Semigroups, amenabiblity and fixed point theorems

Abstract

This master thesis is focused on the study of amenability theory and its connection with fixed point theory for dynamical systems. A semigroup is said to be amenable if it admits a finitely additive probability measure defined in all of its subsets that is invariant by translations. We aim to give a general overview on amenability theory and its relationship with the existence on invariant means, which deeply makes use of techniques from functional analysis. We show techniques to produce examples of amenable and nonamenable groups. During this approach we prove and apply extensively Markov- Kakutani and Day’s fixed point theorems, which provide a characterization of amenability in terms of the existence of a common fixed point for a semigroup action. These results lie within the scope of the so-called “Kakutani-type” fixed point theorems in the context of topological dynamical systems. Last chapter will be devoted to the study of a wide class of this type of theorems for semigroup actions with special focus on the a Ryll-Narzdewski fixed point theorem. Some applications and further linear and non-linear extensions will be considered.