Let T be an ergodic automorphism defined on a standard Borel probability space for which T and T~l 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~l, i.e. S satisfies TS = ST~\ then S 2 = /, 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 = /. 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.

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ABSTRACT. An automorphism T : (X, ~', p) --* (X, ~c,/~) on a standard Borel probability space is said to have uniform rank n, if it has rank n and the columns at each stage can be chosen of equal height. We show that for such an automorphism T, if S is an automorphism conjugating T to its inverse (i.e., ST = T-1S), then S 2m -.~ I for some positive integer rn ~ n (I denotes the identity automorphism). Related results for transformations having local rank one (also called positive ~-rank maps) with no partial rigidity are given. Particular attention is given to the question of whether the Cartesian square T • T can have finite rank, and this question is answered in certain cases. The transformations R(z, y) = (y, Tz) and the symmetric Cartesian square T | are shown to have some analogous properties. INTRODUCTIONLet T : (X, ~',/~) --, (X, ~', p) be an invertible measure preserving tr~n~ formation (automorphism) defined on a standard Borel nonatomic probability space. It is known that if T has simple spectrum and S is an automorphism satisfying ST = T-1S, then S 2 = I (see [4]). We note also that there are automorphisms with simple continuous spectrum which are not conjugate to their inverses; see [13].It is natural to conjecture that if the maximal spectral multiplicity of T is equal to n, then S 2n = I (or S 2m = I for some m ~_ n). However, we shall give examples which show this to be false. Consider instead the analogous question regarding finite rank transformations. Suppose that T has rank 1; then, in fact, T has simple spectrum, so if ST = T-1S, then S 2 = I. Now consider the class of --iform rank n maps. We show that the natural generalization of the rank 1 situation is true, i.e., we prove 1991 Mathematics Subject Classification. 28D05.