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We discuss the Kolmogorov's entropy and Sinai's definition of it; and thendefine a deformation of the entropy, called {\it scaling entropy}; this is alsoa metric invariant of the measure preserving actions of the group, which ismore powerful than the ordinary entropy. To define it, we involve the notion ofthe $\epsilon$-entropy of a metric in a measure space, also suggested by A. N.Kolmogorov slightly earlier. We suggest to replace the techniques of measurablepartitions, conventional in entropy theory, by that of iterations of metrics orsemi-metrics. This leads us to the key idea of this paper which as we hope isthe answer on the old question: what is the natural context in which one shouldconsider the entropy of measure-preserving actions of groups? the same questionabout its generalizations--scaling entropy, and more general problems ofergodic theory.  Namely, we propose a certain research program, called {\it asymptoticdynamics of metrics in a measure space}, in which, for instance, thegeneralized entropy is understood as {\it the asymptotic Hausdorff dimension ofa sequence of metric spaces associated with dynamical system.} As may besupposed, the metric isomorphism problem for dynamical systems as a whole alsogets a new geometric interpretation.

Dynamics of metrics in measure spaces and their asymptotic invariants