Vol. 59, No. 1, 1975

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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

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,T1(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
Received: 31 July 1974
Revised: 28 April 1975
Published: 1 July 1975
Emilio Gagliardo
Clifford Alfons Kottman