Dynamics of Metrics in Measure Spaces and Their Asymptotic Invariants
2010, v.16, Issue 1, 169-184
We discuss the Kolmogorov's entropy and Sinai's definition of it; and then define a deformation of the entropy, called 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 asymptotic dynamics of metrics in a measure space, in which, for instance, the generalized entropy is understood as 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.
Keywords: scaling entropy,metric compact with measure,asymptotic geometry,filtrations