Dynamical Systems Seminar

In 1950, Utz introduced expansive homeomorphisms, a concept widely acknowledged in current literature. Recently, there has been notable progress in exploring this idea from a pointwise dynamics perspective. This not only enhances existing theories but also reveals insights not accessible through a global dynamics approach. This talk focuses on two such notions: uniformly expansive points and minimally expansive points. We will explore the presence of uniformly expansive points in homeomorphisms on bounded intervals and their implications for identifying expansive homeomorphisms on the unit circle. Additionally, we will discuss how minimally expansive points contribute to topological stability of a point by establishing a pointwise form of Walters' stability theorem. We will discuss how this result can be applied for homeomorphisms on intervals which is not the case with other existing forms of Walters' stability theorem. This talk is based on joint work with Tarun Das.

Suppose we have a collection of objects such as such as the group of homeomorphisms or diffeomorphisms. We choose an object from this collection at “random”. What dynamical properties does this object have? The usual notion of a “random” or a “large” collection is that of generic or comeager, i.e., we show that the collection of all objects with certain dynamical properties contain a countable intersection of dense open sets. There is an alternate natural notion of bigness called prevalence and its complement called shy sets or Haar null sets. This notion of smallness often highlights different properties than the usual notion of generic. In this introductory talk, we give a background on prevalence, discuss a variety of results involving prevalence and analysis, prevalence and automorphism groups and state some open problems concerning prevalence and dynamics.