Abstract
We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled tree associated to any such $f$. It is proven that any bi-angled tree is realizable by such an $f$, and moreover, $f$ is essentially uniquely determined by its associated bi-angled tree. This combinatorial classification is used to show that such $f$ are in natur
Results & Lemmas (50)
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Proposition 2.3.
Proposition 2.3. An unbounded simply connected domain Ω⊊bC with ∞/∈∂Ωand int Ω= Ωis a quadrature domain if and only if the Riemann…
Proposition 2.3. An unbounded simply connected domain Ω⊊bC with ∞/∈∂Ωand int Ω= Ωis a quadrature domain if and only if the Riemann uniformization f : bC \ D →Ωextends to a rational map on bC. In this case, the Schwarz reflection map σ of Ωis given by f ◦η ◦ (f|bC\D)−1, and if deg f ≥2, we have σ(Ω) = bC. Moreover, if the degree of the rational map f is d+1, then σ : σ−1(Ω) →Ωis a branched covering of degree d, and σ : σ−1(int Ωc) →int Ωc is a branched covering of degree d + 1.
Proposition 2.7.
Proposition 2.7. Let f ∈Σ∗ d. Then, f has d + 1 distinct simple critical points on T.
Proposition 2.7. Let f ∈Σ∗ d. Then, f has d + 1 distinct simple critical points on T.
Proposition 2.9.
Proposition 2.9. Let f ∈Σ∗ d. Then, the d + 1 cusps of f(T) are of the type (3, 2).
Proposition 2.9. Let f ∈Σ∗ d. Then, the d + 1 cusps of f(T) are of the type (3, 2).
Proposition 2.10.
Proposition 2.10. Let f ∈Σ∗ d, and ζ0 be a double point on f(T). Then, the two distinct non-singular local branches of f(T) have non-zero…
Proposition 2.10. Let f ∈Σ∗ d, and ζ0 be a double point on f(T). Then, the two distinct non-singular local branches of f(T) have non-zero curvature and distinct osculating circles at ζ0 (in particular, they have a contact of order 1). Moreover, in suitable local conformal coordinates near ζ0, the two distinct non-singular local branches of f(T) are of the form (u±(t), v±(t)) = (t + o(t2), c±t2 + o(t3)), for some c± ∈C∗with c+ ̸= c−.
Proposition 2.13.
Proposition 2.13. There is a bijective correspondence between equivalence classes of Suf- fridge polynomials of degree d + 1 and affine…
Proposition 2.13. There is a bijective correspondence between equivalence classes of Suf- fridge polynomials of degree d + 1 and affine equivalence classes of extremal unbounded quadrature domains of order d.
Theorem 2.21.
Theorem 2.21. [KS03] Let q(z), deg(q) = d > 1, be an analytic polynomial. Then # n z ∈C: q(z) = z o ≤3d −2
Theorem 2.21. [KS03] Let q(z), deg(q) = d > 1, be an analytic polynomial. Then # n z ∈C : q(z) = z o ≤3d −2
Proposition 2.23.
Proposition 2.23. Two CS polynomials q1 and q2 are equivalent if and only if the corre- sponding CS anti-polynomials p1 and p2…
Proposition 2.23. Two CS polynomials q1 and q2 are equivalent if and only if the corre- sponding CS anti-polynomials p1 and p2 (respectively) are affinely conjugate. Notation: For an affine map A, we define ˜A(z) := A(z). Clearly, ˜A is also an affine map.
Theorem 2.26.
Theorem 2.26. If a sequence of quadrilaterals Qn converges to the quadrilateral Q, then lim n→∞M(Qn) = M(Q).
Theorem 2.26. If a sequence of quadrilaterals Qn converges to the quadrilateral Q, then lim n→∞M(Qn) = M(Q).
Theorem 2.27.
