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A note on the bounded orbit conjecture

Enhui Shi, Ziqi Yu

TL;DR

The paper proves that an orientation-reversing fixed-point-free homeomorphism $f$ of $\mathbb{R}^2$ with all orbits bounded must have infinitely many periodic orbits. The authors reduce to the orientation-preserving map $f^2$, compactify to the sphere $S^2$, and apply fixed-point index theory together with Lefschetz numbers; a parity argument shows that assuming only finitely many periodic orbits leads to an odd fixed-point-index sum, contradicting the Lefschetz total of $2$. This establishes a strong form of the bounded orbit phenomenon for planar, orientation-reversing dynamics and highlights the power of index-theoretic methods in ruling out finite-periodicity scenarios. The result advances understanding of how absence of unbounded orbits constrains periodic structure in planar dynamical systems.

Abstract

If $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is an orientation reversing fixed point free homeomorphism on the plane $\mathbb{R}^2$ with no unbounded orbit, then $f$ has infinitely many periodic orbits.

A note on the bounded orbit conjecture

TL;DR

The paper proves that an orientation-reversing fixed-point-free homeomorphism of with all orbits bounded must have infinitely many periodic orbits. The authors reduce to the orientation-preserving map , compactify to the sphere , and apply fixed-point index theory together with Lefschetz numbers; a parity argument shows that assuming only finitely many periodic orbits leads to an odd fixed-point-index sum, contradicting the Lefschetz total of . This establishes a strong form of the bounded orbit phenomenon for planar, orientation-reversing dynamics and highlights the power of index-theoretic methods in ruling out finite-periodicity scenarios. The result advances understanding of how absence of unbounded orbits constrains periodic structure in planar dynamical systems.

Abstract

If is an orientation reversing fixed point free homeomorphism on the plane with no unbounded orbit, then has infinitely many periodic orbits.

Paper Structure

This paper contains 5 sections, 6 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notions in dynamical system
  4. Fixed point index and Lefschetz number
  5. Proof of the main theorem \ref{['main']}

Key Result

Theorem 1.1

(Brouwer translation theorem) If $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ is an orientation preserving homeomorphism and has no fixed point, then $f$ is a translation, that is each orbit of $f$ is unbounded.

Theorems & Definitions (9)

  • Theorem 1.1
  • Example 1.2
  • Conjecture 1.3: Bounded orbit conjecture
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof