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Title: Complex bounds for interval maps
Author: Trejo Abad, Sofía
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2013
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In this thesis we give a proof of complex bounds for real analytic interval maps, for all possible combinatorics. We begin by constructing a sequence of intervals, known as the enhanced nest, that covers both non-renormalizable and infinitely renormalizable maps. This is a generalisation of the nest introduced in [KSvS1]. In Chapter 3, we prove the combinatorial and geometric properties of the enhanced nest, known as real bounds. The results from this chapter extend the ones in [KSvS1] to maps with odd critical points. In Chapter 4, we make use of Poincar´e disks based on intervals from the enhanced nest to construct quasi box-mappings associated to (real) first return maps. Key in this part of the proof are the pullbacks along monotone branches, that are controlled with the aid of fundamental domains, and the pullbacks between two consecutive levels from the enhanced nest of different combinatorial type, one non-terminating and one terminating. We use similar techniques to the ones developed in [LevS2] to obtain complex box-mappings from the quasi box-mappings that we constructed. Finally, we follow the arguments in [KSvS1] to prove complex bounds for the complex box-mappings we constructed.
Supervisor: Not available Sponsor: Consejo Nacional de Ciencia y Tecnología (Mexico) (212227) ; Secretaría de Educación Pública ; Mexico ; University of Warwick ; Mathematics Institute
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics