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A new construction of counterexamples to the bounded orbit conjecture

Jiehua Mai, Enhui Shi, Kesong Yan, Fanping Zeng

TL;DR

The paper addresses the bounded orbit conjecture for plane homeomorphisms by presenting a topological construction that yields a counterexample in the orientation-reversing case. It first builds a normally rising orientation-reversing homeomorphism $f$ on the square $J^2$ with explicit $\omega$- and $\alpha$-limit sets, then uses a semi-conjugacy and a global conjugacy to extend the map to the plane. The main contributions are a clearer, topological method for producing counterexamples and a detailed mechanism by which all nonperiodic orbits remain bounded while no fixed points exist. The results demonstrate a fixed-point-free plane homeomorphism with only 2-periodic axis points and all other orbits bounded, highlighting the sharp contrast with the orientation-preserving case.

Abstract

The bounded orbit conjecture says that every homeomorphism on the plane with each of its orbits being bounded must have a fixed point. Brouwer's translation theorem asserts that the conjecture is true for orientation preserving homeomorphisms, but Boyles' counterexample shows that it is false for the orientation reversing case. In this paper, we give a more comprehensible construction of counterexamples to the conjecture. Roughly speaking, we construct an orientation reversing homeomorphisms $f$ on the square $J^2=[-1, 1]^2$ with $ω(x, f)=\{(-1. 1), (1, 1)\}$ and $α(x, f)=\{(-1. -1), (1, -1)\}$ for each $x\in (-1, 1)^2$. Then by a semi-conjugacy defined by pushing an appropriate part of $\partial J^2$ into $(-1, 1)^2$, $f$ induces a homeomorphism on the plane, which is a counterexample.

A new construction of counterexamples to the bounded orbit conjecture

TL;DR

The paper addresses the bounded orbit conjecture for plane homeomorphisms by presenting a topological construction that yields a counterexample in the orientation-reversing case. It first builds a normally rising orientation-reversing homeomorphism on the square with explicit - and -limit sets, then uses a semi-conjugacy and a global conjugacy to extend the map to the plane. The main contributions are a clearer, topological method for producing counterexamples and a detailed mechanism by which all nonperiodic orbits remain bounded while no fixed points exist. The results demonstrate a fixed-point-free plane homeomorphism with only 2-periodic axis points and all other orbits bounded, highlighting the sharp contrast with the orientation-preserving case.

Abstract

The bounded orbit conjecture says that every homeomorphism on the plane with each of its orbits being bounded must have a fixed point. Brouwer's translation theorem asserts that the conjecture is true for orientation preserving homeomorphisms, but Boyles' counterexample shows that it is false for the orientation reversing case. In this paper, we give a more comprehensible construction of counterexamples to the conjecture. Roughly speaking, we construct an orientation reversing homeomorphisms on the square with and for each . Then by a semi-conjugacy defined by pushing an appropriate part of into , induces a homeomorphism on the plane, which is a counterexample.
Paper Structure (4 sections, 5 theorems, 36 equations)

This paper contains 4 sections, 5 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Normally rising orientation reversing homeomorphisms on $J^2$
  4. Homeomorphisms of the plane which have neither fixed point nor unbounded orbit

Key Result

Theorem 1.1

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: Brouwer translation theorem
  • Example 1.2
  • Conjecture 1.3: Bounded orbit conjecture
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof