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A return time invariant for finitary isomorphisms. (English) Zbl 0589.28014

A dynamical system (X,\({\mathcal B},\mu,T)\) is called almost-topological if there exists a topology \({\mathcal T}\) such that (i) \({\mathcal T}\subset {\mathcal B}\) and \(\sigma\) (\({\mathcal T})={\mathcal B}(mod \mu)\), \((ii)\quad T^{-1}{\mathcal T}\subset {\mathcal T}(mod \mu)\) and \((iii)\quad \mu (U)>0\) for all U in \({\mathcal T}\). \(p=p(T)=\inf \{p(Y):\quad U\in {\mathcal T}\}\) is called the return time invariant of the system. This invariant is nondecreasing under finitary homomorphisms. It is shown that for every two real numbers \(h>0\) and \(k>1\) one can construct a countable state mixing Markov shift with entropy h and return time invariant k.
Reviewer: D.Ramachandran

MSC:

28D20 Entropy and other invariants
28D10 One-parameter continuous families of measure-preserving transformations
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References:

[1] Petrov, Sums of Independent Random Variables (1975) · doi:10.1007/978-3-642-65809-9
[2] DOI: 10.1007/BF02761830 · Zbl 0441.28008 · doi:10.1007/BF02761830
[3] Rudolph, Ergod. Th. & Dynam. Sys. 2 pp 85– (1982)
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