Topological entropy, horseshoe and minimal sets of continuous maps of dendrites
Abstract
Let X be a dendrite with a countable set of end points, and let f : X → X be a continuous map. We show that the topological entropy of f is positive if and only if there exists a natural number n ≥ 1 such that fn has an arc horseshoe. We also demonstrate the fundamental role of infinite minimal sets in the dynamics of these maps. In particular, we prove that the positivity of the topological entropy of such maps implies the existence of an infinite minimal set M in X such that the restriction f|M has positive topological entropy.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Advanced Topology and Set Theory
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- Mathematics and Theoretical Computer Science
- Volume
- 4
- Issue
- 4
- Pages
- 65–86
- Publication date
- 5 February 2026