2021/07/31 by Aymen Daghar, Daghar, Aymen, Habib Marzougui +1
Mathematics · #Mathematical Dynamics and Fractals #Advanced Differential Equations and Dynamical Systems #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2108.00182
Let X be a regular curve and let f: X→ X be a monotone map. In this paper, nonwandering set of f and the structure of special α-limit sets for f are investigated. We show that AP(f)= \textrmR(f) =Ω(f), where AP(f), \textrmR(f) and Ω(f) are the sets of almost periodic points, recurrent points and nonwandering of f, respectively. This result extends that of Naghmouchi established, whenever f is a homeomorphism on a regular curve [J. Difference Equ. Appl., 23 (2017), 1485--1490] and [Colloquium Math., 162 (2020), 263--277], and that of Abdelli and Abdelli, Abouda and Marzougui, whenever f is a monotone map on a local dendrite [Chaos, Solitons Fractals, 71 (2015), 66--72] and [Topology Appl., 250 (2018), 61--73], respectively. On the other hand, we show that for every X∖ \textrmP(f), the special α-limit set sαf(x) is a minimal set, where P(f) is the set of periodic points of f and that sαf(x) is always closed, for every x∈ X. In addition, we prove that \textrmSA(f) = \textrmR(f), where \textrmSA(f) denotes the union of all special α-limit sets of f; these results extend, for monotone case, recent results on interval and graph maps obtained respectively by Hantáková and Roth in [Preprint: arXiv 2007.10883.] and Foryś-Krawiec, Hantáková and Oprocha in [Preprint: arXiv:2106.05539.]. Further results related to the continuity of the limit maps are also obtained, we prove that the map ωf (resp. αf, resp. sαf) is continuous on X∖ \textrmP(f) (resp. X∞∖ \textrmP(f)). %In particular, it is continuous on X (resp. X∞) whenever \textrmP(f)=∅.