This is a preprint of: “Julia sets with a wandering branching point”, Xavier Buff, Jordi Canela, Pascale Roesch, Indiana Univ. Math. J., vol. 69(6), 2241–2265, 2020. JULIA SETS WITH A WANDERING BRANCHING POINT XAVIER BUFF, JORDI CANELA, AND PASCALE ROESCH Abstract. According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this article, we reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point even- tually maps to the other critical point which eventually maps to a repelling fixed point. Notations • C is the complex plane, • D is the unit disk, • S1 is the unit circle. • T := R/Z. Introduction In this article, we consider polynomials f : C → C of degree at least 2 as dynamical systems: the orbit of a point z ∈ C is the set O(z) := {f ◦n (z)}n≥0 . The orbit of a point is finite if and only if the point is preperiodic. More precisely, a point α ∈ C is preperiodic if f ◦(r+s) (α) = f ◦r (α) for some integers r ≥ 0 and s ≥ 1. If r and s are minimal integers such that f ◦(r+s) (α) = f ◦r (α), then r is the preperiod and s is the period. The point α is periodic if the preperiod is 0. In this case, the point is repelling if (f ◦s )0 (α) > 1. The filled-in Julia set Kf is the set of points with bounded orbit and the Julia set Jf is its topological boundary. The sets Kf and Jf are compact subsets of C. They are completely invariant: f −1 (Kf ) = f (Kf ) = Kf and f −1 (Jf ) = f (Jf ) = Jf . Preperiodic points are contained in Kf . Repelling periodic points are contained in Jf . In fact, Jf is the closure of the set of repelling periodic points (see e.g. [1, 5]). A point ω ∈ C is a critical point if the derivative of f vanishes at ω. The topology of Kf and Jf is related to the behavior of critical orbits. For example, Kf and Jf are connected if and only if the critical points of f belong to Kf (see e.g. [1, 5]). A dendritic polynomial is a polynomial f for which Jf is a dendrite, i.e., Jf is connected and locally connected and contains no simple closed curve. This is the case whenever each critical point is preperiodic to a repelling periodic point (see [3, Th. V.4.2]). This research was supported in part by the ANR grant Lambda ANR-13-BS01-0002. 1 2 X. BUFF, J. CANELA, AND P. ROESCH Example 1. The Julia set of the quadratic polynomial f (z) = z 2 + i is a dendrite (see Figure 1). The unique critical point is ω = 0 and its orbit is /i / −1 + i ( 0 j −i √ The derivative of f ◦2 at −1 + i is 4 + 4i which has modulus 4 2 > 1. Figure 1. The Julia set Jf for f (z) = z 2 + i is a dendrite. Assume Jf is locally connected. A point ξ ∈ Jf is a branching point if Jf \ {ξ} has (at least) three connected components. According to Thurston [7], if f is a quadratic polynomial, the orbit of such a branching point contains a periodic point or a critical point. A point z ∈ C is precritical if f ◦n (z) = ω for some critical point ω and some integer n ≥ 0. Theorem 2 (Thurston). Let f be a quadratic polynomial with locally connected Julia set Jf . If ξ ∈ Jf is a branching point, then ξ is preperiodic or precritical. Blokh and Oversteegen [2] proved that such a result does not hold for higher degree polynomials. Theorem 3 (Blokh-Oversteegen). There exist dendritic cubic polynomials having a branching point which is neither preperiodic nor precritical. Our goal is to give a new proof of this result. The strategy of our proof consists in exhibiting a sequence {fn } of dendritic polynomials with a sequence {ξn } of branching points in Jfn which are precritical to both critical points of fn and preperiodic to a repelling fixed point of fn , such that • the sequence {fn } converges to a dendritic polynomial f , • the sequence {ξn } converges to a branching point ξ in Jf and • ξ is neither precritical nor preperiodic for f . More precisely, we exhibit sequences satisfying: • fn◦jn (ξn ) = ωn , fn◦kn (ωn ) = ωn0 and fn◦`n (ωn0 ) = αn for some increasing sequences of integers {jn }, {kn } and {`n }, where ωn and ωn0 are the two critical points of fn and αn is a repelling fixed point of fn ; • Jfn \ {ξn } has (at least) three connected components containing points βn , βn0 and βn00 such that fn (βn ) = fn (βn0 ) = fn (βn00 ) = βn . JULIA SETS WITH A WANDERING BRANCHING POINT 3 We shall see that we can choose fn+1 arbitrarily close to fn for each n; this is enough to deduce Theorem 3. The paper is structured as follows. In §1 we introduce the notion of admissible polynomials and state key results used in the proof of Theorem 3: convergence of nodal points, convergence of Carathéodory loops and a Key Proposition regarding the existence of a particular sequence of admissible polynomials. In §2 we prove Theorem 3 assuming those key results. Finally, in §3 we prove the convergence of nodal points, in §4 we prove the convergence of Carathéodory loops and in §5 we prove the Key Proposition. 