Wandering domains in quasiregular dynamics

Nicks, Daniel

doi:10.1090/S0002-9939-2012-11625-9

Wandering domains in quasiregular dynamics
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by Daniel A. Nicks PDF

Proc. Amer. Math. Soc. 141 (2013), 1385-1392 Request permission

Abstract:

We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a sequence of wandering domains contained in a bounded part of the plane. This contrasts with the situation in the analytic case, where wandering domains are impossible for polynomials and, for transcendental entire functions, the existence of wandering domains in a bounded part of the plane has been an open problem for many years.

References

Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
Walter Bergweiler, Iteration of quasiregular mappings, Comput. Methods Funct. Theory 10 (2010), no. 2, 455–481. MR 2791320, DOI 10.1007/BF03321776
Walter Bergweiler, Alastair Fletcher, Jim Langley, and Janis Meyer, The escaping set of a quasiregular mapping, Proc. Amer. Math. Soc. 137 (2009), no. 2, 641–651. MR 2448586, DOI 10.1090/S0002-9939-08-09609-3
D. A. Brannan and W. K. Hayman, Research problems in complex analysis, Bull. London Math. Soc. 21 (1989), no. 1, 1–35. MR 967787, DOI 10.1112/blms/21.1.1
Alastair N. Fletcher and Daniel A. Nicks, Quasiregular dynamics on the $n$-sphere, Ergodic Theory Dynam. Systems 31 (2011), no. 1, 23–31. MR 2755919, DOI 10.1017/S0143385709001072
A. Hinkkanen, Uniformly quasiregular semigroups in two dimensions, Ann. Acad. Sci. Fenn. Math. 21 (1996), no. 1, 205–222. MR 1375517
Pentti Järvi, On the zeros and growth of quasiregular mappings, J. Anal. Math. 82 (2000), 347–362. MR 1799670, DOI 10.1007/BF02791234
Ricardo Pérez Marco, Sur une question de Dulac et Fatou, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 8, 1045–1048 (French, with English and French summaries). MR 1360570
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
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. (2) 122 (1985), no. 3, 401–418. MR 819553, DOI 10.2307/1971308
Dao Chun Sun and Le Yang, Quasirational dynamical systems, Chinese Ann. Math. Ser. A 20 (1999), no. 6, 673–684 (Chinese, with Chinese summary). MR 1741007
D. Sun and L. Yang, Quasirational dynamic system, Chinese Science Bull. 45 (2000), 1277–1279.
Daochun Sun and Lo Yang, Iteration of quasi-rational mapping, Progr. Natur. Sci. (English Ed.) 11 (2001), no. 1, 16–25. MR 1831577
Mini-Workshop: The Escaping Set in Transcendental Dynamics, Oberwolfach Rep. 6 (2009), no. 4, 2927–2963. Abstracts from the mini-workshop held December 6–12, 2009; Organized by Walter Bergweiler and Gwyneth Stallard; Oberwolfach Reports. Vol. 6, no. 4. MR 2724312, DOI 10.4171/OWR/2009/54