Positively regular homeomorphisms of Euclidean spaces |
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| Authors: | Beverly Brechner |
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| Affiliation: | Department of Mathematics, University of Florida, Gainesville, FL32611, USA;Department of Mathematics, North Texas State University, Denton, Texas, USA |
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| Abstract: | A homeomorphism of Rn onto itself is called positively regular (or EC+) iff its family of non-negative iterates is pointwise equicontinuous. For EC+ homeomorphism of Rn such that some point of Rn has bounded positive semi-orbit, the nucleus M is defined, and the following theorems are proved.Theorem 1. If such a homeomorphism h:Rn→Rn has compact nucleus M, then M is a fully invariant compact AR. Further, for n≠4,5,h:Rn/M→Rn/M is conjugate to a contraction on Rn.Theorem 2. In Rn,n≠4,5,M compact iff there existsa disk D such that h(D)?IntD.Theorem 3. In R2, either M is a disk and h|M is a rotation, or h|M is periodic. The relationship between M and the irregular set of ? is also studied. |
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| Keywords: | positively regular homeomorphism invariant continuum Euclidean space action of homeomorphism contraction prime ends |
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