Finitary Codes, a short survey

Serafin, Jacek

doi:10.1214/lnms/1196285827

Cited by 10 publications

(4 citation statements)

References 23 publications

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“…Being finitary is a locality condition, and for other natural notions of locality, we refer the reader to [MS22] (in their more refined framework, what we called finitary above is called stop-finitary ). Note also that the classic example of isomporhism between Bernoulli shifts by Meshalkin ([M59]) is also a finitary one and that in [KS79], Keane and Smorodinsky proved the existence of finitary isomorphisms between Bernoulli shifts of the same entropy over , and thus strengthening Ornstein’s landmark theorem (for a survey of finitary codings in this direction, see [S06]).…”

Section: The Case Of Finitary Smentioning

confidence: 99%

A factor of i.i.d. with uniform marginals and infinite clusters spanned by equal labels

Mester

2022

Ergod. Th. Dynam. Sys.

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Section: The Case Of Finitary Smentioning

confidence: 99%

A factor of i.i.d. with uniform marginals and infinite clusters spanned by equal labels

Mester

2022

Ergod. Th. Dynam. Sys.

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“…Mappings satisfying the property in the question are said to be finitary (see [32] and [15] for more information).…”

Section: Theorem 2 the Time-one Map Of A Topologically Weak Mixing Su...mentioning

confidence: 99%

Weak mixing suspension flows over shifts of finite type are universal

Quas¹,

Soo²

2012

Journal of Modern Dynamics

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Let S be an ergodic measure-preserving automorphism on a nonatomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Hölder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T , then there exists an embedding of the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen [5] and Ratner [31], we also obtain an embedding of the measure-preserving automorphism into a geodesic flow whenever the measure-theoretic entropy of S is strictly less than the topological entropy of the time-one map of the geodesic flow.

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“…Aperiodic finite-state Markov processes are finitarily isomorphic to BSs (are fB ) (see [3]). In unpublished work, Smorodinsky showed that every finitary factor of a BS has exponentially decaying return times (see [9] for more details). Countable state mixing Markov processes with exponentially decaying return times are fB (see [7]).…”

mentioning

confidence: 99%

“…Countable state mixing Markov processes with exponentially decaying return times are fB (see [7]). For an up-to-date survey of the literature on finitary maps, see [9].…”

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confidence: 99%

Finitarily Bernoulli factors are dense

Shea

2013

Fund. Math.

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It is not known if every finitary factor of a Bernoulli scheme is finitarily isomorphic to a Bernoulli scheme (is finitarily Bernoulli). In this paper, for any Bernoulli scheme X, we define a metric on the finitary factor maps from X. We show that for any finitary map f : X → Y , there exists a sequence of finitary maps fn : X → Y (n) that converges to f , where each Y (n) is finitarily Bernoulli. Thus, the maps to finitarily Bernoulli factors are dense. Let (X(n)) be a sequence of Bernoulli schemes such that each Y (n) is finitarily isomorphic to X(n). Let X be a Bernoulli scheme with the same entropy as Y . Then we also show that (X(n)) can be chosen so that there exists a sequence of finitary maps to the X(n) that converges to a finitary map to X .

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Finitary Codes, a short survey

Abstract: In this note we recall the importance of the notion of a finitary isomorphism in the classification problem of dynamical systems.

Cited by 10 publications

References 23 publications

A factor of i.i.d. with uniform marginals and infinite clusters spanned by equal labels

A factor of i.i.d. with uniform marginals and infinite clusters spanned by equal labels

Weak mixing suspension flows over shifts of finite type are universal

Finitarily Bernoulli factors are dense

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