ar X iv : 0 70 5 . 21 48 v 5 [ m at h . D S ] 1 6 Ja n 20 09 PREDICTABILITY , ENTROPY AND INFORMATION OF INFINITE TRANSFORMATIONS

We show that a certain type of quasi finite, conservative, ergodic , measure preserving transformation always has a maximal zero entropy factor, generated by predictable sets. We also construct a conservative, ergodic , measure preserving transformation which is not quasi finite; and consider distribution asymptotics of information showing that e.g. for Boole’s transformation, information is asymptotically mod-normal with normalization ∝ √n. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most 1 2 . §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 it is measurable with respect to its own past in the sense that A ∈ σ({T−nA : n ≥ 1}) (the σ-algebra generated by {T−nA : n ≥ 1}) and let P = PT := {T-predictable sets}. If m(X) < ∞, Pinsker’s theorem ([Pi]) says that • PT is the maximal, zero-entropy factor algebra i.e. P ⊂ B is a factor algebra (T -invariant, sub-σ-algebra), h(T,P) = 0 (see §1) and if C ⊂ B is a factor algebra, with h(T, C) = 0, then C ⊆ P . P is aka the 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, whence σ(PT ) = B. It is not known if there is always a maximal, zero-entropy factor algebra (in case there is some zero-entropy factor algebra). We recall the basic properties of entropy in §1 and define the class of log lower bounded conservative, ergodic , measure preserving transformations in §2. 1991 Mathematics Subject Classification. 37A40, 60F05).