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

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,