Extending rational maps

Author:
Gaven J. Martin

Journal:
Conform. Geom. Dyn. **8** (2004), 158-166

MSC (2000):
Primary 30C60, 30C65, 30F40, 30D50

DOI:
https://doi.org/10.1090/S1088-4173-04-00115-8

Published electronically:
November 16, 2004

MathSciNet review:
2122524

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate when a rational endomorphism of the Riemann sphere extends to a mapping of the upper half-space which is rational with respect to some measurable conformal structure. Such an extension has the property that it and all its iterates have uniformly bounded distortion. Such maps are called *uniformly quasiregular*. We show that, in the space of rational mappings of degree , such an extension is possible in the structurally stable component where there is a single (attracting) component of the Fatou set and the Julia set is a Cantor set.

We show that generally outside of this set no such extension is possible. In particular, polynomials can never admit such an extension.

**1.**A. Beardon,*Iteration of Rational Functions*, Springer-Verlag, 1991. MR**1128089 (92j:30026)****2.**L. Carleson and T. Gamelin,*Complex Dynamics*, Springer-Verlag, 1993. MR**1230383 (94h:30033)****3.**A. Hinkannen,*Semigroups of planar quasiregular mappings*, Ann. Acad. Sci. Fenn. Ser. A I Math.**4.**A. Hinkannen 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. MR**1427078 (97m:30028)****5.**A. Hinkannen, G. J. Martin and V. Mayer,*The dynamics of UQR-mappings*, Math. Scand. (to appear).**6.**T. Iwaniec and G. J. Martin,*Quasiregular Semigroups*, Ann. Acad. Sci. Fenn. Math.**21**(1996), no. 2, 241-254. MR**1404085 (97i:30032)****7.**T. Iwaniec and G. J. Martin,*Geometric Function Theory and Nonlinear Analysis*, Oxford University Press, 2001. MR**1859913 (2003c:30001)****8.**G.J. Martin,*Branch sets of uniformly quasiregular maps*, Conform. Geom. Dyn.**1**(1997), 24-27. MR**1454921 (98d:30032)****9.**G.J. Martin and V. Mayer,*Rigidity in holomorphic and quasiregular dynamics*, Trans. Amer. Math. Soc.**355**(2003), no. 11, 4349-4363. MR**1990755 (2004i:37095)****10.**O. Martio and U. Srebro,*Automorphic quasimeromorphic mapping in*, Acta Math.**135**(1975) 221-247. MR**0435388 (55:8348)****11.**R. Mañè, P. Sad and D. Sullivan,*On the dynamics of rational maps*, Ann. Sci. École Norm. Sup.**16**(1983) 193-217. MR**0732343 (85j:58089)****12.**V. Mayer,*Uniformly quasiregular mappings of Lattès type*, Conform. Geom. Dyn.**1**(1997), 104-111. MR**1482944 (98j:30017)****13.**V. Mayer,*Behaviour of quasiregular semigroups near attracting fixed points*, Ann. Acad. Sci. Fenn. Math.**25**(2000), 31-39. MR**1737425 (2000k:30029)****14.**S. Rickman,*Quasiregular Mappings*, Springer-Verlag 1993. MR**1238941 (95g:30026)****15.**S. Rickman,*The analogue for Picard's theorem for quasiregular mappings in dimension 3*, Acta Math.**154**(1985), 195-242. MR**0781587 (86h:30039)**

Retrieve articles in *Conformal Geometry and Dynamics of the American Mathematical Society*
with MSC (2000):
30C60,
30C65,
30F40,
30D50

Retrieve articles in all journals with MSC (2000): 30C60, 30C65, 30F40, 30D50

Additional Information

**Gaven J. Martin**

Affiliation:
Department of Mathematics, University of Auckland and Massey University, Auckland, New Zealand

Email:
martin@math.auckland.ac.nz

DOI:
https://doi.org/10.1090/S1088-4173-04-00115-8

Keywords:
Rational mapping,
quasiconformal,
quasiregular,
extension

Received by editor(s):
April 15, 2002

Received by editor(s) in revised form:
February 1, 2003

Published electronically:
November 16, 2004

Additional Notes:
Research supported in part by grants from the Australian Research Council, the Marsden Fund and Royal Society (NZ) and Institute Mittag-Leffler (Sweden)

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.