1. Definitions and key results 1.1. Dendrites. A dendrite J ⊂ C is a connected and locally connected compact set containing no simple closed curve. In this article, we assume in addition that J is not reduced to a point. Properties of dendrites are discussed in [9]. In particular, a dendrite is uniquely arcwise connected. Definition 4. Given two points β and β 0 in a dendrite J , we denote [β, β 0 ]J the arc joining β and β 0 in J . Definition 5. The nodal point of three points β, β 0 and β 00 in a dendrite J is the unique point which belongs simultaneously to [β, β 0 ]J , [β 0 , β 00 ]J and [β 00 , β]J . If ξ ∈ J , then any connected component C of J \{ξ} is open in J and C \C = {ξ}. Definition 6. A point ξ in a dendrite J is non-separating if J \ {ξ} is connected. Definition 7. A point ξ in a dendrite J is a branching point if J \ {ξ} has (at least) three connected components. It separates β, β 0 and β 00 in J if β, β 0 and β 00 are in three distinct connected components of J \ {ξ}. Note that a nodal point is not necessarily a branching point. For example, when β 00 ∈ [β, β 0 ]J , then the nodal point of β, β 0 and β 00 in J is the point β 00 . In fact, the nodal point ξ of β, β 0 and β 00 in J separates β, β 0 and β 00 in J if and only if ξ∈ / {β, β 0 , β 00 }. The complement C \ J of a dendrite is connected and simply connected. It follows that there is a (unique) conformal representation φ : C \ D → C \ J such that φ(z)/z → R > 0 as z → ∞. Since J is locally connected, a theorem by Torhorst [8], based on the work on prime-ends of Carathédory (see the discussion in [6]), asserts that φ extends to a continuous map φ : C \ D → C. Since J has empty interior, this map is surjective and restricts to a map ϕ : S1 → J . Definition 8. The Carathéodory loop of J is the restriction ϕ : S1 → J . The Carathéodory loop is a continuous and surjective map from S1 to J . We shall use the following result whose proof is given in §3. Lemma 9 (Convergence of nodal points). Let {Jn } be a sequence of dendrites. Assume the associated sequence {ϕn : S1 → Jn } of Carathéodory loops converges uniformly to some non constant map ϕ : S1 → C. Then, • J := ϕ(S1 ) is a dendrite with Carathéodory loop ϕ; • if {βn }, {βn0 } and {βn00 } are sequences of points in Jn converging to β, β 0 and β 00 in J , then the corresponding sequence of nodal points of βn , βn0 and βn00 in Jn converges to the nodal point of β, β 0 and β 00 in J . 4 X. BUFF, J. CANELA, AND P. ROESCH 1.2. Admissible polynomials. Consider the affine space A of monic cubic poly- nomials fixing 0. We will restrict our study to the open subset V ⊂ A of polynomials f such that: (1) f has three distinct repelling fixed point α = 0, β and γ; (2) f has two distinct critical points ωα , ωβ ; (3) the ray of angle 0 does not bifurcate and lands at α; (4) the ray of angle 1/2 does not bifurcate and lands at β; (5) the rays of angles 1/4 and −1/4 do not bifurcate, land at γ, and separate the plane in two connected components • Uα containing α, ωα and f (ωβ ) and • Uβ containing β, ωβ , f (ωα ) and f ◦2 (ωα ). Figure 2 shows the ray configuration and the position of the points involved in the definition of V. 1 4 β f (ωα ) ωβ ωα f (ωβ ) α γ 1 0 2 f ◦2 (ωα ) Uβ Uα − 14 Figure 2. The ray configuration and the position of the points involved in the definition of V. The angle of each external ray is indicated. Definition 10. A cubic polynomial f ∈ V is admissible if it has critical points ω and ω 0 and a branching point ξ, such that: (1) ξ is precritical to ω, ω is precritical to ω 0 and ω 0 is prefixed to α; (2) ξ separates β, β 0 and β 00 in Jf , where f −1 (β) = {β, β 0 , β 00 }. We set ξf := ξ and denote by jf the integer j ≥ 0 such that f ◦j (ξ) = ω. Our key result, proved in §5, is the following. Proposition 11 (Key proposition). Assume f ∈ V is an admissible polynomial. Then, there is a sequence {gm } of admissible polynomials which converges to f , such that {ξgm } converges to ξf and jgm > jf for all m. This result shall be completed with the following observation. An admissible polynomial is strictly postcritically finite. It follows that its Julia set is a dendrite (see Figure 3) and thus, has an associated Carathéodory loop. We shall prove in §4 that the convergence of polynomials implies the convergence of the associated Carathéodory loops. JULIA SETS WITH A WANDERING BRANCHING POINT 5 β0 0 ◦2 β f (ω) ω = f (ω) γ ω=ξ α = f (ω 0 ) β 00 Figure 3. The Julia set of an admissible polynomial f with crit- ical points ω and ω 0 . We have f ◦2 (ω) = ω 0 and f (ω 0 ) = α. In this example, jf = 0. Lemma 12 (Convergence of Carathéodory loops). Let {gm } be a sequence of cubic polynomials, with locally connected Julia sets, which converges to an admissible polynomial f . Then, the sequence of Carathéodory loops of Jgm converges uniformly to the Carathéodory loop of Jf . 2. Existence of non preperiodic and non precritical branching points We now prove Theorem 3, assuming Lemmas 9 and 12 and Proposition 11. 2.1. An admissible polynomial. We first prove that there exists an admissible polynomial. The polynomial f : C → C defined by f (z) = z(z − 3ω)2 is monic, fixes 0, has critical points at ω and ω 0 = 3ω and satisfies f (ω 0 ) = 0. Then, f (ω) = 4ω 3 and f ◦2 (ω) = 4ω 3 (4ω 3 − 3ω)2 . The equation f ◦2 (ω) = 3ω has a unique real negative solution which is r q 1 √ ω := − 6 + 2 9 + 8 3. 