Non-wandering Sets for Dendrite Maps

Abstract

Let $$(X,d)$$ be a metric space, and $$f$$ be a continuous map from $$X$$ to $$X$$ , and denote by $$P(f)$$ , $$\omega (f)$$ and $$\Omega (f)$$ the sets of periodic points, $$\omega $$ -limit points and non-wandering points of $$f$$ , respectively. It is well known that for an interval map $$f$$ , the following statements hold: (1) $$\omega (f)=\bigcap _{n=1}^{+\infty }f^n(\Omega (f))$$ . (2) Any isolated point of $$P( f )$$ is also an isolated point of $$\Omega ( f )$$ . (3) $$x\in \Omega ( f )$$ if and only if there exist points $$ x_k\longrightarrow x$$ and positive integers $$n_k\longrightarrow \infty $$ such that $$ f^{ n_k} (x_k) = x$$ . 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.