Transformations conjugate to their inverses have even essential values
Geoffrey R. Goodson, Mariusz Lemańczyk · Proceedings of the American Mathematical Society · 1996
Let T T be an ergodic automorphism defined on a standard Borel probability space for which T T and T − 1 T^{-1} are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of T T . It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97–124) that if T T is ergodic having simple spectrum and isomorphic to its inverse, and if S S is a conjugation between T T and T − 1 T^{-1} (i.e. S S satisfies T S = S T − 1 TS=ST^{-1} ), then S 2 = I S^{2}=I , the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation S S , the unitary operator induced by T T on L 2 ( X , μ ) L^{2}(X,\mu )