4 The graph of f : R → R is shown on Figure 4 and the Julia set Jf is shown in Figure 3. The fixed points of f are α := 0, β := 3ω − 1 and γ := 3ω + 1 ∈ (β, α). The respective multipliers at α, β and γ are 9ω 2 > 1, 3 − 6ω > 1 and 3 + 6ω < −1. In particular, the three fixed points are repelling. We have that β = 3ω − 1 < f (ω) = 4ω 3 < f ◦2 (ω) = ω 0 = 3ω < γ = 3ω + 1 < ω < 0 = α. The intersection of the Julia Jf with the real axis is the interval [β, α]. Since f is a real polynomial, the ray of angle 0 is (α, +∞) and lands at α. The ray of angle 1/2 is (−∞, β) and lands at β. Since the multiplier of γ is negative, the orbit of a ray landing at γ must alternate in between the upper half-plane and the lower half- plane. Since f (ω 0 ) = α, the rays of angles 1/3 and −1/3 land at ω 0 and separate β from γ. It follows that the digits of ternary expansion of the rays landing at γ alternate between 0 and 2: those rays have angle 1/4 and −1/4 and separate the plane in two connected components. The one containing α contains ω and f (ω 0 ). The one containing β contains ω 0 = f ◦2 (ω) and f (ω). So, the polynomial f belongs to V. Set ξ := ω. Then, ξ is precritical to ω, ω is precritical to ω 0 and ω 0 is prefixed to α := 0. So, Condition (1) in Definition 10 is satisfied with jf = 0. In addition, 6 X. BUFF, J. CANELA, AND P. ROESCH β f (ω) ω0 γ ω α=0 Figure 4. The graph of the real polynomial f (z) = z(z − 3ω)2 . ξ separates β, β 0 and β 00 in Jf (see Figure 5), so that Condition (2) in Definition 10 is satisfied. β = f (β) f (ω) f (ω 0 ) = f (α) = α f β0 β ω0 ω α β 00 Figure 5. A schematic representation of the preimage of [β, α] by f illustrating why ω separates β, β 0 and β 00 in Jf 2.2. A sequence of admissible polynomials. We now build a Cauchy sequence of admissible polynomials. Given two maps f : X → C and g : X → C defined on some set X ⊆ C, we denote by dX (f, g) the spherical distance between f and g: f (x) − g(x)| dX (f, g) := sup p p . x∈X 1 + |f (x)|2 1 + |g(x)|2 JULIA SETS WITH A WANDERING BRANCHING POINT 7 Let f be an admissible polynomial and let ξf be the associated branching point. According to Definition 10, the points ξf , f (ξf ), . . . , f ◦(jf −1) (ξf ) are not critical points of f and are pairwise distinct. For each integer j, the map V × C → C : (g, z) 7→ g ◦j (z) is continuous at (f, ξf ) and the critical points of g depend continuously on g. Thus, there exists f > 0 such that for |z − ξf | < f and dC (f, g) < f , the points z, g(z), . . . , g ◦(jf −1) (z) are not critical points of f and are pairwise distinct. According to Proposition 11, there is a sequence {gm } of admissible polynomials which converges to f , such that {ξgm } converges to ξf and jgm > jf for all m. According to Lemma 12, the sequence of Carathéodory loops ϕgm of Jgm converges uniformly to the Carathéodory loop ϕf of Jf . So, for all ε > 0, if m is large enough, then |ξgm − ξf | < ε, dC (f, gm ) < ε and dS1 (ϕf , ϕgm ) < ε. Let us now define recursively a sequence {fn } of admissible polynomials and a sequence {εn } of positive numbers as follows. We let f0 be any admissible polyno- mial and set ε0 := f0 /2. Once fn and εn are defined, we let fn+1 be an admissible polynomial such that jfn+1 > jfn , |ξfn+1 − ξfn | < εn , dC (fn , fn+1 ) < εn and dS1 (ϕfn , ϕfn+1 ) < εn . We then set 1  εn+1 = min εn , fn+1 . 2 Figures 6 and 7 illustrate the Julia sets of f0 , f1 , f2 , f3 for such a sequence. Ob- serve that for all n ≥ 0 and all p ≥ 0, εn+p ≤ εn /2p ≤ fn /2p+1 . As a consequence, |ξfn+p − ξfn | < 2εn ≤ fn , dC (fn , fn+p ) < 2εn ≤ fn and dS1 (ϕfn , ϕfn+p ) < 2εn ≤ fn . In particular, the sequences {fn }, {ξfn }, and {ϕfn } are Cauchy sequences. 2.3. Proof of Theorem 3. The sequences {fn }, {ξfn }, and {ϕfn } converge to f , ξ and ϕ. We now prove that ξ is a wandering non-precritical branching point of f . Let βn , βn0 , βn00 be the landing points of the rays of angles 1/2, 1/6 and −1/6 for fn , so that • fn−1 (βn ) = {βn , βn0 , βn00 } and • ξn is the nodal point of βn , βn0 and βn00 in Jfn . The sequences {βn }, {βn0 } and {βn00 } converge to the landing points β, β 0 and β 00 of the rays of angles 1/2, 1/6 and −1/6 for f . According to Lemma 9, J := ϕ(S1 ) is a dendrite with Carathéodory loop ϕ and ξ is the nodal point of β, β 0 and β 00 in J. We claim that Jf = J . Indeed, the critical orbits of f are approximated by the critical orbits of fn . So, they are bounded and Jf is connected. According to [4], Jf ⊆ J := lim Jfn ⊆ Kf . n→+∞ Since J is a dendrite and Jf ⊆ J is connected, Jf itself is a dendrite. It follows that Kf = Jf = J . 8 X. BUFF, J. CANELA, AND P. ROESCH β00 β0 ω00 ω0 β000 (a) f0 (z) = 8.7534421003338z + 5.9172433109798z 2 + z 3 , j = 0, k = 2, ` = 1. β10 β1 ω1 ω10 β100 (b) f1 (z) = (8.6656058283165+0.059672002492800i)z +(5.8731379216063+0.020430827270432i)z 2 +z 3 , j = 2, k = 2, ` = 3. β20 β2 ω20 ω2 β200 (c) f2 (z) = (8.6620018002588 + 0.049185458993292i)z + (5.871351730126 + 0.017466126249776i)z 2 + z 3 , j = 4, k = 5, ` = 5. β30 β3 ω3 ω30 β300 (d) f3 (z) = (8.6620495410606 + 0.049156312358058i)z + (5.871375113635 + 0.017446586001088i)z 2 + z 3 , j = 9, k = 8, ` = 10. Figure 6. A sequence of 3 perturbations of the polynomial f0 in- troduced in §2.1. We draw the external rays landing at the corre- sponding branching points ξ. In all figures f ◦j (ξ) = ω, f ◦k (ω) = ω 0 and f ◦` (ω 0 ) = 0. We claim that ξ is neither preperiodic nor precritical for f , so that it cannot coincide with β, β 0 or β 00 . As a consequence, ξ is a branching point which is neither preperiodic nor precritical, as required. To prove that ξ is neither preperiodic nor precritical, we use that for all n, |ξ − ξfn | < fn and dC (f, fn ) < fn . By definition of fn , the points ξ, f (ξ), . . . , f ◦(jfn −1) (ξ) are not critical points of f and are pairwise distinct. Since the sequence {jfn } is increasing, it takes arbitrarily large values. Therefore, all the points in the f -orbit of ξ are not critical points of f and are pairwise distinct. JULIA SETS WITH A WANDERING BRANCHING POINT 9 β00 β10 ω0 ω10 β000 β100 (a) f0 (b) f1 β20 β30 ω2 ω30 β200 β300 (c) f2 (d) f3 Figure 7. Zooms in Figure 6. This completes the proof of Theorem 3 assuming Lemmas 9 and 12 and Propo- sition 11. 