Equicontinuity of maps on dendrites

Abstract

Given a dendrite $X$ and a continuous map $f:X\to X$, we show the following are equivalent: (i) ${\omega }_f$ is continuous and $\overline{\mathrm{Per}(f)}={\bigcap }_{n\in \mathbb{N}}{f}^n(X)$; (ii) $\omega (x,f)=\Omega (x,f)$ for each $x\in X$; and (iii) $f$ is equicontinuous. Furthermore, we present some examples illustrating our results.