Conjugacy relation and dynamics in the group of homeomorphism of the Cantor Space
Time and place
12:30 PM on Tuesday, October 11th, 2011; NAC 6-113
Udayan Darji (University of Louisville)
Abstract
In this talk we discuss graph theoretic tech- niques to study the dynamics and the structure of the H, homeomorphism group of the Cantor space. We establish an important approximation theorem and give an invariant which determines when two homeomorphisms of the Cantor space are topologically conjugate to each other. Our investigation leads to an uni ed view point from which many well known and new results follow with ease. For example, Kechris and Rosendal proved that H admits generics and an explicit description of the Special Homeomorphism was given by Akin, Glasner and Weiss. Our investigation gives a different and an useful description of the Special Homeomorphism. As a simple corollary to our techniques we show that a generic homeomorphism of the Cantor space has no Li-Yorke pair, which implies the well known result of Glasner and Weiss that a generic homeomorphism of the Cantor space has topological entropy zero. In addition to other applications to the dynamics of homeomorphisms, we also develop similar but simpler techniques in order to study the dynamics of the generic continuous self-map of the Cantor space.
