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

B. S. Pitskel'

Matematičeskie zametki, Tome 23 (1978) no. 3, pp. 447-462.

Résumé

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., $\varlimsup\limits_{n\to\infty}|A_n|^{-1}H\cdot(\bigvee\limits_{g\in R}T_gP)$, 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 $\bar\nu$-dimensional lattice $Z^\nu$, the Folner condidition coincides with the Van Hove condition.