  We discuss the Kolmogorov's entropy and Sinai's definition of it; and then
define a deformation of the entropy, called {\it scaling entropy}; this is also
a metric invariant of the measure preserving actions of the group, which is
more powerful than the ordinary entropy. To define it, we involve the notion of
the $\epsilon$-entropy of a metric in a measure space, also suggested by A. N.
Kolmogorov slightly earlier. We suggest to replace the techniques of measurable
partitions, conventional in entropy theory, by that of iterations of metrics or
semi-metrics. This leads us to the key idea of this paper which as we hope is
the answer on the old question: what is the natural context in which one should
consider the entropy of measure-preserving actions of groups? the same question
about its generalizations--scaling entropy, and more general problems of
ergodic theory.
  Namely, we propose a certain research program, called {\it asymptotic
dynamics of metrics in a measure space}, in which, for instance, the
generalized entropy is understood as {\it the asymptotic Hausdorff dimension of
a sequence of metric spaces associated with dynamical system.} As may be
supposed, the metric isomorphism problem for dynamical systems as a whole also
gets a new geometric interpretation.
