Steklov Institute of Mathematics at St.Petersburg

PREPRINT 11/2009

DYNAMICS OF METRICS IN MEASURE SPACES AND THEIR ASYMPTOTIC INVARIANTS

This preprint was accepted December 11, 2009

ABSTRACT: 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 $\varepsilon$-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.

А.М. Вершик

ДИНАМИКА МЕТРИК В ПРОСТРАНСТВЕ С МЕРОЙ И ИХ АСИМПТОТИЧЕСКИЕ ИНВАРИАНТЫ

АННОТАЦИЯ.
Мы определяем деформацию колмогоровской энтропии,
 как энтропия последовательности
компактов с мерой, строящейся по автоморфизму с инвариантной мерой,
 или по фильтрации сигма-алгебр. Эта деформация, называемая 
масштабированной энтропией есть класс последовательностей натуральных
 чисел, который является метрическим инвариантом автоморфизма или 
соответственно фильтрации. Предполагается, что в этотм контекст -- 
асимптотика последоватекльности компактов, -- 
есть естественный контекст для энтропии и других метрических 
и топологических инвариантов в теории меры и в теории 
динамических систем.