3. Convergence of nodal points Here, we prove Lemma 9. Let {Jn } be a sequence of dendrites. Assume the associated sequence {ϕn : S1 → Jn } of Carathéodory loops converges uniformly to some non constant map ϕ : S1 → C. We first prove that J := ϕ(S1 ) is a dendrite. Since ϕ is continuous, J is connected and locally connected. The Carathéodory loop ϕn : S1 → Jn is the boundary value of a continuous map φn : C \ D → C which is univalent in C \ D. The sequence {ϕn : S1 → Jn } converges uniformly to some ϕ : S1 → C; according to the Maximum Modulus Principle, the sequence {φn : C \ D → C} converges uniformly to some φ : C \ D → C. As a non constant limit of univalent maps, φ : C \ D → C is univalent and J = ∂U with U := φ(C \ D). In particular, J has 10 X. BUFF, J. CANELA, AND P. ROESCH empty interior. Since the Julia set Jn has empty interior, the maps φn : C \ D → C are surjective, and so, the limit φ : C \ D → C is surjective. Indeed, if w ∈ C and φn (zn ) = w, then φ(z) = w for any limit value z of the sequence {zn }. It follows that C = U t J . Therefore, J contains no simple closed loop since, otherwise, either U = C \ J is not connected, or J has non-empty interior. We now assume {βn }, {βn0 } and {βn00 } are sequences of points in Jn converging to β, β 0 and β 00 in J . We must prove that the sequence of nodal points of βn , βn0 and βn00 in Jn converges to the nodal point of β, β 0 and β 00 in J . This is based on the following lemma. Given two distinct points θ and θ0 in T, we set [θ, θ0 ]T := {t ∈ S1 ; θ, t and θ0 are in counterclockwise order on T}. Lemma 13. Let J ⊂ C be a dendrite with Carathéodory loop ϕ : S1 → J . Let θ and θ0 be two distinct points in S1 . Then,     ϕ(θ), ϕ(θ0 ) J = ϕ [θ, θ0 ]T ∩ ϕ [θ0 , θ]T .   Proof. Let S := ϕ(θ), ϕ(θ0 ) J be the arc joining ϕ(θ) and ϕ(θ0 ) in J . Since ϕ is   continuous, ϕ [θ, θ0 ]T and ϕ [θ0 , θ]T are connected. Since they both contain ϕ(θ) and ϕ(θ0 ),   S ⊆ ϕ [θ, θ0 ]T ∩ ϕ [θ0 , θ]T .   Now, assume ξ ∈ ϕ [θ, θ0 ]T ∩ ϕ [θ0 , θ]T , i.e., ξ = ϕ(t) = ϕ(t0 ) with t ∈ [θ, θ0 ]T and t0 ∈ [θ0 , θ]T . The rays of angles t and t0 separate J in two connected components: one contains ϕ(θ) and the other contains ϕ(θ0 ). Thus, ϕ(θ) and ϕ(θ0 ) are in distinct connected components of J \ {ξ}. Since S is an arc joining ϕ(θ) and ϕ(θ0 ) in J , we necessarily have ξ ∈ S. Therefore,   ϕ [θ, θ0 ]T ∩ ϕ [θ0 , θ]T ⊆ S.  Corollary 14. Let J ⊂ C be a dendrite with Carathéodory loop ϕ : S1 → J . Let θ, θ0 and θ00 be three distinct points in S1 . Then, the nodal point of ϕ(θ), ϕ(θ0 ) and ϕ(θ00 ) in J is    ϕ [θ, θ0 ]T ∩ ϕ [θ0 , θ00 ]T ∩ ϕ [θ00 , θ]T . If the three points βn , βn0 and βn00 are not distinct for infinitely many n, then the nodal point in Jn coincides with the corresponding multiple point, and Lemma 9 follows easily. So, without loss of generality, assume the points are distinct and let θn , θn0 and θn00 be three distinct points in S1 with ϕn (θn ) = βn , ϕn (θn0 ) = βn0 and ϕn (θn00 ) = βn00 . Let ξ be a limit value of the sequence {ξn }. We must show that ξ is the nodal point of β, β 0 and β 00 in J . Extracting a subsequence and reordering the points if necessary, we may assume that θn , θn0 and θn00 are in counterclockwise order on S1 . Let ξn be the nodal points of βn , βn0 and βn00 in Jn . According to Lemma 13, there are points tn ∈ [θn , θn0 ]T , t0n ∈ [θn0 , θn00 ]T and t00n ∈ [θn00 , θn ]T with ξn = ϕn (tn ) = ϕn (t0n ) = ϕn (t00n ). Extracting a further subsequence if necessary, we may assume that {ξn }, {θn }, {θn0 }, {θn00 }, {tn }, {t0n } and {t00n } converge to ξ, θ, θ0 , θ00 , t, t0 and t00 . Since {ϕn } converges uniformly to ϕ, we have that ϕ(θ) = β, ϕ(θ0 ) = β 0 , ϕ(θ00 ) = β 00 and ϕ(t) = ϕ(t0 ) = ϕ(t00 ) = ξ. JULIA SETS WITH A WANDERING BRANCHING POINT 11 If θ, θ0 and θ00 are not distinct, let us say θ = θ0 , then t = θ = θ0 and ξ = β = β 0 is the nodal point of β, β 0 and β 00 in J . Otherwise, if θ, θ0 and θ00 are distinct, t ∈ [θ, θ0 ]T , t0 ∈ [θ0 , θ00 ]T and t00 ∈ [θ00 , θ]T ; the proof follows from Corollary 14. 4. Convergence of Carathéodory loops In this section, we prove Lemma 12. Our proof relies on puzzle techniques. 4.1. The puzzle of an admissible polynomial. In the whole section, f ∈ V is an admissible polynomial. We denote by φf : C \ D → C \ Jf the Böttcher coordinate conjugating z 7→ z 3 to f , with φf (z)/z → 1 as z → ∞. This Böttcher coordinate extends as a continuous map φf : C \ D → C and the restriction to S1 is the Carathéodory loop ϕf : S1 → Jf .  