Fixed points for orientation preserving homeomorphisms of the plane which interchange two points

Emilio Gagliardo and Clifford Alfons Kottman

Vol. 59 (1975), No. 1, 27–32

DOI: 10.2140/pjm.1975.59.27

Abstract

Let T be an orientation preserving homeomorphism defined on a subset of the plane which interchanges two points, P and Q. Let Γ be a simple curve joining P and Q and let Ω be a simply connected set contained in the domain and range of T such that Γ ⊂ Ω,T(Γ) ⊂ Ω,T⁻¹(Γ) ⊂ Ω. Then T has a fixed point in Ω. A corollary concerning fixed points of homeomorphisms on S² follows.

The proof would be trivial if T were necessarily an element of a flow on the plane, however an example given in this paper shows that this need not be the case.

Mathematical Subject Classification 2000

Primary: 54H25

Milestones

Received: 31 July 1974

Revised: 28 April 1975

Published: 1 July 1975