Predictability, entropy and information of infinite transformations

TL;DR

This paper investigates the entropy and predictability of infinite measure-preserving transformations, identifying maximal zero entropy factors, constructing non-quasi finite examples, and analyzing information distribution asymptotics.

Contribution

It introduces the concept of maximal zero entropy factors in certain transformations and constructs examples of non-quasi finite transformations with asymptotic information properties.

Findings

Existence of maximal zero entropy factors generated by predictable sets.

Construction of a conservative, ergodic, measure-preserving transformation that is not quasi finite.

Asymptotic distribution of information for Boole's transformation is mod-normal with square root normalization.

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 square root normalization. Lastly we see that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most 1/2.