Mating quadratic maps with the modular group

Bullett, Shaun; Penrose, Christopher

doi:10.1007/bf01231770

Cited by 45 publications

(93 citation statements)

References 8 publications

(3 reference statements)

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“…In [3] it was observed that all quadratic maps with connected Julia sets could be realised in the family of correspondences (1). The advantage of the present analysis is that the surgery approach shows that matings of all quadratic polynomials having connected Julia sets with all faithful discrete representations of C 2 * C 3 having connected regular sets are realised in this family.…”

Section: Making the Correspondence Holomorphicmentioning

confidence: 72%

“…This completes the proof of Theorem 1. Moreover since the projection (z, w) → z of S toĈ is a double cover with one double point, over the fixed point P of ρ, and two branch points, over T and the critical value of q, it follows by a calculation of Euler characteristic that S is of genus zero and hence, from the analysis in [3], that following a change in variable the relation p(z, w) = 0 can be put in the form…”

Section: Making the Correspondence Holomorphicmentioning

confidence: 99%

“…The first examples were described in [3] and more general constructions were presented in [5]. These examples and constructions pick out particular classes of polynomial relations p(z, w) = 0 and then in appropriate circumstances identify the resulting correspondences as matings.…”

Section: Introductionmentioning

confidence: 99%

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Mating quadratic maps with Kleinian groups via quasiconformal surgery

Bullett

Harvey

2000

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. Let q :Ĉ →Ĉ be any quadratic polynomial and r : C 2 * C 3 → P SL(2, C) be any faithful discrete representation of the free product of finite cyclic groups C 2 and C 3 (of orders 2 and 3) having connected regular set. We show how the actions of q and r can be combined, using quasiconformal surgery, to construct a 2 : 2 holomorphic correspondence z → w, defined by an algebraic relation p(z, w) = 0.

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Section: Making the Correspondence Holomorphicmentioning

confidence: 72%

Section: Making the Correspondence Holomorphicmentioning

confidence: 99%

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Mating quadratic maps with Kleinian groups via quasiconformal surgery

Bullett

Harvey

2000

Electron. Res. Announc. Amer. Math. Soc.

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“…A one (complex) parameter family of holomorphic correspondences F a containing matings between the modular group and quadratic polynomials was discovered by the first author and Christopher Penrose nearly twenty years ago [4], and further investigated in [3] and [2]. Two naturally defined subsets of interest in the parameter space are:…”

Section: Introductionmentioning

confidence: 99%

A holomorphic correspondence at the boundary of the Klein combination locus

Bullett¹,

Curtis²

2013

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…This boundary should contain both the Julia set of the polynomial and the limit set of the group. The first examples of such matings are due to S. Bullett and C. Penrose [6], who mated representations of C 2 * C 3 in P SL(2, C) (in particular the modular group) with quadratic polynomials.…”

Section: Marianne Freibergermentioning

confidence: 99%

Mating Kleinian groups isomorphic to 𝐶₂∗𝐶₅ with quadratic polynomials

Freiberger

2003

Conform. Geom. Dyn.

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Abstract. Given a quadratic polynomial q :Ĉ →Ĉ and a representation G : C →Ĉ of C 2 * C 5 in P SL(2, C) satisfying certain conditions, we will construct a 4 : 4 holomorphic correspondence on the sphere (given by a polynomial relation p(z, w)) that mates the two actions: The sphere will be partitioned into two completely invariant sets Ω and Λ. The set Λ consists of the disjoint union of two sets, Λ + and Λ − , each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial P . This filled Julia set contains infinitely many copies of the filled Julia set of q. Suitable restrictions of the correspondence are conformally conjugate to P on each of Λ + and Λ − . The set Λ will not be connected, but it can be joined up using a family C of completely invariant curves. The action of the correspondence on the complement of Λ ∪C will then be conformally conjugate to the action of G on a simply connected subset of its regular set. Background and motivationThe theories of iterated rational maps [3], [7] and Kleinian groups [2], [10], both acting on the Riemann sphereĈ = C ∪ ∞ exhibit a number of striking similarities, which arise from the fact that in both casesĈ is partitioned into two completely invariant sets, namely the regular set Ω and the limit Λ in the case of a Kleinian group, and the Fatou set F and the Julia set J in the case of a rational map. Orbits of points under the group or under backward iteration of the rational map accumulate on the limit or Julia set respectively, whereas the action of the group or rational map on the regular or Fatou set is discontinuous and equicontinuous.One can mate two abstractly isomorphic Fuchsian groups G 1 and G 2 which are topologically conjugate on the upper half plane by gluing them together at their limit sets. This is realised by a third quasi-Fuchsian group G whose regular set consists of two simply connected components. On each of these components the action of G is conformally conjugate to one of the G i . Similarly, one can mate two hyperbolic quadratic polynomials q 1 and q 2 (which both lie in the main cardiod of the Mandelbrot set) via a third rational map R by gluing them together at their Julia sets. The Fatou set of R will consist of two completely invariant components,

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Mating quadratic maps with the modular group

Cited by 45 publications

References 8 publications

Mating quadratic maps with Kleinian groups via quasiconformal surgery

Mating quadratic maps with Kleinian groups via quasiconformal surgery

A holomorphic correspondence at the boundary of the Klein combination locus

Mating Kleinian groups isomorphic to 𝐶₂∗𝐶₅ with quadratic polynomials

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