On nearly abelian polynomial semigroups

Authors:
Xiantao Wang and Zhigang Huang

Journal:
Proc. Amer. Math. Soc. **133** (2005), 83-89

MSC (2000):
Primary 30D05

DOI:
https://doi.org/10.1090/S0002-9939-04-07669-5

Published electronically:
August 10, 2004

MathSciNet review:
2085156

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a polynomial semigroup containing an element with degree at least 2 with the semigroup operation being functional composition. We prove that is nearly abelian if and only if the smallest completely invariant closed subset of the Riemann sphere is not equal to the Riemann sphere. We also give a positive answer to Conjecture 7.1 in Hinkkanen and Martin's paper on the dynamics of semigroups of rational functions.

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Additional Information

**Xiantao Wang**

Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China

Email:
xtwang@mail.hunnu.edu.cn

**Zhigang Huang**

Affiliation:
Department of Mathematics, Tsinghua University, Beijing 100080, People’s Repulic of China

Address at time of publication:
Department of Mathematics, University of Science and Technology of SuZhou, Suzhou, Jiangsu 215011, People’s Republic of China

Email:
huang.z.g@263.sina.com

DOI:
https://doi.org/10.1090/S0002-9939-04-07669-5

Keywords:
Polynomial semigroup,
nearly abelian,
completely invariant,
Green's function

Received by editor(s):
August 15, 2003

Published electronically:
August 10, 2004

Additional Notes:
This research was partly supported by FNS of China (No. 10271043), Soft Project of Science and Technology of Hunan Province and the Foundation for Scholars back from Foreign Countries.

Communicated by:
Linda Keen

Article copyright:
© Copyright 2004
American Mathematical Society

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