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 ∝ √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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-1,
author = {Jon Aaronson and Kyewon Koh Park},
title = {Predictability, entropy and information of infinite transformations},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {1-21},
zbl = {1187.37014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-1}
}
Jon Aaronson; Kyewon Koh Park. Predictability, entropy and information of infinite transformations. Fundamenta Mathematicae, Tome 205 (2009) pp. 1-21. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm206-0-1/