Vol. 59, No. 2, 1975

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Homeomorphisms of the plane

Beverly L. Brechner and R. Daniel Mauldin

Vol. 59 (1975), No. 2, 375–381

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
Received: 20 September 1974
Published: 1 August 1975
Beverly L. Brechner
R. Daniel Mauldin
Department of Mathematics
University of North Texas
Denton TX 76203-1430
United States