Dynamical Systems with Invariant Measure

Abstract

The theory of transformations with invariant measure or the metric theory of dynamical systems or ergodic theory is extensively related to various branches of mathematics — to the theory of classical dynamical systems, i.e., to classical mechanics, to probability theory, to functional analysis, to algebra, to number theory, to topology, etc. These diverse and steadfast relationships are rooted in the following two factors: first, the basic object of study, namely, a transformation with invariant measure (in other words, an automorphism of a space with measure), frequently encountered in mathematics, has proved to be a topic which is very meaningful and which lends itself to profound study; second, ergodic theory itself has repeatedly turned out to be an area in which there have been applied and verified new powerful general mathematical ideas and methods such as operator theory, general measure theory, and quite recently, information theory and probability theory, etc.