New York Journal of Mathematics
Volume 27 (2021), 393-416


Terrence Adams

Genericity and rigidity for slow entropy transformations

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Published: February 25, 2021.
Keywords: entropy, codewords, complexity, rigidity, weak mixing, category.
Subject: 37A05, 37A25, 37A35.

The notion of slow entropy, both upper and lower slow entropy, was defined by Katok and Thouvenot as a more refined measure of complexity for dynamical systems, than the classical Kolmogorov-Sinai entropy. For any subexponential rate function an(t), we prove there exists a generic class of invertible measure preserving systems such that the lower slow entropy is zero and the upper slow entropy is infinite. Also, given any subexponential rate an(t), we show there exists a rigid, weak mixing, invertible system such that the lower slow entropy is infinite with respect to an(t). This gives a general solution to a question on the existence of rigid transformations with positive polynomial upper slow entropy, Finally, we connect slow entropy with the notion of entropy covergence rate presented by Blume. In particular, we show slow entropy is a strictly stronger notion of complexity and give examples which have zero upper slow entropy, but also have an arbitrary sublinear positive entropy convergence rate.


We wish to thank Adam Kanigowski, Daren Wei, Karl Petersen and an anonymous referee for their feedback. Also, this article is dedicated to the memory of Nat Friedman, who was a pioneer of ergodic theory, an accomplished artist and ultimately a giving teacher.

Author information

Terrence Adams:
Department of Mathematics and Statistics
State University of New York at Albany
Albany, NY 12222, USA