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Title: Dynamics of surface homeomorphisms : braid types and coexistence of periodic orbits
Author: Guaschi, John
ISNI:       0000 0001 3521 6838
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1991
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In this Thesis, we discuss the following general problem in dynamical systems: given a surface homeomorphism, and some information about its periodic orbits, what else can we deduce about its periodic orbit structure? Using the concept of the ‘braid type’ of a periodic orbit, its relation to Artin’s braid group, and the Nielsen-Thurston classification of surface homeomorphisms, we examine problems pertaining to the coexistence of periodic orbits, in particular for homeomorphisms of the disc, annulus and 2-torus. We aim to elucidate the underlying geometry and topology in such systems. The main original results are the following: • classification of braid types for periodic orbits of diffeomorphisms of genus one surfaces with topological entropy zero (Theorems 2.5 and 2.6). • lower bounds on the size of the rotation sets of annulus homeomorphisms which possess certain periodic orbits or finite invariant sets (Theorems 3.17 and 3.19, Theorem 3.20). • bounds on the size and shape of rotation sets of torus homeomorphisms possessing certain periodic orbits. (Theorems 3.24 and 3.25). • the coexistence of periodic orbits in the disc, for periodic orbits of prime period (Theorem 4.2), of period 4 (Theorem 4.10), and for 3-point invariant sets (Theorem 4.11). • the coexistence of periodic orbits in the annulus (Theorem 4.4), and of the sphere with a 4-point invariant set (Theorem 4.12). • given a torus homeomorphism isotopic to the identity which possesses a fixed point, it is isotopic to the identity relative to that fixed point (Theorem 5.6). • given a periodic orbit of a disc homeomorphism of period 3, the coexistence of a strongly linked fixed point (Theorem 5.10). • given a periodic orbit of the annulus homeomorphism of pseudo-Anosov braid type, its rotation number lies in the interior of the rotation set (Theorem 6.1). • amongst certain sets of braid types of the annulus and disc, the existence of minimal elements, which any other element dominates (Theorems 7.4 and 7.15).
Supervisor: Not available Sponsor: University of Warwick ; Science and Engineering Research Council ; British Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics