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Hollow quasi-Fatou components of quasiregular maps

Published online by Cambridge University Press:  23 September 2016

DANIEL A. NICKS  and
DAVID J. SIXSMITH
DANIEL A. NICKS
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD. e-mails: Dan.Nicks@nottingham.ac.uk; David.Sixsmith@open.ac.uk
DAVID J. SIXSMITH
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD. e-mails: Dan.Nicks@nottingham.ac.uk; David.Sixsmith@open.ac.uk
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Abstract

We define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.

Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 3 , May 2017 , pp. 561 - 574
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Baker, I. N. Multiply connected domains of normality in iteration theory. Math. Z. 81 (1963), 206–214.Google Scholar
[2] Baker, I. N. An entire function which has wandering domains. J. Austral. Math. Soc. Ser. A 22 2 (1976), 173–176.Google Scholar
[3] Baker, I. N. Wandering domains in the iteration of entire functions. Proc. London Math. Soc. (3) 49 3 (1984), 563–576.CrossRefGoogle Scholar
[4] Bergweiler, W. Iteration of meromorphic functions. Bull. Amer. Math. Soc. (N.S.) 29 2 (1993), 151–188.Google Scholar
[5] Bergweiler, W. Fixed points of composite entire and quasiregular maps. Ann. Acad. Sci. Fenn. Math. 31 2 (2006), 523–540.Google Scholar
[6] Bergweiler, W. Fatou-Julia theory for non-uniformly quasiregular maps. Ergodic Theory Dynam. Systems 33 1 (2013), 1–23.Google Scholar
[7] Bergweiler, W., Drasin, D. and Fletcher, A. The fast escaping set for quasiregular mappings. Anal. Math. Phys. 4 1–2 (2014), 83–98.Google Scholar
[8] Bergweiler, W., Fletcher, A., Langley, J. and Meyer, J. The escaping set of a quasiregular mapping. Proc. Amer. Math. Soc. 137 2 (2009), 641–651.Google Scholar
[9] Bergweiler, W., Fletcher, A. and Nicks, D. A. The Julia set and the fast escaping set of a quasiregular mapping. Comput. Methods Funct. Theory 14 2–3 (2014), 209–218.Google Scholar
[10] Bergweiler, W. and Nicks, D. A. Foundations for an iteration theory of entire quasiregular maps. Israel J. Math. 201 1 (2014), 147–184.Google Scholar
[11] Bergweiler, W., Rippon, P. J. and Stallard, G. M. Multiply connected wandering domains of entire functions. Proc. Lond. Math. Soc. (3) 107 6 (2013), 1261–1301.Google Scholar
[12] Drasin, D. and Sastry, S. Periodic quasiregular mappings of finite order. Rev. Mat. Iberoamericana. 19 3 (2003), 755–766.Google Scholar
[13] Eremenko, A. E. On the iteration of entire functions. Dynamical systems and ergodic theory (Warsaw 1986). 23 (1989), 339–345.Google Scholar
[14] García-Máynez, A. and Illanes, A. A survey on unicoherence and related properties. An. Inst. Mat. Univ. Nac. Autónoma México 29 (1989), 17–67.Google Scholar
[15] Järvi, P. On the zeros and growth of quasiregular mappings. J. Anal. Math. 82 (2000), 347–362.CrossRefGoogle Scholar
[16] Kuratowski, K. Topology, Vol. II. Translated from the French by Kirkor, A. (Academic Press, New York-London; PWN Polish Scientific Publishers, Warsaw, 1968).Google Scholar
[17] Nadler, S. B. Jr., Continuum theory. Monographs and Textbooks in Pure and Applied Mathematics, vol. 158 (Marcel Dekker, Inc., New York, 1992).Google Scholar
[18] Napier, T. and Ramachandran, M. An Introduction to Riemann Surfaces. Cornerstones (Birkhäuser/Springer, New York, 2011).Google Scholar
[19] Nicks, D. A. and Sixsmith, D. J. The size and topology of quasi-Fatou components of quasiregular maps. To appear in Proc. Amer. Math. Soc. arXiv:1601.03308.Google Scholar
[20] Rickman, S. Quasiregular mappings. Ergeb. Math. Grenzgeb. (3), vol. 26 (Springer-Verlag, Berlin, 1993).CrossRefGoogle Scholar
[21] Rippon, P. J. and Stallard, G. M. Fast escaping points of entire functions. Proc. London Math. Soc. (3) 105 4 (2012), 787–820.Google Scholar
[22] Sixsmith, D. J. Entire functions for which the escaping set is a spider's web. Math. Proc. Camb. Phil. Soc. 151 3 (2011), 551–571.CrossRefGoogle Scholar
[23] Sixsmith, D. J. On fundamental loops and the fast escaping set. J. Lond. Math. Soc. (2) 88 3 (2013), 716–736.Google Scholar
[24] Töpfer, H. Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sin z und cos z . Math. Ann. 117 (1939), 65–84.Google Scholar
[25] Vuorinen, M. Conformal geometry and quasiregular mappings. Lecture Notes in Math., vol. 1319 (Springer-Verlag, Berlin, 1988).CrossRefGoogle Scholar
[26] Zheng, J.-H. On multiply-connected Fatou components in iteration of meromorphic functions. J. Math. Anal. Appl. 313 1 (2006), 24–37.Google Scholar