Non-wandering Sets for Dendrite Maps

• Taixiang Sun ^[1] ; Qiuli He ^[1] ; Jing Liu ^[1] ; Chunyan Tao ^[1] ; Hongjian Xi ^[1]
  1. [1] Guangxi University

     Guangxi University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 14, Nº 1, 2015, págs. 101-108
• Idioma: inglés
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• Resumen

  • Let (X, d) be a metric space, and f be a continuous map from X to X, and denote by P( f ), ω( f ) and ( f ) the sets of periodic points, ω-limit points and non-wandering points of f , respectively. It is well known that for an interval map f , the following statements hold: (1) ω( f ) = +∞ n=1 f n(( f )). (2) Any isolated point of P( f ) is also an isolated point of ( f ). (3) x ∈ ( f ) if and only if there exist points xk −→ x and positive integers nk −→ ∞ such that f nk (xk ) = x. In [Mai and Sun (Topol Appl, 154, 2306–2311, 2007); Mai et al. (J Math Anal Appl, 383, 553–559, 2011)], we generalized those results to graph maps. It is natural to ask whether those results can be generalized to dendrite maps. The aim of this paper is to show that the answer is negative.