Guaschi, John (1991) Dynamics of surface homeomorphisms : braid types and coexistence of periodic orbits. PhD thesis, University of Warwick.

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Abstract

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).

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Homeomorphisms, Braid theory, Combinatorial dynamics
Official Date:	September 1991
Dates:	Date Event September 1991 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	MacKay, Robert S., 1956- ; Manning, Anthony, 1946- ; Franks, John M., 1943-
Sponsors:	University of Warwick. Mathematics Institute ; Science and Engineering Research Council (Great Britain) ; British Council
Format of File:	pdf
Extent:	[146] leaves : illustrations
Language:	eng

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