Branch sets of uniformly quasiregular maps
HTML articles powered by AMS MathViewer
- by G. J. Martin
- Conform. Geom. Dyn. 1 (1997), 24-27
- DOI: https://doi.org/10.1090/S1088-4173-97-00016-7
- Published electronically: June 19, 1997
- PDF | Request permission
Abstract:
Let $n\geq 2$ and $f: {\Bbb S}^n\to {\Bbb S}^n$ be a quasiregular mapping with branch set $B_f$, the set where $f$ fails to be locally injective. We show that there is a quasiregular mapping $g: {\Bbb S}^n\to {\Bbb S}^n$ with $B_g = B_f$ and such that $g$ can be chosen to be conformal (rational) with respect to some measurable Riemannian structure on ${\Bbb S}^n$. Hence $g$ is uniformly quasiregular. That is, $g$ and all its iterates are quasiregular with a uniform bound on the dilatation.References
- F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393. MR 139735, DOI 10.1090/S0002-9947-1962-0139735-8
- A. Hinkkanen and G. J. Martin, Attractors in quasiregular semigroups, Proc. XVI Nevanlinna colloquium, Eds. I. Laine and O. Martio, de Gruyter, Berlin–New York, 1996, 135–141.
- T. Iwaniec and G. J. Martin, Quasiregular semigroups, Ann. Acad. Sci. Fenn. Math. 21 (1996) 241–254.
- G. J. Martin, The dynamics of uniformly quasiregular mappings, to appear.
- V. Mayer, Uniformly quasiregular mappings of Lattès type, Preprint.
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
- P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982). MR 658932, DOI 10.5186/aasfm.1981.0626
Bibliographic Information
- G. J. Martin
- Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
- MR Author ID: 120465
- Email: martin@math.auckland.ac.nz
- Received by editor(s): January 5, 1997
- Received by editor(s) in revised form: April 16, 1997
- Published electronically: June 19, 1997
- Additional Notes: Research supported in part by a grant from the N.Z. Marsden Fund.
- © Copyright 1997 American Mathematical Society
- Journal: Conform. Geom. Dyn. 1 (1997), 24-27
- MSC (1991): Primary 30C60
- DOI: https://doi.org/10.1090/S1088-4173-97-00016-7
- MathSciNet review: 1454921