On homeomorphisms of the plane which have no fixed points

Abstract

Homeomorphisms of the planeR ² onto itself are studied, subject to the restriction that they should preserve the sense of orientation and have no fixed points. The results of this investigation are then applied to the problem of determining which homeomorphisms can be embedded in flows, i.e., in one-parameter subgroups of the full homeomorphism group of the plane.

A “free mapping” ofR ² onto itself is defined to be a homeomorphismT, without fixed points, such thatC∩TC=0 impliesC∩T ⁿ C=0 for alln≠0 wheneverC is a compact connected subset ofR ². Free mappings turn out to be just those homeomorphisms ofR ² onto itself that preserve orientation and have no fixed points.

A fundamental property of free mappingsis the fact that ifT is a free mapping andA is any compact subset ofR ² then$\mathop U\limits_{ - \infty }^{ + \infty } T^n A$ does not meet some unbounded connected subsetB ofR ². The proof of this theorem is lengthy, and will appear elsewhere. The theorem can be weakened by adding the extra assumption thatT be embedded in a flow; the proof of this weakened version is much easier, and is included in the present article.

It is found that for an arbitrary free mappingT there exists a natural partition of the plane into a collection of “fundamental regions”, with the property that ifT is embedded in a flow then each of the fundamental regions is invariant under the flow. An example is given of a free mapping whose fundamental regions are bad enough so that the mapping cannot be embedded in a flow.

It is proved, on the other hand, that if a free mappingT has just one fundamental region thenT is equivalent to a translation, i.e., there is a homeomorphismU ofR ² onto itself such thatUTU ⁻¹ is just the translation(x, y)→(x+1, y). Indeed,T is equivalent to a translation if and only ifT has just one fundamental region.