Generic Transformations Have Zero Lower Slow Entropy and Infinite Upper Slow Entropy

06/27/2020

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by   Terry Adams, et al.

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The notion of slow entropy, both upper and lower slow entropy, were 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,