A new construction of counterexamples to the bounded orbit conjecture

TL;DR

This paper presents a clearer construction of counterexamples to the bounded orbit conjecture, demonstrating that certain orientation reversing homeomorphisms on the plane lack fixed points despite bounded orbits.

Contribution

It provides a more comprehensible method to construct counterexamples to the bounded orbit conjecture for orientation reversing homeomorphisms.

Findings

Constructed explicit counterexamples on the square

Demonstrated bounded orbits without fixed points

Used semi-conjugacy to extend to the plane

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 $\omega(x, f)=\{(-1. 1), (1, 1)\}$ and $\alpha(x, f)=\{(-1. -1), (1, -1)\}$ 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.