Predictability, entropy and information of infinite transformations. (English) Zbl 1187.37014

Aaronson, Jon; Park, Kyewon Koh

Fundam. Math. 206, 1-21 (2009).

Summary: We show that a certain type of quasifinite, 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 quasifinite; and consider distribution asymptotics of information showing that e.g. for Boole’s transformation, information is asymptotically mod-normal with normalization \(\propto\sqrt n\). Lastly, we show that certain ergodic, probability preserving transformations with zero entropy have analogous properties and consequently entropy dimension of at most \(1/2\).

MSC:

37A40	Nonsingular (and infinite-measure preserving) transformations
60F05	Central limit and other weak theorems

Keywords:

measure preserving transformation; conservative; ergodic; entropy; quasifinite; predictable set; entropy dimension