Predictability, Entropy and Information of Infinite Transformations

We show that a certain type of conservative, ergodic, measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also consider distribution asymptotics of information; e.g. for Boole’s transformation, information is asymptotically mod-normal, a property shared by certain ergodic, probability preserving transformations with zero entropy. §0 Introduction Let (X,B, m, T ) be a conservative, ergodic, measure preserving transformation and let F := {F ∈ B : m(F ) < ∞}. Call a set A ∈ F T -predictable if A ∈ σ({T−nA : n ≥ 1}) and let P = PT := {T-predictable sets}. If m(X) < ∞, then Pinsker’s theorem ([Pi]) says that • PT is the maximal, zero-entropy factor algebra i.e. P ⊂ A is a factor algebra (T -invariant, sub-σ-algebra), h(T,P) = 0 and if C ⊂ A is a factor algebra, with h(S, C) = 0, then C ⊆ P. P is aka Pinsker algebra of (X,B, m, T ). When (X,B, m, T ) is a conservative, ergodic, measure preserving transformation with m(X) = ∞, the above statement fails and indeed σ(P) = B: Krengel has shown ([K2]) that: • ∀ A ∈ F , ǫ > 0, ∃ B ∈ F , m(A∆B) < ǫ, a strong generator in the sense that σ({T−nB : n ≥ 1}) = B. In particular, σ(PT ) = B and it is not known if there is always a maximal, zero-entropy factor algebra (in case there is some zero-entropy factor algebra). We define the class of log lower bounded conservative, ergodic, measure preserving transformations (in §2). These are quasi finite in the sense of [K1] (see §2, also for examples). A log lower bounded conservative, ergodic, measure preserving transformation with some zero-entropy factor algebra has a maximal, zero-entropy factor algebra generated by a specified hereditary subring of predictable sets (see §5). We obtain information convergence (in §4) for quasi finite transformations (extending [KS]). 1991 Mathematics Subject Classification. 37A40, 60F05). c ©2007 Preliminary version.