zbMATH — the first resource for mathematics

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