Chain recurrent points of monotone graph maps and CR-Pm property on local dendrites

ABSTRACT

In this paper, the set of chain recurrent points of monotone maps on graphs is investigated. A compact metric space X has the CR-Pm property provided that for each monotone self-mapping f of X, $CR\left(f\right)=\overline{P\left(f\right)}$, where $P\left(f\right)$ and $CR\left(f\right)$ are the sets of periodic points and chain recurrent points of f, respectively. We show that a local dendrite X has the CR-Pm property if and only if X is a dendrite containing neither the infinite star nor the dendrite $G_{\alpha }^{+}$.

KEYWORDS:

• Periodic point
• chain recurrent point
• ϵ-chain
• monotone map
• graph
• dendrite
• local dendrite

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37B45
• 37B20

Acknowledgments

The authors would like to thank the referees and the editor for their valuable comments and corrections.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the research unit ‘Dynamical systems and their applications’ (UR17ES21) of Higher Education and Scientific Research, Tunisia.