Finitarily Bernoulli factors are dense
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 $X'$.