Monotone Maps on dendrites and their induced maps

Abstract

A continuum $X$ is a dendrite if it is locally connected and contains no simple closed curve, a self mapping $f$ of $X$ is called monotone if the preimage of any connected subset of $X$ is connected. If $X$ is a dendrite and $f:X\to X$ is a monotone continuous map then we prove that any $\omega$-limit set is approximated by periodic orbits and the family of all $\omega$-limit sets is closed with respect to the Hausdorff metric. Second, we prove that the equality between the closure of the set of periodic points, the set of regularly recurrent points and the union of all $\omega$-limit sets holds for the induced maps $\mathcal{F}_n(f):\mathcal{F}_n(X)\to \mathcal{F}_n(X)$ and $\mathcal{T}_n(f):\mathcal{T}_n(X)\to \mathcal{T}_n(X)$ where $\mathcal{F}_n(X)$ denotes the family of finite subsets of $X$ with at most $n$ points, $\mathcal{T}_n(X)$ denotes the family of subtrees of $X$ with at most $n$ endpoints and $\mathcal{F}_n(f)=2^f_{\mid\mathcal{F}_n(X)}$, $\mathcal{T}_n(f)=2^f_{\mid\mathcal{T}_n(X)}$, in particular there is no Li-Yorke pair for these maps. However, we will show that this rigidity in general is not exhibited by the induced map $\mathcal{C}(f):\mathcal{C}(X)\to \mathcal{C}(X)$ where $\mathcal{C}(X)$ denotes the family of sub-continua of $X$ and $\mathcal{C}(f)=2^f_{\mid\mathcal{C}(X)}$, we will discuss an example of a homeomorphism $g$ on a dendrite $S$ which is dynamically simple whereas its induced map $\mathcal{C}(g)$ is $\omega$-chaotic and has infinite topological entropy.