DYNAMICAL DESSINS ARE DENSE

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1 DYNAMICAL DESSINS ARE DENSE CHRISTOPHER J. BISHOP AND KEVIN M. PILGRIM Abstract. We apply a recent result of the first author to prove the following result: any continuum in the plane can be approximated arbitrarily closely in the Hausdorff topology by the Julia set of a critically finite polynomial with two finite postcritical points. 1. Introduction Given compact subsets A, B C their Hausdorff distance d(a, B) is given by d(a, B) = inf{r : A N r (B), B N r (A)} where N r (A), N r (B) denote the r-neighborhoods of A and B, respectively. Given a polynomial g C[z], we denote by g n the nth iterate of g, and define its filled-in Julia set K(g) = {z : g n (z) }, basin of infinity A(g) = {z : g n (z) }, and Julia set J(g) = K(g). K. Lindsey and W. Thurston [LT, Theorem 1.4] have shown the following: Theorem 1. Given any bounded, simply connected set A C and any ɛ > 0, there exists a polynomial g C[z] such that (1) d(k(g), A) < ɛ (2) d(j(g), A) < ɛ (3) d(a(g), C A) < ɛ. Their proof is constructive; they apply their method to a simple Jordan domain A outlining the figure of a cat, yielding a polynomial g of degree 301. In this note, we generalize conclusion (2) of their theorem. Though not yet as practically implementable, our result strengthens the conclusion by showing that the polynomial g may be taken to be very special. Recall that a continuum is a compact connected set; it is a dendrite if in addition it is locally connected and contains no simple closed curves. A polynomial g C[z] is critically finite if P (g) = k 1 g n (C(g)) is finite, where C(g) = {c : g (c) = 0}. If g is critically finite, the following facts are known (see e.g. [M]): J(g) is connected and locally connected, and is a dendrite if and only if no element of C(g) is periodic. Our main result is 2010 Mathematics Subject Classification. MSC primary 37F10, secondary 14H57, 37B45. Key words and phrases. dessin, Julia set. 1

2 2 C. J. BISHOP AND K. M. PILGRIM Theorem 2. Given any continuum K C and any ɛ > 0, there is a critically finite polynomial g C[z] such that #P (g) = 2, the Julia set J(g) is a dendrite, and d(k, J(g)) < ɛ. The proof has three steps. Suppose K C is a continuum and ɛ > 0 is given. 1. A polynomial f is a Belyi polynomial if deg(f) > 1 and if its set of critical values f(c(f)) is contained in the set {0, 1}. We denote by BP the set of Belyi polynomials. If f BP, its dessin is D(f) := f 1 ([0, 1]). We regard D(f) as a planar tree with vertices V (f) := f 1 ({0, 1}); its edges e are the closures of the components of f 1 ((0, 1)). The first author has recently shown the following, which is the main result of [B]. Theorem 3. Given any continuum K C and any ɛ > 0, there exists f BP for which d(k, D(f)) < ɛ. Moreover, one may assume that for each v V (f), deg(f, v) Next, we apply the preceding theorem to obtain a polynomial f BP. Let q(z) = 4z(1 z). Note that q BP and that q([0, 1]) = q 1 ([0, 1]) = [0, 1]. For each n N, we have q n f BP and D(q n f) = D(f) as subsets of C. Their tree structures differ: each edge of D(f) is a union of 2 n edges of D(q n f). 3. By replacing f with q f we may assume that the local degree of f at each point in f 1 (1) is equal to two. Suppose v 0, v 1 V (f) are leaves of D(f), that is, vertices of valence 1. Note that by our assumption, f(v 0 ) = f(v 1 ) = 0. Fix n N. Following the second author [P] we turn q n f into a dynamical system. There is a unique complex affine map A : C C for which A(0) = v 0 and A(1) = v 1. Let g = A q n f; thus g has two branch values, namely {v 0, v 1 }. Abusing notation slightly, we denote V (g) = g 1 ({v 0, v 1 }). Each critical point of g maps either to v 0 or to v 1 ; by construction, v 0 = g(v 0 ) = g(v 1 ), and g (v 0 ) 0. It follows that P (g) = {v 0, v 1 }, so that g is critically finite, and that every critical point lands on the fixed point v 0 under iteration of g. It is a general fact that all fixed points of a critically finite map g are either critical points or they lie in the Julia set. We conclude v 0 J(g) and that J(g) is a dendrite. Figure 1. At left: the dessin of f(z) = z 3 subdivided n = 5 times. At right: g 1 ([v 0, v 1 ]), which is an approximation of J(g); its greater apparent thickness is an artifact of plotting the approximately vertices. Images courtesy of Don Marshall.

