 

Terrence Adams
Genericity and rigidity for slow entropy transformations
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print


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


Abstract
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 KolmogorovSinai entropy.
For any subexponential rate function a_{n}(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 a_{n}(t),
we show there exists a rigid, weak mixing, invertible system such that
the lower slow entropy is infinite with respect to a_{n}(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.


Acknowledgements
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
terry@ieee.org

