Abstract
Let (X,μ, T) be an ergodic dynamic system and let ξ = (C1, C2, ...) be a discrete decomposition of X. Conditions are considered for the existence almost everywhere of
whereC ξn(x) is the element of the decomposition ξn = ξ V T ξ V ... < Tn-1ξ containing x. It is proved that the condition H(ξ) < ∞ is close to being necessary. If T is a Markov automorphism and ξ is the decomposition into states, then the limit exists, even if H(ξ) = ∞, and is equal to the entropy of the chain.
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Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 93–103, January, 1971.
I wish to express my gratitude to B. M. Gurevich, D. V. Anosov, and Ya. G. Sinai for their useful comments and for their interest in this work.
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Pitskel', B.S. Some remarks concerning the individual ergodic theorem of information theory. Mathematical Notes of the Academy of Sciences of the USSR 9, 54–60 (1971). https://doi.org/10.1007/BF01405054
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DOI: https://doi.org/10.1007/BF01405054