A Family of Cubic Rational Maps and Matings of Cubic Polynomials

“…A discussion of different types of Levy cycles can be found in [23]. The following result, the culmination of work by Rees, Shishikura and Tan, greatly simplifies the search for Thurston obstructions in the bicritical case.…”

Section: Thurston Equivalencementioning

confidence: 89%

“…We finish with a slight generalization of Theorem 1.4, which we shall use in the next section. Lemma 1.5 [23]. Let F be a mating.…”

Section: Thurston Equivalencementioning

confidence: 99%

Constructing rational maps with cluster points using the mating operation

Sharland

2012

Journal of the London Mathematical Society

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In this article, we show that all admissible rational maps with fixed or period 2 cluster cycles can be constructed by the mating of polynomials. We also investigate the polynomials that make up the matings that construct these rational maps. In the one-cluster case, one of the polynomials must be an n-rabbit and in the two-cluster case, one of the maps must be either f , a 'double rabbit', or g, a secondary map that lies in the wake of the double rabbit f . There is also a very simple combinatorial way of classifying the maps which must partner the aforementioned polynomials to create rational maps with cluster cycles. Finally, we also investigate the multiplicities of the shared matings arising from the matings in the paper.This research was funded by a grant from EPSRC. Lemma 1.1. There does not exist a rational map F with a period 2 cluster cycle such that the critical points are in the same cluster.Definition 1.2. Let F be a rational map with a fixed cluster point. Label the endpoints of the star as follows. Let e 0 be the first critical point, and label the remaining arms in anticlockwise order by e 1 , e 2 , . . . , e 2n−1 . Then the second critical point is one of the e j , and we call j the critical displacement of the cluster of F . We denote the critical displacement by δ.We will sometimes use the fact that the critical displacement can equally well be calculated as the combinatorial distance between the critical values, as opposed to the critical points. Also, note that if one choice of marking for the critical points gives δ = k, then the alternative marking will give a critical displacement of δ = −k. Clearly, in light of Lemma 1.1, any attempt to use the above definition to define the critical displacement in the period 2 case is impossible, hence we give the following definition for the period 2 case. Definition 1.3. Let F be a rational map with a period 2 cluster cycle. Choose one of the critical points to be c 1 , and label the cluster containing it to be C 1 . Then (by Lemma 1.1) the other critical point c 2 is in the second cluster C 2 . We define the critical displacement δ as follows. Label the arms in the star of C 1 , starting with the arm with endpoint c 1 , in anticlockwise order 0 , 1 , . . . , 2n−1 . Then F (c 2 ) is the endpoint of one of the k . This integer k is the critical displacement, which we again denote by δ.

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“…(However, (Shishikura and Tan Lei, 2000) have described a cubic example where the geometric mating does not exist, even though K 1 ⊥ ⊥ K 2 is a topological sphere.) 2.7 When Does Mating Exist?…”

Section: Existence and Uniquenessmentioning

confidence: 99%

Pasting Together Julia Sets: A Worked Out Example of Mating

Milnor¹

2004

Experimental Mathematics

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The operation of "mating" two suitable complex polynomial maps f 1 and f 2 constructs a new dynamical system by carefully pasting together the boundaries of their filled Julia sets so as to obtain a copy of the Riemann sphere, together with a rational map f 1 ⊥ ⊥ f 2 from this sphere to itself. This construction is particularly hard to visualize when the filled Julia sets K(f i ) are dendrites, with no interior. This note will work out an explicit example of this type, with effectively computable maps from K(f 1 ) and K(f 2 ) onto the Riemann sphere.

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A Family of Cubic Rational Maps and Matings of Cubic Polynomials

Cited by 45 publications

References 10 publications

Antipode preserving cubic maps: the Fjord theorem

Antipode preserving cubic maps: the Fjord theorem

Constructing rational maps with cluster points using the mating operation

Pasting Together Julia Sets: A Worked Out Example of Mating

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