We introduce and present the basic facts about the Thompson group, a widely studied object in Geometric group theory. This group was many times used to give new examples of phenomena or counter examples to various conjectures. We also give a number of easy to state open problems. No prerequisites are required to understand this talk.
We introduce a new concept of finite time entropy which is a local version of the classical concept of metric entropy. Based on that, a finite time version of Pesin's entropy formula and also an explicit formula of finite time entropy for 2-D systems are derived. We also discuss about how to apply the finite time entropy field to detect special dynamical structures such as Lagrangian coherent structures.
The talk covers the long-term behaviour of measure preserving dynamical systems induced by amenable groups acting on probability spaces. We start with an introduction in the world of classic ergodic theorems. Some of those latter statements can be interpreted as "laws of large numbers" from statistics applied in a general setting of group dynamics. This leads to the notion of almost-additive Banach space-valued set functions, with their normalized version converging along Folner sequences. We present the corresponding abstact mean ergodic theorem and as an application, we mention the pointwise almost everywhere convergence of bounded, additive processes.
We study actions of finitely-generated groups on metric spaces. The goal of the talk is to introduce the notion of shadowing property for group actions and study its basic properties. Special attention will be given to nilpotent, Baumslag-Solitar and free groups.