These notes axe based on lectures given in 1994 by the first author at a Summer School in Tuczno (Poland) and at the University of Metz.

The purpose is to give a quick introduction to ergodic theory, to study interesting examples as illustration and to present some recent and spectacular developments in topological dynamics of group actions. More precisely, the focus will be on the following two types of systems of a geometrical–algebraic nature:

The geodesic flow on the unit tangent bundle of a locally symmetric space and unipotent actions on homogeneous spaces. Classical examples are the geodesic flow and the horocyclic flow on the unit tangent bundle of a compact Riemann surface of constant negative curvature. These flows are among the most studied dynamical systems. Of particular interest are their ergodic (or mixing) properties and the asymptotic behaviour of their orbits.

Unipotent actions on homogeneous spaces enjoy remarkable regularity properties. A striking illustration of this regularity is Hedlund's minimality theorem: for any lattice Γ in G = SL(2,ℝ), any orbit in the homogeneous space Γ\ G under a unipotent subgroup of SL(2,ℝ) is either periodic or dense. Such actions have close connections to problems in Number Theory. For instance, one of the most spectacular application of unipotent actions is the solution in 1987 by Margulis of Oppenheim's conjecture which was open for more than 40 years (see [Ma3]).

Let G be a locally compact group acting on a probability space X, and let π be the associated unitary representation. Recall that the action of G is strongly mixing if every matrix coefficient of π vanishes at infinity on G (see Chap. I, Definition 2.10).

The Vanishing Theorem of Howe and Moore states that matrix coefficients of arbitrary (non trivial) unitary representations of a simple Lie group vanish at infinity. This is a powerful theorem showing that many interesting flows are ergodic (and even strongly mixing). A typical application is as follows. Let G be a simple Lie group. Let Γ be a lattice and let H be a non-compact closed subgroup of G. Then the action of H on Γ \ G by right translations is ergodic. As we saw in the previous chapter, the geodesic flow of a locally symmetric space is related to flows of this type.

A more general application is Moore's ergodicity theorem which gives optimal conditions under which the action of a subgroup of a semisimple Lie group is ergodic or even strongly mixing (see Theorems 2.1, 2.5).

The theorem of Howe and Moore is first proved for SL(2, ℝ). The proof we present, due to R. Howe, is remarkable in that it essentially uses no representation theory. Again, a crucial rôle is played by Mautner's lemma (Chap. II, 3.6)

The extension to general semisimple Lie groups is a technical matter: one has to use sufficienly many copies of SL(2,ℝ) inside the given group. Here we follow [Vel].

Ergodic Systems

Ergodic theory may be viewed as the study of measure (or, more generally, measure class) preserving actions of groups (or semigroups) on measure spaces.

The main examples to be treated throughout these notes arise as follows. Let G be a locally compact group, and let H, L be closed subgroups of G. The homogeneous space G/H carries a unique G-invariant measure class. Now, L acts on G/H by left translations. An interesting and important problem is to study, for specific G, H, L this action of L on G/H from a measure theoretic point of view. Usually, H is a lattice in G (see Chap. II, §2) so that G/H carries a G-invariant probability measure. So, we shall almost always deal with measure preserving actions on a probability space.

This chapter is a quick introduction to ergodic theory. We discuss mainly material which is relevant for later chapters.

Our exposition is incomplete as several important topics, such as entropy, have been omitted. Section 1 contains some standard examples of ergodic actions. In Section 2, ergodicity is formulated in terms of unitary group representations (the so-called Koopmanism). The classical ergodic theorem of von Neumann is proved and M. Keane's elegant proof of Birkhoff's ergodic theorem is reproduced. Moreover, strong mixing and weak mixing are introduced and discussed from the point of view of unitary representations. In Section 3, we state the theorem about the decomposition of general measure preserving group actions into ergodic pieces.