Genericity and Rigidity for Slow Entropy Transformations

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 Kolmogorov-Sinai 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.