Remarks on minimal sets on dendrites and finite graphs

ABSTRACT

Let X be a dendrite and let $f:X\to X$ be a continuous map that has an infinite minimal set M in X. In the article we obtain conditions on the structure of a dendrite X and the set M, under which f has an arc horseshoe, and, hence, f has a positive topological entropy. We show that this result is not correct both for continuous maps with an empty set of periodic points, and for continuous maps with a non-empty set of periodic points on finite graphs.

KEYWORDS:

• Dendrite
• finite graph
• continuous map
• minimal set
• horseshoe
• topological entropy

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 37B40
• 37B45
• 37E25
• 54F50

Disclosure statement

No potential conflict of interest was reported by the author(s).