Homeomorphisms of the plane

Beverly L. Brechner and R. Daniel Mauldin

Vol. 59 (1975), No. 2, 375–381

DOI: 10.2140/pjm.1975.59.375

Abstract

This paper is concerned with homeomorphisms of Euclidean spaces onto themselves, with bounded orbits. The following results are obtained. (1) A homeomorphism of E² onto itself has both bounded orbits and an equicontinuous family of iterates iff it is a conjugate of either a rotation or a reflection; (2) An example of Bing is modified to produce a fixed point free, orientation preserving homeomorphism of E³ onto itself, such that orbits of bounded sets are bounded; and (3) There is no homeomorphism of E² onto itself such that the orbit of every point is dense.

Mathematical Subject Classification 2000

Primary: 54H20

Milestones

Received: 20 September 1974

Published: 1 August 1975