2020/06/27 by Adams, Terry
Computer Science · Mathematics · #37A35 #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2006.15462
openalex publication_date 2020/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.