Periodic orbits of period 3 in the disc

TL;DR

This paper investigates the dynamics of orientation-preserving homeomorphisms of the disc with a period-3 orbit, showing they are either conjugate to a rotation or have periodic points of all periods.

Contribution

It characterizes the structure of such homeomorphisms, establishing a dichotomy based on their isotopy class and periodic point distribution.

Findings

Homeomorphisms with a period-3 orbit are either conjugate to a rotation or have all periods.

The paper provides a classification of periodic orbits for these homeomorphisms.

It extends understanding of periodic dynamics in planar topological systems.

Abstract

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.