Handel’s fixed point theorem revisited.

The author gives a new proof of Handel’s fixed point theorem [Topology 38, No. 2, 235–264 (1999; Zbl 0928.55001)] thereby slightly generalizing the result. Denote by D the plane open unit disk and let P⊂D be a compact convex n-gon with vertices {vi|i∈Zn} and edges ei joining vi and vi+1 and orient the ei in such a way that P is either to the right or left of ei. One says that the orientations of ei and ej coincide if P is to the same side of both ei and ej. For i∈Z let δi=0 if the orientations of ei−1 and ei coincide and δi=1 else. Define the index of P by i(P)=1−12∑i∈Znδi. Denote by αi (ωi) the first (last) point of intersection of the straight line containing ei with ∂D where we orient the line according to ei. A homeomorphism f:D→D is said to realize P if there exists a family (zi)i∈Z such that limk→−∞fk(zi)=αi and limk→∞fk(zi)=ωi for all i∈Zn. The author then proves the following theorem: Assume that f:D→D is an orientation-preserving homeomorphism realizing a compact convex polygon P⊂D where the points αi,ωi are different for all i∈Zn. Assume further that f can be extended as a homeomorphism of D∪⋃i∈Zn{αi,ωi}. If i(P)≠0 then f has a fixed point. If, in addition, i(P)=1 then there exists a simple closed curve in D of index 1.