On Limit sets of Monotone maps on Regular curves

2021/06/23 by Daghar, Aymen, Marzougui, Habib
#2000 Mathematics Subject Classification. 37B20 #37B45 #54H20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.12418

Abstract

We investigate the structure of ω-limit (resp. α-limit) sets for a monotone map f on a regular curve X. %Let X be a regular curve and let f: X\longrightarrowX be a monotone map. We show that for any x∈ X (resp. for any negative orbit (x_n)_{n≥ 0} of x), the ω-limit set ω_f(x) (resp. α-limit set α_f((x_n)_{n≥ 0})) is a minimal set. This also hold for α-limit set α_f(x) whenever x is not a periodic point. These results extend those of Naghmouchi \citen %[J. Difference Equ. Appl., 23 (2017), 1485--1490] established whenever f is a homeomorphism on a regular curve and those of Abdelli \citea %[Chaos, Solitons Fractals, 71 (2015), 66--72] , whenever f is a monotone map on a local dendrite. Further results related to the basin of attraction of an infinite minimal set are also obtained.

Related

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• Monotone Maps on dendrites and their induced maps
• Dynamics of homeomorphisms of regular curves
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