Entropy of stochastic processes homogeneous with respect to a commutative group of transformations
B. S. Pitskel' Moscow Institute of Railroad Engineering,
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
$\{A_n\}$ 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.
UDC:
513.6
Received: 18.12.1975
Fulltext:
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