Theorem 2.27. Let Q be a quadrilateral. Then: 1 π (log(1 + 2sb/sa))2 1 + 2 log(1 + 2sb/sa) ≤M(Q) ≤π 1 + 2 log(1 + 2sa/sb) (log(1 +…
Theorem 2.27. Let Q be a quadrilateral. Then: 1 π (log(1 + 2sb/sa))2 1 + 2 log(1 + 2sb/sa) ≤M(Q) ≤π 1 + 2 log(1 + 2sa/sb) (log(1 + 2sa/sb))2 , where sa, sb denote the Euclidean path distance in Q between the a-sides, b-sides, respec- tively, of the topological quadrilateral Q. 3. Dynamics of Schwarz Reflections Arising from Σ∗ d The function f of Theorem 4.1 is obtained via a sequence of quasiconformal deforma- tions applied to a canonical map in Σ∗ d. Subsection 3.1 introduces this canonical ma
Proposition 3.1.
Proposition 3.1. The map f0(z):= z −1/dzd is injective on bC. In particular, f0 ∈Σ∗ d.
Proposition 3.1. The map f0(z) := z −1/dzd is injective on bC\D. In particular, f0 ∈Σ∗ d.
Proposition 3.2.
Proposition 3.2. 1) The tiling set T ∞(σ) is open. Its closure T ∞(σ) is a compact, con- nected set. 2) The basin of infinity B∞(σ) is a…
Proposition 3.2. 1) The tiling set T ∞(σ) is open. Its closure T ∞(σ) is a compact, con- nected set. 2) The basin of infinity B∞(σ) is a simply connected, completely invariant domain.
Proposition 3.3.
Proposition 3.3. Let g ∈Σ∗ d, Ω:= g(bC D), and σ the Schwarz reflection map of Ω. Further, let µ be a σ-invariant Beltrami coefficient on bC,…
Proposition 3.3. Let g ∈Σ∗ d, Ω:= g(bC \ D), and σ the Schwarz reflection map of Ω. Further, let µ be a σ-invariant Beltrami coefficient on bC, and Φ : (bC, ∞) →(bC, ∞) be any quasiconformal map satisfying Φz/Φz = µ a.e.. Then Φ(Ω) is a simply connected unbounded quadrature domain. There exists a normalization for Φ (specified in the proof) with which we have Φ(Ω) = h(bC \ D) for some h ∈Σ∗ d, and Φ ◦σ ◦Φ−1 is the Schwarz reflection map of Φ(Ω).
Proposition 3.4.
Proposition 3.4. bC = T ∞(σ) ⊔B∞(σ).
Proposition 3.4. bC = T ∞(σ) ⊔B∞(σ).
Corollary 3.5.
Corollary 3.5. T ∞(σ) is a full continuum.
Corollary 3.5. T ∞(σ) is a full continuum.
Proposition 3.6.
Proposition 3.6. The limit set L(σ) has zero area.
Proposition 3.6. The limit set L(σ) has zero area.
Theorem 4.1.
Theorem 4.1. Let T be a bi-angled tree. There exists an extremal unbounded quadrature domain Ωwhose associated bi-angled tree is isomorphic…
Theorem 4.1. Let T be a bi-angled tree. There exists an extremal unbounded quadrature domain Ωwhose associated bi-angled tree is isomorphic to T .
Proposition 4.2.
Proposition 4.2. There exists a conformal map: Ψ: Q →[−1, 1] × [−1, 1] mapping the critical values of f0 to the four vertices of [−1, 1] ×…
Proposition 4.2. There exists a conformal map: Ψ : Q →[−1, 1] × [−1, 1] mapping the critical values of f0 to the four vertices of [−1, 1] × [−1, 1].
Proposition 4.3.
Proposition 4.3. There exist a family of quasiconformal maps (φt)t∈[1,∞): bC →bC, and a family of rational maps (ft)t∈[1,∞) such that…
Proposition 4.3. There exist a family of quasiconformal maps (φt)t∈[1,∞) : bC →bC, and a family of rational maps (ft)t∈[1,∞) such that (φt)z/(φt)z = µt a.e., φt(∞) = ∞, ft ∈Σ∗ 3, and ft(bC \ D) = φt(Ω0) for all t ∈[1, ∞).