3 DYNAMICAL DESSINS ARE DENSE 3 The proof of Theorem 2 then rests upon establishing the closeness that Figure 2 suggests: Lemma 1. The Hausdorff distance d(d(f), J(g)) 0 as n. 2. Proof of Lemma 1 Suppose f, q, n, g are as in Step 3 of the outline given in the Introduction. Lemma 2. The maximum diameter of an edge e of D(q n f) tends to zero as n. Proof. An easy exercise shows the conclusion holds for the polynomial q itself. Suppose f BP. Since the inverse branches of f are uniformly continuous on (0, 1), the general conclusion holds. Denote D = D(f) and M = diam(d). In the following paragraph, we cover D by a pair of Jordan domains: one containing v 0 but not v 1, the other containing v 1 but not v 0. We will show that as n the diameters of the preimages of these domains under g tend to zero. We will need some Koebe space around these domains. See Figure 2. W 1 v 0 v 1 Figure 2. Caricature of W 1. The domain W 0 is similar. The domain Ŵ 1 is the portion of the disk to the right of the vertical segment. Notation: B(a, r) = { z a < r}. Let v 0 = 7v 0 + v 1, v 0 = 3v 0 + v v 1 = v 0 + 7v 1, v 1 = v 0 + 3v 1 8 ( 4 ) v1 + v 0 W := B, 10M 2 Ŵ 1 i = W { z v i < z v i }, i = 0, 1 ( ) v0 + v 1 W 1 i = B, 9M { z v i < z v i }, i = 0, 1. 2 By construction, Ŵi {v 0, v 1 } = v i, i = 0, 1;

4 4 C. J. BISHOP AND K. M. PILGRIM D W 0 W 1 ; Ŵi \ W i is an annulus, i = 0, 1; each component of g 1 (Ŵ0), g 1 (Ŵ1) contains precisely one element ṽ of V (g). Thus for each ṽ V (g) we have a proper map of pairs g : ( Ŵ ṽ, Wṽ) (Ŵ, W ) where W = W 0 or W 1. Lemma 3. max{diam( Wṽ) : ṽ V (g)} 0 as n. Proof. Put ( Ŵ, W ) = ( Ŵ ṽ, Wṽ). The control on the local degrees of the polynomial f in Theorem 3 shows that k = deg(g, ṽ) {1, 2, 4}. Thus the restriction g : Ŵ Ŵ is proper and is ramified only at ṽ, hence has degree k. Up to precomposition with a rotation, there is a unique Riemann map φ : (B(0, 1), 0) (Ŵ, ṽ). Hence there exist 0 < r < s < 1 independent of i, n, ṽ such that if U = φ 1 (W ), then B(0, r) U B(0, s) B(0, 1). Put Ũ = {z B(0, 1) zk U} so that B(0, r 1/k ) Ũ B(0, s1/k ) B(0, 1). The Riemann map φ : B(0, 1) Ŵ lifts to a Riemann map φ : B(0, 1) Ŵ such that φ(ũ) = W : φ : (B(0, 1), Ũ, 0) ( Ŵ, W, ṽ). The rescaled map ψ = φ (0) 1 ( φ φ(0)) is an element of the class S of Schlicht functions: injective holomorphic maps ψ : B(0, 1) C with the normalization ψ(0) = 0, ψ (0) = 1. By [A, Theorem 5.3], for all z B(0, 1) and all Schlicht functions ψ, z (1 + z ) 2 ψ(z) z (1 z ) 2. Hence upon setting we have ρ = r 1/k (1 + r 1/k ) 2, σ = s 1/k (1 + s 1/k ) 2, δ = φ (0) B(ṽ, ρδ) W B(ṽ, σδ). Let e be any one of the k components of g 1 ((v 0, v 1 )) whose closure meets ṽ. Since (0, 1) W, we have e W, so which implies ρδ < diam(e) σδ < diam(e) σ ρ and so diam( W ) 2σδ < 2 diam(e) σ ρ 0 as n, by Lemma 2. The constants ρ, σ are independent of n and of the choice of ṽ, so the proof is complete.

5 DYNAMICAL DESSINS ARE DENSE 5 Denote J = J(g). Pick ɛ < 1 2 sup{ a b : a D, b C \ W 0 W 1 }. Apply Lemma 3 to obtain n so that diam( Wṽ) < ɛ for all ṽ V (g). On the one hand, g 1 (W 0 W 1 ) = Wṽ }{{} N ɛ (D) W 0 W 1 ṽ V Lemma 2 and so W 0 W 1 is backward-invariant under g. It is a general fact that J may be equivalently defined as the smallest closed subset of C satisfying #J > 1 and g 1 (J) J; see [M]. Thus J W 0 W 1. By invariance of J we have then J g 1 (W 0 W 1 ) = Wṽ N ɛ (D). ṽ V On the other hand, V J, so N ɛ (J) N ɛ (V ) ṽ V Wṽ D. This completes the proof of Lemma 1 and establishes Theorem Acknowledgements The first author was supported by NSF grant DMS The second author was supported by Simons foundation grant The authors thank the Institute for Pure and Applied Mathematics for hosting the workshop which led to this collaboration. References [A] Lars V. Ahlfors. Conformal Invariants. McGraw-Hill Series in Higher Mathematics, [LT] Kathryn A. Lindsey and William P. Thurston, Shapes of polynomial Julia sets. Preprint. arxiv: , [B] Christopher J. Bishop, Approximation by critical points of generalized Chebyshev polynomials. Preprint. [M] John Milnor. Dynamics in one complex variable, 3rd edition. Princeton University Press,???. [P] Kevin M. Pilgrim. Dessins d enfants and Hubbard trees. Ann. Sci. L Ecole Normale Superieure 4e serie t. 33(2000), address: bishop@math.sunysb.edu Department of Mathematics, Stony Brook University, Stony Brook, NY USA address: pilgrim@indiana.edu Dept. Math., Indiana University, Bloomington, IN USA