Finitary coding for the one-dimensional ${T,T^{-1}}$ process with drift

Abstract

We show that there is a finitary isomorphism from a finite state independent and identically distributed (i.i.d.) process to the $T,T^{-1}$ process associated to one-dimensional random walk with positive drift. This contrasts with the situation for simple symmetric random walk in any dimension, where it cannot be a finitary factor of any i.i.d. process, including in $d\ge 5$, where it becomes weak Bernoulli.