The bounded orbit conjecture for plane homeomorphisms

Let $f:\mathbb R^2\rightarrow\mathbb R^2$ be a homeomorphism, and suppose that every orbit of $f$ is bounded. Bounded orbit conjecture. Then $f$ has a fixed point. This conjecture extends the fixed-point conclusion of Brouwer's translation theorem to homeomorphisms that are not necessarily orientation preserving. It was disproved in 1981 by Boyles, who constructed a fixed-point-free orientation-reversing homeomorphism of the plane with every orbit bounded; the present paper gives a more comprehensible construction of such counterexamples.