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A criterion to generate carpet Julia sets

Author: Fei Yang
Journal: Proc. Amer. Math. Soc. 146 (2018), 2129-2141
MSC (2010): Primary 37F45; Secondary 37F10, 37F30
Published electronically: December 18, 2017
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Abstract: It has been known that the Sierpiński carpets can appear as the Julia sets in the families of some rational maps. In this article we present a criterion that guarantees the existence of the carpet Julia sets in some rational maps having exactly one fixed (super-) attracting or parabolic basin. We show that this criterion can be applied to some well-known rational maps such as McMullen maps and Morosawa-Pilgrim family. Moreover, we give also some special examples whose Julia sets are Sierpiński carpets.

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Additional Information

Fei Yang
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

Keywords: Julia set, Sierpi\'nski carpet, Fatou components
Received by editor(s): January 11, 2017
Received by editor(s) in revised form: May 4, 2017, and July 25, 2017
Published electronically: December 18, 2017
Additional Notes: This work is supported by the National Natural Science Foundation of China (grant Nos.11401298 and 11671092) and the Fundamental Research Funds for the Central Universities (grant No. 0203-14380013).
Communicated by: Nimish Shah
Article copyright: © Copyright 2017 American Mathematical Society

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