Ergodic transformations conjugate to their inverses by involutions

Abstract

AbstractLetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.