Ergodic transformations conjugate to their inverses by involutions

Let T be an ergodic automorphism defined on a standard Borel probability space for which T and T-1 are isomorphic. We investigate the form of the conjugating automorphism. It is well known that if T is ergodic having a discrete spectrum and S is the conjugation between T and T-1, i.e. S satisfies TS = ST-1, then S-2 = I, the identity automorphism. We show that this result remains true under the weaker assumption that T has a simple spectrum. If T has the weak closure property and is isomorphic to its inverse, it is shown that the conjugation S satisfies S-4 = 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.