The Centre and the Depth of the Centre for Continuous Maps on Dendrites with Finite Branch Points

Abstract

Let $\mathbf{D}$ be a dendrite with finite branch points and $f:\mathbf{D}\rightarrow \mathbf{D}$ be continuous. Denote by R(f) and $\Omega (f)$ the set of recurrent points and the set of non-wandering points of f respectively. Let $\Omega _0 (f)=\mathbf{D}$ and $\Omega _n (f)=\Omega (f|_{\Omega _{n-1} (f)})$ for all $n\in \mathbf{N}$. The minimal $m\in \mathbf{N}\cup \{\infty \}$ such that $\Omega _{m} (f)=\Omega _{m+1} (f)$ is called the depth of f. In this note, we show that $\Omega _3(f)=\overline{R(f)}$ and the depth of f is at most 3. Furthermore, we show that there exist a dendrite $\mathbf{D}$ with finite branch points and $f\in C^0(\mathbf{D})$ such that $\Omega _3(f)=\overline{R(f)}\ne \Omega _2(f)$.