Finitarily Bernoulli factors are dense (Q2856663)

The author essentially proves that the maps defined from a Bernoulli scheme to a finitarily Bernoulli scheme are dense in the set of maps defined on a Bernoulli scheme (BS). For this, he considers a symbolic space \(X=\{x_0x_1\dots\}\), with finite or countable alphabet, endowed with the Bernoulli measure. An isomorphism \(\psi\) between Bernoulli schemes \(X,Y\) is defined to be finitarily if for any sequence \(x\in X\) there are integers \(m\leq n\) such that the zero coordinates of \(\psi(x)\) and \(\psi(x')\) agree for almost \(x'\), with the block \(x'_m,\dots,x'_n\) equal to the block \(x_m,\dots,x_n\). A BS is an \(r\)-process when any element \(a\) in the alphabet of \(X\) with \(P(x_n=a)\) has \(0\)-Bernoulli distribution. A metric \(d\) is introduced to prove the main results. One of them (Theorem 3.1) states that if \(f:X\to Y\) is finitarily factor map, with \(X\) a finite-state BS, then there exists a sequence of finitarily factor maps \(f_n\) with values in a finitarily BS such that \(\lim\limits_{n\to\infty}d(f_n,f)=0\). Therefore, maps with values in a finitarily BS are dense in the space of finitarily factor maps. In the other result (Theorem 3.2) states the existence of processes finitarily isomorphic arbitrary close to BS with equal entropy.