Entropy of stochastic processes homogeneous with respect to a commutative group of transformations

In order to define the entropy of a stochastic field homogeneous with respect to a countable commutative group of transformations G, one fixes a sequence {A_n} of finite subsets of the group G and considers the upper limit of the sequence of mean entropies of the iterates of the decomposition P. i.e., , where ¦A_n¦ is the number of elements in A_n. It is proved that for a fixed stochastic field and all sequences {A_n} satisfying the Folner condition, the limit of the means exists and is unique. If the sequence {A_n} is such that for all stochastic fields invariant under G, the entropy calculated in terms of it is the same as that calculated for a Folner-sequence, then {A_n} satisfies the Folner condition. In the case when G is a -dimensional lattice Z, the Folner condidition coincides with the Van Hove condition.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 447–462, March, 1978.The author thanks D. V. Anosov for valuable advice, and also B. M. Gurevich and A. M. Stepin for helpful discussions concerning this note.