On the Embedding of Homeomorphisms of the Plane in Flows

Citation

Andrea, Stephen Alfred (1964) On the Embedding of Homeomorphisms of the Plane in Flows. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/REXT-WM28. https://resolver.caltech.edu/CaltechETD:etd-09062002-104631

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Homeomorphisms of the plane onto itself are studied, subject to the restriction that they should preserve the sense of orientation and have no fixed points. The author tries to determine which mappings in this general class can be embedded in one-parameter subgroups of the full homeomorphism group of the plane. Such subgroups are called flows. By a theorem of Brouwer, [...] as [...] for any point p in the plane, if T is in the general class being studied. As a consequence, it is shown that if T is embedded in a flow then [...] is a proper subset of the plane for any compact set A. The author suspects that this property might be shared by all homeomorphisms in the general class. It is found that for an arbitrary T there exists a natural partition of the plane into a collection of "fundamental regions", with the property that the restriction of T to any fundamental region must be embedded in a flow within that region whenever T over the whole plane is embedded in a flow. An example is given of a homeomorphism which, for this very reason, cannot be embedded in a flow over the whole plane [...]. The author proves that if T satisfies the above condition that [...] and if T has exactly one fundamental region, that being [...] itself, then T can be embedded in a flow, and indeed is equivalent to a translation. Finally, it is shown by an example that even if the restrictions of T to its fundamental regions are all equivalent to translations, it might still be impossible to create a flow for T over all of [...].

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