Centralizer of an Ergodic Measure Preserving Transformation

§ 1. Let T be an ergodic measure preserving transformation of a Lebesgue measure space (Q, 23, P), P(O)=1, that is, Tis a one to one mapping from Q onto itself, bimeasurable (T9J = 93), measure preserving (P(T"1A) = P(A) for A in S3) and ergodic (every measurable function /(«) with f(Tco)=f((o) a.e. is constant a.e.). For measure preserving transformations 17 and 17' we write 17 = 17' if P(UcQ^U'o)) = Q and U^U' otherwise. A measure preserving transformation 17 of (O5 S, P) is called a p-ih root of TQ?^2) if l/* = T. A 1-parameter group {If,} of measure preserving transformations of (Q, S3, P) (i.e. 17,+s= 17,17S for — oo < f, s < -f oo) is called a measurable flow if (co, i)-»Uta) is a measurable mapping from £2 x R onto £2. If there exists a measurable flow {Ut} with 17! = T, Tis said to be embeddable in a measurable flow. The existence of a p-th root or an embedding measurable flow has been one of problems in ergodic theory. It is obvious that the existence of an embedding measurable flow of T implies the existence of a p-th root of Tfor every p^2 and also that a p-ih root of T(if exists) and an embedding measurable flow of T(if exists) are ergodic. A measure preserving transformation 17 of (Q, 23, P) is said to commute with T if UT= TU. We denote by C(T) the group consisting of all measure preserving transformations each of which commutes with T and call it the centralizer of T. Since a p-th root of T(if exists) and a transformation Ut for fixed t in an embedding measurable flow {Ut} of T (if exists) are in C(T), we may expect to solve the existence problem of roots and an embedding measurable