Let Γ0f be the union of the equipotential φf (2e2πiθ ) ; θ ∈ T , the external rays of angles 1/4 and −1/4 and their landing point γ. For m ≥ 0, set −m Γm f := f (Γ0f ). Definition 15. The puzzle pieces of depth m ≥ 0 are the bounded connected components of C \ Γm f . If z ∈ Jf is not an iterated preimage of γ, we denote by pm (z) the puzzle piece of depth m which contains z. There are two puzzle pieces of depth 0: one contains α, ωα and f (ωβ ); the other contains β, ωβ , f (ωα ) and f ◦2 (ωα ). By construction, the puzzle pieces of depth m ≥ 1 are the connected components of the preimages of the puzzle pieces of depth m − 1. Therefore, if p is a puzzle piece of depth m ≥ 1, then f (p) is a puzzle piece of depth m − 1 and f : p → f (p) is a ramified covering. The union of the rays of angles 1/4 and −1/4 is invariant by f . It follows that the puzzle pieces of depth 1 do not intersect those rays; therefore, each puzzle piece of depth 1 is entirely contained in a puzzle piece of depth 0. By induction on the depth, each puzzle piece of depth m is entirely contained in a puzzle piece of depth m − 1. In particular, a puzzle piece contains at most one critical point. So, if p is a puzzle piece of depth m ≥ 1, then f : p → f (p) is an isomorphism if p does not contain a critical point, and a ramified covering of degree 2 otherwise. Figure 8 shows the puzzle pieces of depth m ∈ {0, 1, 2, 3} for some admissible polynomial f ∈ V. We use one color for the puzzle pieces which are iterated preim- ages of p0 (α) and another one for the puzzle pieces which are iterated preimages of p0 (β). The main result from which we deduce Lemma 12 is the following. Proposition 16. The maximum diameter of a puzzle piece of depth m tends to 0 as m tends to +∞. Proof of Lemma 12 assuming Proposition 16. First, if g ∈ V is sufficiently close to f , the Böttcher coordinate φg tangent to identity at infinity is defined on the circle of radius 2, and we may still define Γ0g as the union of the equipotential  φg (2e2πiθ ) ; θ ∈ T , the external rays of angles 1/4 and −1/4 and their landing point γg . And for m ≥ 0, we may define −m Γm g := g (Γ0g ). Then, for each fixed m, Γm g depends holomorphically on g in some neighborhood of f. 12 X. BUFF, J. CANELA, AND P. ROESCH p0 (β) p0 (α) Figure 8. The puzzle pieces of depth m ∈ {0, 1, 2, 3}, for some admissible polynomial f ∈ V. Second, according to Proposition 16, given  > 0, we may choose m ≥ 0 suffi- ciently large so that the puzzle pieces of f of depth m have diameters at most /2. Let U be a sufficiently small neighborhood of f in V so that the puzzle of depth m depends holomorphically on g in U: for g ∈ U, there is a continuous map ψ : U × C 3 (g, z) 7→ ψg (z) ∈ C such that for each g ∈ U, ψg : C → C is a homeomorphism sending Γm m f to Γg . Shrinking U if necessary, we may assume that for g ∈ U, dC (ψg , id) < /2. Third, assume p is a puzzle piece of f of depth m, θ ∈ T and g ∈ U has a locally connected Julia set. Let ϕg : S1 → Jg be the corresponding Carathéodory loop. Then, ϕf (e2πiθ ) ∈ p if and only if ϕg (e2πiθ ) ∈ ψg (p). Thus, dS1 (ϕf , ϕg ) < .  Proof of Proposition 16. We will use the fact that f is expanding for a suitable orbifold metric (see [5, Th. 19.6]). Assume k ≥ 1 and ` ≥ 1 are integers such that JULIA SETS WITH A WANDERING BRANCHING POINT 13 f ◦k (ω) = ω 0 and f ◦` (ω 0 ) = α. Consider the function ν : C → {1, 2, 4} defined by   1 if z ∈ / Pf , ν(z) := 2 if z = f ◦n (ω) with 1 ≤ n ≤ k and   4 if z = f ◦n (ω 0 ) with 1 ≤ n ≤ `. The orbifold (C, ν) has Euler characteristic 1 3 χ(C, ν) = 1 − k − ` < 0, 2 4 so that there is a universal covering of orbifold π : D → C which ramifies precisely with local degree ν(z) above z ∈ C. Then, there is a metric of orbifold µ which is smooth outside Pf and blows up at points in Pf , such that π ∗ µ is the hyperbolic metric on D. There are constants K1 and K2 such that for all puzzle piece p, diam(p) ≤ K1 diamµ (p) and diamµ (p) ≤ K2 , where diamµ (p) stands for the diameter of p for the metric µ. In addition, there is a holomorphic map F : D → D such that the following diagram commutes: F Do D π π  f  C / C. The map F is contracting for the hyperbolic metric on D. It follows that there is a constant κ < 1 such that 1 ∀z ∈ p0 (α) ∪ p0 (β), kDz f kµ > κ (this norm may blow up at points in f −1 (Pf )). In particular, if p is a puzzle piece, then (  κdiamµ f (p) if p ∩ Cf = ∅ and diamµ (p) ≤  2κdiamµ f (p) if p ∩ Cf 6= ∅. Lemma 17. The piece p1 (α) maps isomorphically to p0 (α). Proof. Recall that the external rays of angles 1/4 and −1/4 separate the plane in two connected components (see Figure 2): Uα containing α and Uβ containing β. Each ray in Uα has angle in (−1/4, 1/4). It has two preimages in Uβ , one with angle in (1/4, 5/12) and one with angle in (−5/12, −1/4), and one preimage in Uα with angle in (−1/12, 1/12). As a consequence, the component of f −1 (Uα ) containing α is contained in Uα and maps isomorphically to Uα . The other component is contained in Uβ and maps to Uα with degree 2.  Since p0 (α) ⊂ Uα , it follows that the component p1 (α) of f −1 p0 (α) containing α maps isomorphically to p0 (α).  It follows by induction on m ≥ 1 that pm (α) is the image of pm−1 (α) by the inverse branch f −1 : p0 (α) → p1 (α) and that f ◦m : pm (α) → p0 (α) is an isomor- phism. As a consequence,  diamµ pm (α) ≤ K2 · κm . 