The \(\omega \)-limit function on dendrites (Q2215648)

Let $X$ be a compact metric space, and let $f:X\to X$ be a map. For each point $x$ in $X$, $\omega(x,f)=\{y\in X \mid$ there exists an increasing sequence of natural numbers $(n_i)$ such that $\lim_{i \to \infty} f^{n_i}(x)=y\}$, and $\Omega(x,f)=\{y\in X\mid$ there exists an increasing sequence of natural numbers $(n_i)$ and a sequence $(x_i)$ in $X$ such that $\lim_{i \to \infty} x_i=x$ and $\lim_{i \to \infty} f^{n_i}(x_i)=y\}$. The $\omega$-limit function $\omega_f:X\to 2^X$ is defined for each $x$ in $X$ as $\omega_f(x)=\omega(x,f)$. The authors answer affirmatively to a question posed in [\textit{J. Camargo} et al., Chaos Solitons Fractals 126, 1--6 (2019; Zbl 1448.54023)] by proving that for a compact metric space $X$ and a map $f:X\to X$, if $\omega(x,f)=\omega(x,f)$ for all $x$ in $X$ then $\omega_f$ is continuous. A dendrite is a locally connected continuum without simple closed curves. The authors study the continuity of $\omega_f$ on a dendrite. Let Per$(f)$ denote the collection of periodic points of $f$, i.e., such that $f^k(x)=x$, and let Fix$(f)$ denote the set of all fixed points of $f$. A point $x$ in $X$ is called recurrent if $x\in\omega(x,f)$. Rec$(f)$ denotes the set of all recurrent points of $X$. Let $X$ be a dendrite, and $f:X\to X$ be a map. The authors show that the continuity of $\omega_f$ implies that Per$(f)$ is connected and Fix$(f^n)$ is connected for all natural $n$. This answers, in the affirmative, a question posed in [\textit{I. Vidal-Escobar} and \textit{S. Garcia-Ferreira}, Appl. Gen. Topol. 20, No. 2, 325--347 (2019; Zbl 1426.37021)]. Under the same assumptions, Rec$(f)$ is a continuum. A map $f:X\to X$ on a metric space is called equicontinuous if and only if the collection $\{f^n\}$, for all natural $n$, is equicontinuous. The authors provide a characterization of equicontinuous maps on a dendrite. They show that for a map $f:X\to X$ on a dendrite its equicontinuity is equivalent to each of the following conditions: (1) Per$(f)$ is connected and its closure in $X$ equals $X$; (2) $\omega_f$ is continuous and $X=\mathrm{Rec}(f)$; (3) $\omega(x,f)=\Omega(x,f)$, for each $x$ in $X$; (4) $\omega_f$ is continuous and the closure in $X$ of Per$(f)$ equals $X$. Finally, the authors show that for a homeomorphism $f:X\to X$ on a dendrite the following are equivalent: (1) $\omega_f$ is continuous; (2) Per$(f)$ is connected; (3) Rec$(f)$ is connected.