Dynamics of Groups of Homeomorphisms

摘要

These notes provide a survey of certain results concerning dynamical and spectral properties of some countable subgroups of homeomorphisms of a metric space. Most of them were obtained in recent years. For many results we only give the references or at most a glimpse of the method used in the proof. Let E be a metric space and Homeo(E) be its group of homeomorphisms. We study in this chapter certain results concerning dynamical and spectral properties of some countable subgroups G of the group Homeo(E). We shall point out in a first paragraph some general topological notions which will be used in the following paragraphs. We shall present through the second paragraph some classical dynamical properties of the group G when E is the line R ; in order to state several properties of the existence of minimal sets and of exceptional orbits, we precise the nature of orbits under additional assumptions on the group G or on the class of its elements. The class of an orbit O is the union of all orbits O' which have the same closure as O. We denote by X = E/G the space of classes of orbits (called quasi-orbits space). In the third paragraph, we shall study some of the relations between these groups G on one hand, and approximately finite-dimensional C~*-algebra's and unitary commutative rings on the other hand. We shall look at the case when E is the line R. The purpose of the fourth paragraph will be to give some dynamical properties when these groups are equicontinuous. Throughout this chapter we will give many examples which illustrate the studied situation and show that the hypothesis are necessary. I would expect that this chapter only assumes knowledge of general topology.