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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
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