On local properties ofk-parameter semiflows of nonsingular point transformations

Abstract

In this paper Krengel's result [3] concerning local ergodic properties of oneparameter measurable semiflows of nonsingular point transformations will be generalized to k-parameter semiflows, k being a positive integer. Let R0=[0, ~) and R+=(0, co). Throughout this paper, (0t: tER~) will be a k-parameter measurable semiflow of nonsingular point transformations on a a-finite measure space (~2, o~, #). Thus each 0t is a measurable point transformation of ~2 into itself such that # (E)=0 implies #(Oi-IE)=O, and the transformation (t, o))~0tco of R0kXg2 into f2 is measurable. (Kt: tERko) will be the k-parameter semigroup of operators on the space of all measurable functions f on (f2, o~, p), defined by Ktf(co)=f(O~co). (We do not distinguish between two functions l a n d g provided that f=g almost everywhere on g2.) As is well-known, (Kt: tERko) induces a k-parameter semigroup (T,: tER~) of contraction operators on L1((2)= ---L1(~2,~-,#) by the relation f(T,f)gd~=ff(Ktg)d~ for fELl(f2) and gEL~.(g2). Without assuming the separability condition on LI(f2) we shall first prove that (Tt: tERko) is strongly continuous on R~+, and then that strong-lira ~ Tt=I, I being the identity operator, if and only if p (E)>0 implies #(Ot~E)>O for some tERk+. In case k = l , this was proved by Krengel [3] (see also Lin [4]). Our proof uses techniques in [6] and [7]. Second, we shall prove an L~(f2) individual local ergodic theorem for the semiflow (0t: tERko) under certain additional hypotheses.