Title:

Dynamics of surface homeomorphisms : braid types and coexistence of periodic orbits

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 NielsenThurston classification of surface homeomorphisms, we examine problems pertaining to the coexistence of periodic orbits, in particular for homeomorphisms of the disc, annulus and 2torus. 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 3point invariant sets (Theorem 4.11). • the coexistence of periodic orbits in the annulus (Theorem 4.4), and of the sphere with a 4point 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 pseudoAnosov 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).
