On entropy and generators of measure-preserving transformations

Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy h(T).We prove that T has a generator of size k where 1. Introduction.In this paper we are concerned with ergodic invertible measurepreserving transformations of a Lebesgue measure space {E, 93, p).By a partition {An : n e 0} of E we shall mean a finite or countably infinite collection of disjoint sets An e 33 of positive measure such that E = U An. nee We call a partition {An : n e 0} a generator of an i.m.p.t.T of {E, 93, p) if 93 is generated by Ü {T'An:ned}.i= -00 For the theory of entropy and generators of i.m.p.t.we refer to [1], [4], [5] and [6].It was proved by V. A. Rohlin that every aperiodic i.m.p.t. with finite entropy has a generator with finite entropy [6, 10.7].We shall prove in §2 that every ergodic i.m.p.t. with finite entropy has a finite generator, thereby solving a problem that was posed by V. A. Rohlin [6, p. 30].Throughout most of this paper we shall be given a finite or countably infinite state space Q.For finite Q we shall prove in §3 an approximation theorem for probability measures on Q.z that are invariant under the shift S, {Sx)i = xi+1, ieZ, x = (x,)," _"" e Q.z.This theorem will enable us to derive in §4 from the work of A. H. Zaslavskil [7] a formula for the minimal number of elements that a generator of an ergodic i.m.p.t.can contain.Denote this number by A(r).If the entropy h{T) of T is infinite then ACT) is also infinite, if h{T) < oo, then A(T) ^ eh(T\ Our result is A(D è eÄ(r)-r-l.