Proposition 4.4.
Proposition 4.4. There exist a map f∞∈Σ∗ 3 and a sequence of positive real numbers (tn) ↗+∞such that (ftn) n→∞ −−−−→f∞uniformly on bC with…
Proposition 4.4. There exist a map f∞∈Σ∗ 3 and a sequence of positive real numbers (tn) ↗+∞such that (ftn) n→∞ −−−−→f∞uniformly on bC with respect to the spherical metric.
Theorem 1.10
Theorem 1.10], the restrictions of ft on bC D form a normal family, and a normal limit also has the same properties. Note that by…
Theorem 1.10], the restrictions of ft on bC \ D form a normal family, and a normal limit also has the same properties. Note that by Propositions 4.4 and 2.3, the image Ω∞:= f∞(bC \ D) is an unbounded simply connected quadrature domain.
Proposition 4.5.
Proposition 4.5. The boundary of the quadrature domain Ω∞has 4 cusps and 1 double point.
Proposition 4.5. The boundary of the quadrature domain Ω∞has 4 cusps and 1 double point.
Proposition 2.7
Proposition 2.7 also implies that each ft has 4 distinct critical points on T. Let ξn 1 denote the critical point of ftn in the…
Proposition 2.7 also implies that each ft has 4 distinct critical points on T. Let ξn 1 denote the critical point of ftn in the upper-right-half plane, and enumerate the remaining critical points in counter-clockwise order as ξn 2 , ξn 3 , ξn 4 . Denote the corresponding critical values by ζn 1 , · · · , ζn 4 . We may assume, by taking a subsequence in n if necessary, that ξn j →ξ∞ j ∈∂D for 1 ≤j ≤4. Since ftn →f∞uniformly on bC, it follows that f ′ ∞(ξ∞
Lemma 4.7.
Lemma 4.7. Two bi-angled trees T1 and T2 are isomorphic if and only if the corresponding augmented trees bT1 and bT2 are isomorphic. With…
Lemma 4.7. Two bi-angled trees T1 and T2 are isomorphic if and only if the corresponding augmented trees bT1 and bT2 are isomorphic. With the introductory remarks of this section in mind, we will now record, given a tree T , which edges (f0(Ij))d+1 j=1 we will need to “pinch” in order to obtain an extremal quadrature domain whose bi-angled tree is isomorphic to T . See Figure 11 for an illustration of Definition 4.8. Definition 4.8. Given a tree T , we define a subset (16) S = ST ⊂{1, · · · , d + 1
Proposition 4.9.
Proposition 4.9. There exist a family of quasiconformal maps (φt)t∈[1,∞): bC →bC, and a family of rational maps (ft)t∈[1,∞) such that…
Proposition 4.9. There exist a family of quasiconformal maps (φt)t∈[1,∞) : bC →bC, and a family of rational maps (ft)t∈[1,∞) such that (φt)z/(φt)z = µt a.e., φt(∞) = ∞, ft ∈Σ∗ d, and ft(bC \ D) = φt(Ω0) for all t ∈[1, ∞).
Proposition 4.10.
Proposition 4.10. There exist a map f 1 ∞∈Σ∗ d and a sequence of positive real numbers (tn) ↗∞such that (ftn) n→∞ −−−−→f 1 ∞uniformly on bC…
Proposition 4.10. There exist a map f 1 ∞∈Σ∗ d and a sequence of positive real numbers (tn) ↗∞such that (ftn) n→∞ −−−−→f 1 ∞uniformly on bC with respect to the spherical metric.
Proposition 4.11.
Proposition 4.11. The boundary of the quadrature domain Ω1 ∞has d + 1 cusps and 1 double point.
Proposition 4.11. The boundary of the quadrature domain Ω1 ∞has d + 1 cusps and 1 double point.
Proposition 4.12.
