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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1978, Volume 23, Issue 3, Pages 447–462 (Mi mz8160)

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.

Full text: PDF file (1073 kB)

English version:
Mathematical Notes, 1978, 23:3, 242–250

Bibliographic databases:

UDC: 513.6

Citation: B. S. Pitskel', “Entropy of stochastic processes homogeneous with respect to a commutative group of transformations”, Mat. Zametki, 23:3 (1978), 447–462; Math. Notes, 23:3 (1978), 242–250

Citation in format AMSBIB
\Bibitem{Pit78} \by B.~S.~Pitskel' \paper Entropy of stochastic processes homogeneous with respect to a~commutative group of transformations \jour Mat. Zametki \yr 1978 \vol 23 \issue 3 \pages 447--462 \mathnet{http://mi.mathnet.ru/mz8160} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=494449} \zmath{https://zbmath.org/?q=an:0405.60013|0394.60011} \transl \jour Math. Notes \yr 1978 \vol 23 \issue 3 \pages 242--250 \crossref{https://doi.org/10.1007/BF01651440}