14 X. BUFF, J. CANELA, AND P. ROESCH If m is large enough so that pm (α) ∩ Pf = {α}, then f ◦` : pm+` (ω 0 ) → pm (α) is a ramified cover of degree 2 and   diamµ pm+` (ω 0 ) ≤ 2κ` diamµ pm (α) . It follows that there is a constant K3 such that  diamµ pm (ω 0 ) ≤ K3 · κm . If m is large enough so that pm (ω 0 ) ∩ Pf = {ω 0 }, then f ◦k : pm+k (ω) → pm (ω 0 ) is a ramified cover of degree 2 and   diamµ pm+k (ω) ≤ 2κk diamµ pm (ω) . It follows that there is a constant K4 ≥ K3 such that  diamµ pm (ω) ≤ K4 · κm . Finally, assume p is a piece of depth m and let n ∈ [0, m] be the least integer such that q := f ◦n (p) contains a critical point. Then, q is a piece of depth m − n and f ◦n : p → q is an isomorphism, so that diamµ (p) ≤ κn diamµ (q) ≤ κn · K4 · κm−n = K4 · κm .  5. The key proposition The goal of this section is to prove the Key Proposition (Proposition 11). 5.1. Ray configuration. We shall first describe some general configuration of external rays for polynomials f ∈ V. Lemma 18. Assume f ∈ V and assume the ray of angle θ lands on f (ω) for some critical point ω. Then, the two preimage rays landing at ω separate α and β. One has angle in (0, 5/12)T and one has angle in (−5/12, 0)T Proof. Each ray contained in Uα (with angle in (−1/4, 1/4)T ) has a single preimage in Uα with angle in (−1/12, 1/12)T , and two preimages in Uβ , one with angle in (1/4, 5/12)T and one with angle in (−5/12, −1/4)T . So, if a ray lands at f (ωβ ) ∈ Uα , then one preimage ray with angle in (1/4, 5/12)T and one preimage ray with angle in (−5/12, −1/4)T land at ωβ ∈ Uβ . Those rays separate α and β (see Figure 9). Similarly, if a ray lands at f (ωα ) ∈ Uβ , then one preimage ray with angle in (1/12, 1/4)T and one preimage ray with angle in (−1/4, −1/12)T land at ωα ∈ Uα . Those rays separate α and β (see Figure 9).  Lemma 19. For f ∈ V, the rays of angles 5/12 and −5/12 land at a common point and bound an open set Wf containing α; the rays of angles 5/36 and −5/36 land at a common point and bound an open set Vf containing α. Moreover, f (Vf ) = Wf and f : Vf → Wf is an isomorphism. The result is illustrated on Figure 10. Proof. The rays of angles 1/4 and −1/4 land at a common fixed point γ. This point has three distinct preimages, including itself. The rays of angles 5/12 and −5/12 are contained in Uβ and have to land at common preimage of γ contained in Uβ , while the rays of angles 1/12 and −1/12 are contained in Uα and have to land at the other preimage of γ contained in Uα . As in the proof of Lemma 17, the component Uα0 of f −1 (Uα ) contained in Uα maps isomorphically to Uα . Similarly, the component Uβ0 of f −1 (Uβ ) contained in JULIA SETS WITH A WANDERING BRANCHING POINT 15 1 4 β f (ωα ) γ f (ωβ ) α Uβ Uα − 41 6 f 5 1 5 12 4 36 β ωβ γ ωα α 5 − 41 5 − 12 − 36 Figure 9. Sketch of the configuration of external rays described in the proof of Lemma 18. Uβ maps isomorphically to Uβ . The first is bounded by the rays of angles 1/12 and −1/12 and the second is bounded by the rays of angles 5/12 and −5/12. By assumption, f (ωα ) ∈ Uβ and f ◦2 (ωα ) ∈ Uβ , so that f (ωα ) ∈ Uβ0 . It follows that the region Wf bounded by the rays of angle 5/12 and −5/12 and containing α contains a single critical value f (ωβ ). Note that f (ωβ ) ∈ Uα ⊂ Wf . As a consequence, f −1 (Wf ) has two connected components. One contains ωβ and maps with degree 2 to Wf . The other, Vf , contains Uα0 and maps isomorphically to Wf . This last component Vf contains the ray of angle 0 and so, is bounded by the preimages of the rays of angles 5/12 and −5/12 whose angles are closest to 0, i.e., the rays of angles 5/36 and −5/36.  5.2. Polynomials with (k, `)-configuration. Here, we assume f ∈ V has (k, `)- configuration, i.e., f ◦k (ω) = ω 0 and f ◦` (ω 0 ) = α with k ≥ 1 and ` ≥ 1. As g varies in V the two critical points of g depend holomorphically on g and 0 remains the landing point of the ray of angle 0. Denote by ω : V → C and ω 0 : V → C holomorphic maps following the critical points of g ∈ V with ω(f ) = ω 16 X. BUFF, J. CANELA, AND P. ROESCH 5 12 f (ωα ) γ f (ωβ ) α Uβ0 Wf Uα 5 − 12 6 f 5 36 ωβ ωα γ α Uα0 Vf 5 − 36 Figure 10. The open set Vf containing α and bounded by the rays of angles 5/36 and −5/36 maps isomorphically to the open set Wf containing α and bounded by the rays of angles 5/12 and −5/12. and ω 0 (f ) = ω 0 . In addition, for m ≥ 0, let {ω −m : V → C}m≥0 be defined recursively by  ω 0 := ω and ω −m−1 (g) ∈ Vg with g ω −m−1 (g) = ω −m (g). The sequence is well defined since g : Vg → Wg is an isomorphism and Vg ⊂ Wg . As m → +∞, the sequence {ω −m } converges locally uniformly to 0 on V. We now exhibit a particular sequence {gm }m≥m0 of polynomials converging to f in V. Lemma 20. Assume f ∈ V has (k, `)-configuration. Then, there is a sequence {gm }m≥m0 converging to f in V such that gm has (m + `, k + `)-configuration with 0 critical points ωm and ωm satisfying: ωm = ω 0 (gm ), 0 ωm := ω(gm ) and ◦` gm (ωm ) = ω −m (gm ). Remark 21. The roles of the two critical points are exchanged: the sequence {ωm } converges to ω 0 and the sequence {ωm 0 } converges to ωm (see Figure 11). JULIA SETS WITH A WANDERING BRANCHING POINT 17 f (ω 0 ) ω 0 = f ◦2 (ω) ω ω−1 α = f (ω 0 ) ω−2 (a) f ω1 g1◦2 (ω1 ) = ω10 g1 (ω1 ) g1 (ω10 ) α = g1◦3 (ω10 ) g1◦2 (ω10 ) (b) g1 ω2 g2◦3 (ω2 ) = ω20 g2◦2 (ω2 ) α = g2◦3 (ω20 ) g2 (ω20 ) g2 (ω2 ) g2◦2 (ω20 ) (c) g2 Figure 11. An admissible polynomial f with a (2, 1)-configur- ation together with two perturbations g1 and g2 . Proof. For (a, b) ∈ C2 , let fa,b ∈ A be the cubic polynomial defined by 3 fa,b (z) = z 3 − (a + b)z 2 + 3abz. 2 The critical points of fa,b are a and b, so that f = fω,ω0 . Consider the analytic sets n o  ◦(k+`) Σ := (a, b) ∈ C2 ; fa,b (a) = 0 and Σ0 := (a, b) ∈ C2 ; fa,b ◦` (b) = 0 . 