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CONSTRUCTING HERMAN RINGS BY TWISTING ANNULUS HOMEOMORPHISMS

Part of: Entire and meromorphic functions, and related topics Holomorphic mappings and correspondences Complex dynamical systems

Published online by Cambridge University Press:  01 February 2009

XIUMEI WANG  and
GAOFEI ZHANG
XIUMEI WANG
Affiliation:
Department of Computer Science and Information Technology, JiangSu Teachers University of Technology, Changzhou, 213001, PR China (email: wxmei@jstu.edu.cn)
GAOFEI ZHANG*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, 210093, PR China (email: zhanggf@hotmail.com)
*
✉For correspondence; e-mail: zhanggf@hotmail.com
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Abstract

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Let F(z) be a rational map with degree at least three. Suppose that there exists an annulus such that (1) H separates two critical points of F, and (2) F:H→F(H) is a homeomorphism. Our goal in this paper is to show how to construct a rational map G by twisting F on H such that G has the same degree as F and, moreover, G has a Herman ring with any given Diophantine type rotation number.

Keywords

Herman rings

MSC classification

Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations 32H50: Iteration problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 139 - 143
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The second author is partially supported by NJU-0203005116.

References

[1]Carleson, L. and Gamelin, T. W., Complex Dynamics (Springer, New York, 1993).CrossRefGoogle Scholar
[2]Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54 (Cambridge University Press, New York, 1995).CrossRefGoogle Scholar
[3] S. Marmi and J.-C. Yoccoz (eds.), Dynamical Systems and Small Divisors, Lecture Notes in Mathematics, 1784 (Springer, Berlin, 2002).Google Scholar
[4]Zhang, G., ‘On the dynamics of e 2πiθsin (z)’, Illinois J. Math. 49(4) (2005), 1171–1179.Google Scholar