On the cycles of components of disconnected Julia sets
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Mathematische Annalen
On the cycles of components of disconnected Julia sets Guizhen Cui1,2 · Wenjuan Peng3 Received: 11 October 2019 / Revised: 5 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract For any integers d ≥ 3 and n ≥ 1, we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its Julia set except single points and Jordan curves. Mathematics Subject Classification 37F10 · 37F20
1 Introduction Let f : C → C be a rational map of the Riemann sphere with deg f ≥ 2. Denote by J f and F f the Julia set and the Fatou set of f respectively. Refer to [12] for their definitions and basic properties. It is classical that J f is a non-empty compact set. Assume that J f is disconnected. Let K be a Julia component of f , i.e., a connected component of J f . Then each component of f −1 (K ) is also a Julia component. Thus K is either periodic if f p (K ) = K for some integer p ≥ 1, or eventually periodic
Communicated by Ngaiming Mok. Guizhen Cui was supported by the NSFC Grant No. 11688101 and Key Research Program of Frontier Sciences, CAS, Grant No. QYZDJ-SSW-SYS005. Wenjuan Peng was supported by the NSFC Grant No. 11471317.
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Wenjuan Peng [email protected] Guizhen Cui [email protected]
1
College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, People’s Republic of China
2
HCMS and NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
3
HLM and NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
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G. Cui, W. Peng
if f k (K ) is periodic for some integer k ≥ 0, or wandering if f n (K ) is disjoint from f m (K ) for any integers n > m ≥ 0. If K is periodic with period p ≥ 1, then either deg( f p | K ) = 1 and K is a single point, or deg( f p | K ) > 1 and there exists a rational map g with deg g = deg( f p | K ) such that ( f p , K ) is quasi-conformally conjuagate to (g, Jg ) in a neighborhood of K (see [13]). If K is wandering and f is a polynomial, then K is a single point (refer to [2,11,17]). The situation for general rational maps is more complicated. There are examples of rational maps whose wandering components of their Julia sets are Jordan curves [13]. In fact, a wandering Julia component of a geometrically finite rational map is either a single point or a Jordan curve [15]. A periodic Julia component K is called simple if either K is a single point or each component of f −n (K ) is a Jordan curve for any n ≥ 0. It is called complex otherwise. Denote N ( f ) = #{cycles of complex periodic Julia components of f }. Refer to [15] for the next theorem. Theorem A Let f be a geometrically finite rational map with disconnected Julia set. Then N ( f ) < ∞ and each wandering Julia component of f is either a single point or a Jordan curve. A natural problem is whether Theorem A holds for general rational maps. It is easy to see that N ( f ) ≤ deg f − 2 when f is a
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