2019/06/09 by Sylvie Ruette, Ruette, Sylvie
Mathematics · Physics and Astronomy · #37B40 #37E05 #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1906.03649
openalex publication_date 2019/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that the topological entropy of a continuous interval map f is positive if and only if the type of f for Sharkovskii's order is 2d p for some odd integer p≥ 3 and some d≥ 0; and in this case the topological entropy of f is greater than or equal to (logλp)/(2d), where λp is the unique positive root of Xp-2Xp-2-1. For every odd p≥ 3, every d≥ 0 and every λ≥λp, we build a piecewise monotone continuous interval map that is of type 2dp for Sharkovskii's order and whose topological entropy is (logλ)/(2d). This shows that, for a given type, every possible finite entropy above the minimum can be reached provided the type allows the map to have positive entropy. Moreover, if d=0 the map we build is topologically mixing.