2020/05/31 by David Burguet · 4 citations
Computer Science · Mathematics · #Algorithm #Applied mathematics #Artificial intelligence #Cellular Automata and Applications #Cellular automaton #Computability, Logic, AI Algorithms #Computer science #Discrete mathematics #Entropy (arrow of time) #Exponent #Lyapunov exponent #Mathematical Dynamics and Fractals #Mathematics #Physics #Pure mathematics #Statistical physics #Thermodynamics #math.CO #math.DS
paper · pdf · doi:10.1088/1361-6544/abfeab
published in Nonlinearity 34(7), 4897-4922 (IOP Publishing) · Some inaccuracies in Section 4 and 5 corrected
arxiv created 2020/07/07 · openalex publication_date 2021/06/25 · arxiv updated 2021/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract For a d -dimensional cellular automaton with d ⩾ 1 we introduce a rescaled entropy which estimates the growth rate of the entropy at small scales by generalizing previous approaches [1, 7]. We also define a notion of Lyapunov exponent and proves a Ruelle inequality as already established for d = 1 in [16, 18]. Finally we generalize the entropy formula for one-dimensional permutative cellular automata [19] to the rescaled entropy in higher dimensions. This last result extends recent works [17] of Shinoda and Tsukamoto dealing with the metric mean dimensions of two-dimensional symbolic dynamics.