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

Abstract

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 {An} 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.,\(\mathop {\lim }\limits_{n \to \infty } |A_n |{}^{ - 1}H \cdot (\mathop V\limits_{g \in A_n } TgP)\), where ¦An¦ is the number of elements in An. It is proved that for a fixed stochastic field and all sequences {An} satisfying the Folner condition, the limit of the means exists and is unique. If the sequence {An} 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 {An} satisfies the Folner condition. In the case when G is a ν-dimensional lattice Zν, the Folner condidition coincides with the Van Hove condition.