The theory of differentiate dynamical systems is a rich and diversified field, emerging somewhere between mathematics and the sciences, where it constitutes an important conceptual scheme for the ‘decodification’ of many dynamical processes in a unified way. Indeed, an attentive analysis shows that many different systems, coming from physics, biology, chemistry, economics and technology, have time evolutions that, although irregular and chaotic, nevertheless make them similar under many qualitative aspects. However, this similarity does not appear in a straightforward manner. Rather, it emerges within a conceptual frame where many mathematical ideas and techniques, pertaining number theory, topology, probability theory, etc. are brought together to form a complex and fascinating theoretical corpus.

This review, which collects together a series of lectures given by David Ruelle at the Accademia dei Lincee (Rome, May, 1987), deals with those aspects of dynamical systems which are more closely related with ergodic theory, namely with the study of the properties of the invariant measures generated by the time evolutions themselves.

In writing this work, I strove to make it as structured and progressive as possible in order to make it accessible to readers not yet quite at their ease with these topics.

With this in mind I divided the text into two parts. The first part attempts to clarify the interpretative frame in which several dynamical processes (natural as well as ‘artificial’) can be approached by means of mathematical models of a deterministic type. In the second part the concept of invariant probability measure is introduced, along with some ergodic quantities such as characteristic exponents, entropy, dimensions, resonances, etc., which make it possible to extract useful information on the asymptotic statistical properties of the time evolutions we are dealing with.