0 There√are parameters (a, √ b) which belong neither to Σ nor to Σ (for example, when a := i 2/2 and b = −i 2/2, then a is a fixed point of fa,b ). As a consequence, Σ and Σ0 are 1-dimensional complex curves. By assumption, (ω, ω 0 ) ∈ Σ∩Σ0 . The intersection Σ∩Σ0 consists of postcritically finite polynomials, thus is bounded in C2 . It follows that (ω, ω 0 ) is an isolated point of Σ ∩ Σ0 . Let a : (D, 0) → 0  (C, ω) and b : (D, 0) → (C, ω ) be non constant analytic germs so that a(t), b(t) ∈ Σ ∩ V for t ∈ D. Set Ft := fa(t),b(t) ∈ V and consider the sequence of functions {σm : D → C}m≥0 defined by  σm (t) := Ft◦` b(t) − ω −m (Ft ).  As m tends to +∞, the sequence {σm } converges to σ : D 3 t 7→ Ft◦` b(t) ∈ C. Note that σ vanishes at 0 but does not identically vanish since otherwise, the curve t 7→ Ft would take its values in Σ ∩ Σ0 , contradicting the previous observation that (ω, ω 0 ) is an isolated point of Σ ∩ Σ0 . It follows from the Rouché Theorem that for m large enough, σm vanishes at some point tm ∈ D with tm → 0 as m → +∞. The result follows with gm := Ftm , ωm := b(tm ) and ωm 0 := a(tm ).  5.3. Admissible perturbations. Here, we prove the Key Proposition (Proposi- tion 11). Its proof follows directly from Lemma 22. 18 X. BUFF, J. CANELA, AND P. ROESCH We assume f ∈ V is admissible, β is the landing point of the ray of angle 1/2 for f , ξ is the nodal point of β, β 0 and β 00 in Jf with f −1 (β) = {β, β 0 , β 00 }, and f ◦j (ξ) = ω, f ◦k (ω) = ω 0 and f ◦` (ω 0 ) = α. Let {gm }m≥m0 be a sequence of cubic polynomials provided by Lemma 20. For m ≥ m0 , let βm be the landing point of the ray of angle 1/2 for gm , and let ξm be 0 00 −1 0 00 the nodal point of βm , βm and βm in Jgm with gm (βm ) = {βm , βm , βm }. ◦(j+k) Lemma 22. If m is large enough, then gm (ξm ) = ωm . The idea of the proof is the following. For any polynomial in V, the fixed point α = 0 is the landing point of a unique external ray (of angle 0). If the polynomial has (k, `)-configuration, then f ◦` sends the critical point ω 0 to α with local degree 2 and f ◦k sends the critical point ω to ω 0 with local degree 2. It follows that ω 0 is the landing point of exactly two external rays and ω is the landing point of exactly four external rays. If in addition f is admissible, then f ◦j sends ξ to ω with local degree 1, and ξ is the landing point of exactly four external rays separating β, β 0 and β 00 . Note that f ◦(j+k) has a critical point at ξ with critical value ω 0 . For the perturbed map gm , the critical point ωm is the landing of exactly four ◦(j+k) external rays. The map gm has a critical point close to ξ with critical value ± close to ωm , but different from ωm . It follows that there are two points ξm close to ◦(j+k) ξ which are mapped to ωm by gm . Exactly four rays land at each of these two points. We shall see that those eight rays converge to the four rays landing at ξ for ± 0 00 f , and that one of the two points ξm separates βm , βm and βm in Jgm . Proof. Note that f ◦j : (C, ξ) → (C, ω) has local degree 1 at ξ, f ◦k : (C, ω) → (C, ω 0 ) has local degree 2 at ω and f ◦` : (C, ω 0 ) → (C, α) has local degree 2 at ω 0 . Let D b D b be sufficiently small disks around α = 0 so that Pf ∩ D b = {0}. Let D b0 −` b 0 b be the component of 00 be the component of f (D) which contains ω , and let D b 0 ) which contains ξ. For m large enough, f −j−k (D • gm−` (D) has a component Dm 0 bD b 0 containing ωm and gm ◦k 0 (ωm ), • gm−j−k (Dm0 ) has a component Dm 00 b Db 00 containing a point ζm and two ± points ξm such that ◦j 0 ◦(j+k) + ◦(j+k) − gm (ζm ) = ωm and gm (ξm ) = gm (ξm ) = ωm . ± As m tends to +∞, the sequence {ζm } and {ξm } converge to ξ. A summary of the dynamics within the preimages of D before and after perturbation is provided in Figure 12. Given θ ∈ T, denote by R(θ) the ray of angle θ for f , and by Rm (θ) the ray of angle θ for gm . There is a single ray landing at α: R(0). So, there are two rays landing at ω 0 , four rays landing at ω and four rays landing at ξ. Let θ1 , θ2 , θ3 and θ4 be the angles of the four rays landing at ξ, cyclically ordered counterclockwise. Then, modulo 1, we have that 3j+k θ1 = 3j+k θ3 =: η1 and 3j+k θ2 = 3j+k θ4 =: η2 , and the rays R(η1 ) and R(η2 ) land at ω 0 . In addition, modulo 1, we have that 3` η1 = 3` η2 = 0 and R(0) lands at α = 0. ◦` Since gm (ωm ) 6= 0 and Rm (0) lands at α = 0, for m large enough, the rays 0 ◦(k+`) 0 Rm (η1 ) and Rm (η2 ) land at two distinct points in Dm . Since gm (ωm ) = 0, ◦k 0 one of those rays lands at gm (ωm ). Without loss of generality, relabelling the JULIA SETS WITH A WANDERING