Finitarily Bernoulli factors are dense

Volume 223 / 2013

Stephen Shea Fundamenta Mathematicae 223 (2013), 49-54 MSC: Primary 37A35; Secondary 28D20, 37A50, 60G10. DOI: 10.4064/fm223-1-3

Abstract

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 $f: X \to Y$, there exists a sequence of finitary maps $f_n:X \to Y(n)$ 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 $X'$.

Authors

  • Stephen SheaDepartment of Mathematics
    St. Anselm College
    100 St. Anselm Drive #1792
    Manchester, NH 03102, U.S.A.
    e-mail

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