Finitarily Bernoulli factors are dense

Stephen Shea¹

¹ Department of Mathematics St. Anselm College 100 St. Anselm Drive #1792 Manchester, NH 03102, U.S.A.

Fundamenta Mathematicae, Tome 223 (2013) no. 1, pp. 49-54

Résumé

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 $X'$.

Détail

DOI : 10.4064/fm223-1-3

Keywords: known every finitary factor bernoulli scheme finitarily isomorphic bernoulli scheme finitarily bernoulli paper bernoulli scheme define metric finitary factor maps finitary map there exists sequence finitary maps converges where each finitarily bernoulli maps finitarily bernoulli factors dense sequence bernoulli schemes each finitarily isomorphic bernoulli scheme entropy chosen there exists sequence finitary maps converges finitary map