Proposition 4.12. With notation as above, there exist a family of quasiconformal maps (φt)t∈[1,∞): bC →bC, and a family of rational maps…
Proposition 4.12. With notation as above, there exist a family of quasiconformal maps (φt)t∈[1,∞) : bC →bC, and a family of rational maps (ft)t∈[1,∞) such that (φt)z/(φt)z = µt a.e., φt(∞) = ∞, f1 = f 1 ∞, ft ∈Σ∗ d, and ft(bC \ D) = φt(Ω1 ∞) for all t ∈[1, ∞). Moreover, there exist a map f 2 ∞∈Σ∗ d and a sequence of positive real numbers (tn) ↗∞such that (ftn) n→∞ −−−−→f 2 ∞uniformly on bC with respect to the spherical metric. We define Ω2
Proposition 4.13.
Proposition 4.13. The boundary of the quadrature domain Ω2 ∞has d + 1 cusps and 2 double points.
Proposition 4.13. The boundary of the quadrature domain Ω2 ∞has d + 1 cusps and 2 double points.
Theorem 4.14.
Theorem 4.14. (Pinching Theorem) Let f ∈Σ∗ d. Choose a counter-clockwise labelling ξ1, · · ·, ξd+1 of the (necessarily distinct) critical…
Theorem 4.14. (Pinching Theorem) Let f ∈Σ∗ d. Choose a counter-clockwise labelling ξ1, · · · , ξd+1 of the (necessarily distinct) critical points of f on T. Suppose that f(> ξjξj+1), f(> ξkξk+1) do not intersect, but both intersect the boundary of a common bounded component of bC\f(T). Then there exist f∞∈Σ∗ d and a counter-clockwise labelling ξ∞ 1 , · · · , ξ∞ d+1 of the (neces- sarily distinct) critical points of f∞on T such that the arcs f∞(> ξ∞ j ξ∞
Theorem 5.1.
Theorem 5.1. Let eΩand Ωbe two unbounded extremal quadrature domains such that T (eΩ) and T (Ω) are isomorphic as bi-angled trees. Then,…
Theorem 5.1. Let eΩand Ωbe two unbounded extremal quadrature domains such that T (eΩ) and T (Ω) are isomorphic as bi-angled trees. Then, there exists an affine map A with A(eΩ) = Ω.
Lemma 5.2.
Lemma 5.2. Let eΩand Ωbe two unbounded extremal quadrature domains such that T (eΩ) and T (Ω) are isomorphic as bi-angled trees. Then,…
Lemma 5.2. Let eΩand Ωbe two unbounded extremal quadrature domains such that T (eΩ) and T (Ω) are isomorphic as bi-angled trees. Then, there exists a cusp-preserving homeo- morphism Ψ : eT →T that maps int eTi conformally onto int Ti for 1 ≤i ≤d −1.
Lemma 5.3.
Lemma 5.3. Ψ is asymptotically linear near the singular points on ∂eT. More precisely, if eζ0 is a singular point on ∂eT with ζ0 = Ψ(eζ0),…
Lemma 5.3. Ψ is asymptotically linear near the singular points on ∂eT. More precisely, if eζ0 is a singular point on ∂eT with ζ0 = Ψ(eζ0), then (27) Ψ(eζ) = ζ0 + c1(eζ −eζ0) + o((eζ −eζ0)) as eζ →eζ0 where c1 ∈C∗.
Proposition 2.9
Proposition 2.9, eζ0 (respectively, ζ0) is a (3, 2) cusp. Thus, we can send eζ0 (respectively, ζ0) to ∞by a Möbius map such that the image…
Proposition 2.9, eζ0 (respectively, ζ0) is a (3, 2) cusp. Thus, we can send eζ0 (respectively, ζ0) to ∞by a Möbius map such that the image of eT near eζ0 (respectively, the image of T near ζ0) is a curvilinear strip eS (respectively, S) bounded by the non-singular real-analytic curves y = v1(x) = k1 + k2 √x + k3/√x + o(1/√x) and y = v2(x) = k1 −k2 √x −k3/√x + o(1/√x), where x ≥x0 for some large x0 > 0 (with possibly different constants for S). Note that as vi is a real-algebraic curve (of negativ
Lemma 5.4.
