Entropy of conservative transformations

Abstract

The definition of entropy of a measure-preserving transformation (called: endomorphism) of a finite measure space into itself makes no sense for σ-finite measure spaces. Using induced transformations (introduced by Kakutani [1]) we give a definition which applies to conservative endomorphisms in σ-finite measure spaces. (This covers all cases of interest, since dissipative endomorphisms have a rather simple structure.) A theorem of Abramov [2] implies that for finite measure spaces the new definition is equivalent to the old one. Entropy as a metric invariant of conservative transformations has many, but not all of the properties discovered by Kolmogorov, Sinai, Rokhlin and others in the finite case. Major differences between the finite and the σ-finite case occur in the investigation of transformations with entropy 0. After giving the basic definitions in section 1 we first prove a theorem on antiperiodic transformations, which will be needed in all other sections, unless the reader is willing to assume that all transformations are ergodic. In section 3 we define entropy and prove a theorem which permits its computation. As an example the entropy of the Markov shift for null-recurrent Markov chains is computed in section 4. We then investigate simple properties such as h(T n )=nh(T) (section 5) and give the ergodic decomposition of h(T) in section 6. Section 7 is devoted to the investigation of transformations with entropy zero, especially an example is given which shows that a known necessary and sufficient condition for a transformation with finite invariant measure to have entropy zero is not sufficient for transformations with a σ-finite invariant measure unless they satisfy an additional assumption. Finally section 8 is devoted to the proof of category statements about the set of conservative transformations and the subset of those among them which have entropy zero.