${\bi T},{\bi T}^{\bf -1}$ is not standard
Deborah Heicklen, Christopher Hoffman · Ergodic Theory and Dynamical Systems · 1998
A sequence of random variables, $Y_0,Y_1,Y_2,\ldots\,$, is called standard if there exists a one-sided isomorphism between it and a sequence of independent random variables. In this paper it is demonstrated that the sequence arising from the past of the $T,T^{-1}$ map is not standard.