Lemma 5.4. Ψ can be extended to a quasiconformal map on bC.
Lemma 5.4. Ψ can be extended to a quasiconformal map on bC.
Lemma 5.4.
Lemma 5.4. Note that eσ (respectively, σ) fixes ∂eT (respectively, ∂T) point-wise. Using the (d + 1): 1 covering maps eσ: eσ−1( eT 0) →eT 0…
Lemma 5.4. Note that eσ (respectively, σ) fixes ∂eT (respectively, ∂T) point-wise. Using the (d + 1) : 1 covering maps eσ : eσ−1( eT 0) →eT 0 and σ : σ−1(T 0) →T 0, we can now lift Ψ0 : eT 0 →T 0 to a homeomorphism Ψ1 : eσ−1( eT 0) →σ−1(T 0) that is conformal on the interior. Moreover, we can choose the lift so that Ψ1 : eσ−1( eT 0) →σ−1(T 0) and Ψ0 : eT 0 →T 0 match on ∂eT 0 to produce a homeomorphism Ψ1 : eE1 →E1 (that is conformal on the interior). In fact, it conjugates eσ : ∂eE1 →∂eT 0 to σ
Theorem 5.5.
Theorem 5.5. There is a bijection between affine equivalence classes of extremal unbounded quadrature domains of order d and isomorphism…
Theorem 5.5. There is a bijection between affine equivalence classes of extremal unbounded quadrature domains of order d and isomorphism classes of bi-angled trees with d−1 vertices.
Proposition 6.1.
Proposition 6.1. Let q(z) be a CS polynomial, and denote by U1, · · ·, Ud−1 the d −1 immediate basins of attraction of the d −1 finite…
Proposition 6.1. Let q(z) be a CS polynomial, and denote by U1, · · · , Ud−1 the d −1 immediate basins of attraction of the d −1 finite critical fixed points of p(z) = q(z). Let i, j ∈{1, · · · , d −1} with i ̸= j. Then either Ui ∩Uj = ∅, or Ui ∩Uj = {ζ}, where ζ is a non-critical fixed point of p. Moreover, in the latter situation, Ui and Uj are the only bounded Fatou components touching at ζ.
Proposition 6.2.
Proposition 6.2. Let q(z) be a CS polynomial, and notation as in Proposition 6.1. Then d−1 [ i=1 Ui is connected.
Proposition 6.2. Let q(z) be a CS polynomial, and notation as in Proposition 6.1. Then d−1 [ i=1 Ui is connected.
Proposition 6.3.
Proposition 6.3. Let q(z) be a CS polynomial of degree d. Then p has exactly 3d −2 fixed points in C; d −1 of which are super-attracting,…
Proposition 6.3. Let q(z) be a CS polynomial of degree d. Then p has exactly 3d −2 fixed points in C; d −1 of which are super-attracting, and the remaining 2d −1 are repelling.
Proposition 6.4.
Proposition 6.4. The angled Hubbard tree T (p) of a CS anti-polynomial p of degree d is a bi-angled tree with d −1 vertices.
Proposition 6.4. The angled Hubbard tree T (p) of a CS anti-polynomial p of degree d is a bi-angled tree with d −1 vertices.
Proposition 6.5.
Proposition 6.5. Each bi-angled tree T with d −1 vertices is isomorphic to the angled Hubbard tree of a CS anti-polynomial of degree d.
Proposition 6.5. Each bi-angled tree T with d −1 vertices is isomorphic to the angled Hubbard tree of a CS anti-polynomial of degree d.
Theorem 6.6.
