Sharland, Thomas Joseph (2010) Rational maps with clustering and the mating of polynomials. PhD thesis, University of Warwick.

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Abstract

The main focus of this thesis is the study of a special class of bicritical rational maps of the Riemann sphere. This special property will be called clustering; which informally is when a subcollection of the immediate basins of the two (super-)attracting periodic orbits meet at a periodic point p, and so the basins of the attracting periodic orbits are clustered around the points on the orbit of p. Restricting ourselves to the cases where p is fixed or of period 2, we investigate the structure of such maps combinatorially; in particular showing a very simple collection of combinatorial data is enough to define a rational map uniquely in the sense of Thurston. We also use the language of symbolic dynamics to investigate pairs (f, g) of polynomials such that f - g has a fixed or period two cluster point. We find that that the internal addresses of such maps follow very definite patterns which can be shown to hold in general.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Mappings (Mathematics)
Date:	December 2010
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Epstein, Adam
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC)
Extent:	xiv, 264 leaves : ill.
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/35776

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