Predictability, entropy and information of infinite transformations

Abstract

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 .1991 Mathematics Subject Classification. 37A40, 60F05).

Poisson suspensions and entropy for infinite transformations

“…Translating this result into the Poisson framework yields the following criterion for distality. We now apply our previous results to a question of Aaronson and Park from [2], regarding the existence of a Krengel-Pinsker factor for quasi-finite transformations. First, we note that the assumptions of Theorem 11.3 hold in particular for quasifinite systems:…”

“…In [2] and [10] co-finite partitions were considered. With our terminology, these are finite, local partitions.…”

“…Equality of the entropies. Recall the definition of a quasi-finite transformation from [11] (also see [2]): Let (X, B, µ, T ) be conservative measure-preserving. A ∈ F + is a quasi-finite set if H µ (ρ A ) < ∞, where ρ A is the first-return-time partition of A:…”

“…Using terminology similar to Aaronson and Park [2], we say that a local partition α is quasi-finite if it has a quasi-finite core A, and ρ A ≺ α. We point out that conservative transformations which are not quasi-finite have been constructed in [2]. Parry has proved that, for quasi-finite transformations (called "pseudo-finite" in [13]), Krengel's definition of entropy coincides with Parry's.…”

“…However, this is the case for LLB systems which are shown to be LLB on any of their factors (see [2] again). As a consequence of Theorem 11.3, we get:…”