Ergodic Properties that Lift to Compact Group Extensions

Let T and R be measure preserving, T weakly mixing, R ergodic, and let S be conservative ergodic and nonsingular. Let T be a weakly mixing compact abelian group extension of T. UTxS is ergodic then T x S is ergodic. A corollary is a new proof that if T is mildly mixing then so is T. A similar statement holds for other ergodic multiplier properties. Now let T be a weakly mixing type a compact affine G extension of T where or is an automorphism of G. If T and R are disjoint and a or R has entropy zero, then T and R are disjoint. T is uniquely ergodic if and only if T is uniquely ergodic and a has entropy zero. If T is mildly mixing and T is weakly mixing then T is mildly mixing. We also provide a new proof that if T is weakly mixing then T has the /(-property if T does. I. Statement of results. This paper is concerned with a general class of theorems called lifting theorems. A lifting theorem is a theorem of the following sort: Let T be a weakly mixing measure preserving transformation which satisfies an additional property P. If T is a weakly mixing extension ofT, then T also satisfies P. Throughout this paper an extension will be a compact affine G extension of type a, where G is a compact metrizable group and a is a continuous automorphism of G (the definition is given in §11). The case where a is trivial is called a compact group extension (abelian extension if G is also abelian). Lifting theorems are useful for inductively constructing examples of transformations with various properties by lifting these properties to extensions. In the past, lifting theorems have been proven for various kinds of extensions for the following properties P: mixing, Thouvenot (unpublished); r-fold mixing, Rudolph [RI]; the /Í-property, Parry [P] (cf. also [T]); and the Bernoulli property, Rudolph [R2]. In [Wl], Walters proved a general lifting theorem for abelian extensions concerning the absence of certain invariant sub-tr-algebras. An easy corollary of this is a lifting theorem for mild mixing in abelian extensions (cf. Corollary 2.1). Berg [B] proved a lifting theorem for group extensions involving quasi-disjointness. In this paper we prove two new general lifting theorems, and discuss several corollaries. The proofs appear in §11. We first consider measure spaces (X, p) and (Y, v), where X and Y are standard Borel spaces and p and v are nonatomic Borel probability measures. (We will Received by the editors January 24, 1986 and, in revised form, September 3, 1986. Some of the results in this paper were presented at the Conference on Smooth Ergodic Theory at the University of Warwick, Coventry, England, July 13, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 28D05, 28D20. Partially supported by NSF Grant DMS 85-04025. ©1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page