BRANCHING POINT 19 1−1 2−1 2−1 D ◦j ξ f ω f ◦k ω0 α f ◦` 1−1 2−1 + ξm ◦j + gm (ξm ) ◦k gm (ω 0 ) 2−1 D ◦j ◦k gm gm ζm ω0 ◦` gm α 0 00 Dm ◦j − Dm gm (ξm ) ω ◦` − gm (ω) ξm Figure 12. Dynamics within the preimages of the disk D con- taining α before and after perturbation. rays if necessary, we may assume that this ray is Rm (η1 ). Then, Rm (θ1 ) and 00 Rm (θ3 ) land at ζm , whereas Rm (θ2 ) and Rm (θ4 ) land at two distinct points in Dm . Note that the four rays R(θ1 ), R(θ2 ), R(θ3 ) and R(θ4 ) land at ξ and separate the plane in four connected components. The points β, β 0 and β 00 are in 3 distinct connected components. So, 1/2, 1/6 and −1/6 must belong to distinct components of T \ {θ1 , θ2 , θ3 , θ4 }. Without loss of generality, relabelling the rays if necessary, we may assume that one of the angles 1/2, 1/6 and −1/6 belongs to (θ1 , θ2 )T , one belongs to (θ2 , θ3 )T and one belongs to (θ3 , θ1 )T (see Figure 13). 0 According to Lemma 18, the two rays landing at ωm separate α = 0 and βm . One has angle in (0, 5/12)T and the other has angle in (−5/12, 0)T (see Lemma 18). ◦` 0 Let us recall that gm (ωm ) = ω −m (gm ) is the m-th iterated preimage of ωm by the univalent branch gm : Vgm → Wgm . It follows from Lemma 19 that, if a ray in Wgm has angle in (0, 5/12)T (respectively (−5/12, 0)T ), then the preimage ray in Vgm has angle in (0, 5/36)T (respectively (−5/36, 0)T ). Consequently, the rays landing at ◦` gm (ωm ) have angles with representatives     + 5 − 5 m ∈ 0, and m ∈ − ,0 . 12 · 3m 12 · 3m They separate α = 0 and βm . ± Set δm := ± ` ± ± m /3 and ηm := δm /3 j+k . Then, for m large enough, each of the four rays + − + − Rm (η1 + δm ), Rm (η1 + δm ), Rm (η2 + δm ) and Rm (η2 + δm ) 0 ◦` ◦` land in Dm at a point z satisfying gm (z) = gm (ωm ). This point is necessarily ωm itself (see Figure 13). Similarly, each of the eight rays ± ± ± ± Rm (θ1 + ηm ), Rm (θ2 + ηm ), Rm (θ3 + ηm ) and Rm (θ4 + ηm ) 00 ◦(j+k) + land in Dm at a point z satisfying gm (z) = ωm . This point in necessarily ξm − + − or ξm . So, four of those rays land at ξm and four of them land at ξm . + ± − + If m is large enough, θ1 +ηm , θ2 +ηm and θ3 +ηm belong to (θ1 , θ3 )T and θ3 +ηm , ± − θ4 + ηm and θ1 + ηm belong to (θ3 , θ1 )T . The rays of angles θ1 and θ3 separate the ± plane in two connected components. So, relabelling the points ξm if necessary, we 20 X. BUFF, J. CANELA, AND P. ROESCH θ3 β0 η1 f ◦` - β α = f ◦` (ω 0 ) ξ θ2 0 ◦(j+k) ω =f (ξ) θ4 f ◦(j+k) β 00 - η2 θ1 − θ3 + ηm 0 − η1 + δm ◦` + m βm gm - βm + + θ2 + ηm ◦(j+k) ξm + gm (ζm ) η1 + δm α ζm + − ωm η2 + δm ◦` gm (ωm ) − ξm θ2 + ηm ◦(j+k) gm − + 00 βm - − m θ1 + ηm η2 + δm Figure 13. External rays of f landing near ξ, ω 0 and α and external rays of gm landing near ζm , ωm and α. The angles of the external rays are indicated. may assume that + − + − Rm (θ1 + ηm ), Rm (θ2 + ηm ), Rm (θ2 + ηm ) and Rm (θ3 + ηm ) + land at ξm and that + − + − Rm (θ3 + ηm ), Rm (θ4 + ηm ), Rm (θ4 + ηm ) and Rm (θ1 + ηm ) − land at ξm (see Figure 13). Finally, if m is large enough, one of the angles 1/2, 1/6 and −1/6 belongs + − + − to (θ1 + ηm , θ2 + ηm )T , one belongs to (θ2 + ηm , θ3 + ηm )T and one belongs to + − 0 00 (θ3 + ηm , θ1 + ηm )T . Then, the branching point separating βm , βm and βm in Jgm + is ξm = ξm .  References [1] A. F. Beardon. Iteration of rational functions, volume 132 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. Complex analytic dynamical systems. [2] A. Blokh and L. Oversteegen. Wandering gaps for weakly hyperbolic polynomials. In Complex dynamics, pages 139–168. A K Peters, Wellesley, MA, 2009. [3] L. Carleson and T. W. Gamelin. Complex dynamics. Universitext: Tracts in Mathematics. Springer-Verlag, New York, 1993. JULIA SETS WITH A WANDERING BRANCHING POINT 21 [4] Adrien Douady. Does a Julia set depend continuously on the polynomial? In Complex dynam- ical systems (Cincinnati, OH, 1994), volume 49 of Proc. Sympos. Appl. Math., pages 91–138. Amer. Math. Soc., Providence, RI, 1994. [5] J. Milnor. Dynamics in one complex variable, volume 160 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, third edition, 2006. [6] L. Rempe-Guillen. On prime ends and local connectivity. arXiv:math/0309022v6 (a preivious version of the manuscript appeared in Bull. Lond. Math. Soc. (2008)), 2014. [7] W. Thurston. On the geometry and dynamics of iterated rational maps. In Complex dynamics, pages 3–137. A K Peters, Wellesley, MA, 2009. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher (original manuscript ‘The combinatorics of iterated ratio- nal maps’ (1985)). [8] M. Torhorst. Über der Rand der einfach zusammenhägenden eben Gebiete. Math. Z., 9:44–65, 1921. [9] G. T. Whyburn. Analytic Topology. American Mathematical Society Colloquium Publications, v. 28. American Mathematical Society, New York, 1942. E-mail address: xavier.buff@math.univ-toulouse.fr Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France E-mail address: Jordi.Canela Sanchez@math.univ-toulouse.fr Laboratoire d’Analyse et de Mathématiques Apliquées, Université Paris-Est Marne- la-Vallée, 5, boulevard Descartes, 77454 Champs-sur-Marne, France E-mail address: pascale.roesch@math.univ-toulouse.fr Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France