On Commuting Transformations and Roots1

Let (X, (B, m) denote the measure space consisting of the unit interval with Lebesgue measure. We let 3 denote the class of measurable nonsingular invertible transformations mapping X onto X. Given tG3, <r commutes with r if <r(?-(x)) =r(a(x)) for a.e. xGA and a is a pth root of r if ap(x) =r(x) for a.e. xEX. If a is a root of r then a commutes with r, hence it is of interest to find what properties of r are inherited by a if <r does commute with t and in particular if a is a root of r. We consider some basic results along these lines in §1. Concerning roots, it has been of interest to construct ergodic measure preserving transformations t such that (1) r has no roots of any order. More generally we consider (2) Let P denote the primes p ^ 2 and let P = PA-JP2 where Pi and P2 are disjoint, r has no pth root for pEPi but r has a pth root for pEPiErgodic measure preserving transformations with discrete spectrum satisfying (1) were constructed in [l]. In [4] ergodic measure preserving transformations with continuous spectrum satisfying (1) were constructed. We remark that the continuous spectrum case is quite difficult and it was only recently that transformations with continuous spectrum and no square root were constructed in [3]. In §2 of this note we obtain ergodic measure preserving transformations with discrete spectrum satisfying (2). Our method depends on generalizing certain results in [6]. The question of whether every invertible mixing transformation has a square root is still open. In order to illuminate this problem, in §3 we give some transparent examples of invertible mixing transformations with roots of all orders and also mention a class of mixing transformations for which square roots may not exist.