text.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Circle, Torus (Geometry), Hausdorff measures, Endomorphisms (Group theory) | ||||
| Official Date: | September 1984 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Manning, Anthony, 1946- | ||||
| Sponsors: | Science and Engineering Research Council (Great Britain) | ||||
| Format of File: | |||||
| Extent: | iii, ii-iii, 70 leaves : illustrations | ||||
| Language: | eng |
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