Abstract
In this article, we study the dynamics of the following family of rational maps with one parameter:
where \(n\ge 3\) and \(\lambda \in \mathbb {C}^*\). This family of rational maps can be viewed as a singular perturbations of the simple polynomial \(P_n(z)=z^n\). We give a characterization of the topological properties of the Julia sets of the family \(f_\lambda \) according to the dynamical behaviors of the orbits of the free critical points.
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Notes
The number \(\sqrt{6}/9\) in this inequality will be used in the construction of an example in next section.
Although U and V are not domains in \(\mathbb {C}\), one can use a coordinate transformation to obtain a polynomial-like mapping since as a rational map, \(f_\lambda \) is holomorphic on whole \(\widehat{\mathbb {C}}\).
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Acknowledgments
The first author is supported by the NSFC (Nos. 11301165, 11371126, 11571099) and the program of CSC (2015/2016). He also wants to acknowledge the Department of Mathematics, Graduate School of the City University of New York for its hospitality during his visit in 2015/2016. The second author is supported by the NSFC (No. 11401298) and the NSF of Jiangsu Province (No. BK20140587). We would like to thank the referee for careful reading and useful suggestions.
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Xiao, Y., Yang, F. Singular Perturbations with Multiple Poles of the Simple Polynomials. Qual. Theory Dyn. Syst. 16, 731–747 (2017). https://doi.org/10.1007/s12346-016-0205-0
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DOI: https://doi.org/10.1007/s12346-016-0205-0