Theorem 6.6. There exists a bijection between equivalence classes of CS polynomials of degree d (respectively, affine conjugacy classes of CS…
Theorem 6.6. There exists a bijection between equivalence classes of CS polynomials of degree d (respectively, affine conjugacy classes of CS anti-polynomials of degree d) and iso- morphism classes of bi-angled trees with d −1 vertices.
Theorem 5.1
Theorem 5.1] now implies that p1 and p2 are affinely conjugate. Hence, by Proposition 2.23, the CS polynomials q1 and q2 are equivalent in…
Theorem 5.1] now implies that p1 and p2 are affinely conjugate. Hence, by Proposition 2.23, the CS polynomials q1 and q2 are equivalent in the sense of Definition 2.22. This completes the proof of injectivity of the map between equivalence classes of CS polynomials (of degree d) and isomorphism classes of bi-angled trees (with d −1 vertices).
Lemma 4.3
Lemma 4.3] (see [Buf03, Theorem 4] for a similar surgery procedure in the holomorphic case), one now concludes that ˇσ: ˇσ−1(ˇΩ) →ˇΩextends…
Lemma 4.3] (see [Buf03, Theorem 4] for a similar surgery procedure in the holomorphic case), one now concludes that ˇσ : ˇσ−1(ˇΩ) →ˇΩextends to an anti-quasiregular map (of bC) of degree d with a unique pole at ∞and a simple fixed critical point in each component of the droplet. Moreover, this anti-quasiregular map is quasiconformally conjugate to an anti-rational map of degree d. Clearly, this straightened map is an anti-polynomial of degree d with d −1 distinct fixed critical points in the plane
Proposition 7.4.
Proposition 7.4. The tiling set T ∞(σ) is open. Its closure T ∞(σ) is a compact, connected set. The following result states that the family…
Proposition 7.4. The tiling set T ∞(σ) is open. Its closure T ∞(σ) is a compact, connected set. The following result states that the family of Schwarz reflections arising from S∗ d is quasi- conformally closed.
Proposition 7.5.
Proposition 7.5. Let g ∈S∗ d, Ω:= g(D), and σ the Schwarz reflection map of Ω. Further, let µ be a σ-invariant Beltrami coefficient on bC, and…
Proposition 7.5. Let g ∈S∗ d, Ω:= g(D), and σ the Schwarz reflection map of Ω. Further, let µ be a σ-invariant Beltrami coefficient on bC, and Φ : (bC, 0) →(bC, 0) be a quasiconformal map satisfying Φz/Φz = µ a.e.. Then Φ(Ω) is a simply connected bounded quadrature domain. If the quasiconformal map Φ is normalized appropriately, there exists h ∈S∗ d with Φ(Ω) = h(D), and Φ ◦σ ◦Φ−1 is the Schwarz reflection map of Φ(Ω).
Proposition 7.6.
Proposition 7.6. bC = T ∞(σ).
Proposition 7.6. bC = T ∞(σ).
Theorem 7.7.
Theorem 7.7. Let T be a rooted binary tree. Then there exists an extremal bounded quad- rature domain Ωsuch that T (Ω) is isomorphic to T.
Theorem 7.7. Let T be a rooted binary tree. Then there exists an extremal bounded quad- rature domain Ωsuch that T (Ω) is isomorphic to T .
Theorem 7.8.
Theorem 7.8. Let eΩand Ωbe two bounded extremal quadrature domains such that T (eΩ) and T (Ω) are isomorphic as rooted binary trees. Then,…
Theorem 7.8. Let eΩand Ωbe two bounded extremal quadrature domains such that T (eΩ) and T (Ω) are isomorphic as rooted binary trees. Then, there exists an affine map A with A(eΩ) = Ω.
Theorem 7.9.
Theorem 7.9. Let d ≥2. Then (34) # f ∈S∗ d: f has d −2 double points Zd−1 = 1 d −1 2(d −2) d −2
Theorem 7.9. Let d ≥2. Then (34) # f ∈S∗ d : f has d −2 double points Zd−1 = 1 d −1 2(d −2) d −2
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