Narrow Results

Let $X$ be a local dendrite and let $f:X\to X$ be a monotone map. Denote by $P(f)$, RR$(f)$, UR$(f)$, $R(f)$ the set of periodic (resp., regularly recurrent, uniformly recurrent, recurrent) points and $\mathrm{\Lambda }(f)$ the union of all $\omega$-limit sets of $f$. We show that if $P(f)$ is nonempty, then (i) $\mathrm{\Lambda }(f)=R(f)=\mathrm{\text{UR}}(f)=\mathrm{\text{RR}}(f)=\overline{P(f)}$. (ii) R$(f)=X$ if and only if every cut point is a periodic point. If $P(f)$ is empty, then (iii) $\mathrm{\Lambda }(f)=R(f)=\mathrm{\text{UR}}(f)$. (iv) R$(f)=X$ if and only if $X$ is a circle and $f$ is topologically conjugate to an irrational rotation of the unit circle ${𝕊}^1$. On the other hand, we prove that $f$ has no Li–Yorke pair. Moreover, we show that the family of all $\omega$-limit sets of $f$ is closed with respect to the Hausdorff metric.