Abstract

We give a review of basic properties of invariant σ-algebras for ℤd-actions on a Lebesgue probability space and some applications of them.

Introduction

Invariant σ-algebras are useful tools for solving a series of important problems in ergodic theory. In one – dimensional case they have been applied, among other things, for classification problems and for the investigation of dynamical systems with completely positive entropy.

In this paper we present a review of applications of invariant σ-algebras in the multidimensional case. Now the role of time plays the lexicographical order in the group ℤd, d ≥ 2.

The direct generalization of the concept of invariance to the multidimensional case is not satisfactory for valuable applications. The proper analogue is the so called strong invariance including, beyond the simple extension of the one–dimensional invariance, the continuity condition strictly connected with the rank of the group of multidimensional integers.

The central place in the theory of invariant σ-algebras is taken by perfect σ-algebras. The proof of their existence is based on the methods of relative ergodic theory.

In the review we give applications of perfect σ-algebras to the theory of Kolmogorov ℤd-actions, to the description of the spectrum of ℤd-actions with positive entropy and to find a connection between the monequililorium entropy and the Conze–Katznelson–Weiss entropy.

We also give examples of applications of relatively perfect σ–algebras to describe maximal factors and principal factors.