Dynamics of metrics in measure spaces and scaling entropy

This survey is dedicated to a new direction in the theory of dynamical systems, the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric which are naturally consistent with each other (a metric triple, or an mm-space) defines automatically the notion of its entropy class, thus allowing one to construct a theory of scaling entropy for dynamical systems with invariant measure, which is different and more general in comparison to the Shannon-Kolmogorov theory. This possibility was hinted at by Shannon himself, but the hint went unnoticed. The classification of metric triples in terms of matrix distributions presented in this paper was proposed by Gromov and Vershik. We describe some corollaries obtained by applying this theory.