Notes on Tan s theorem on similarity between the Mandelbrot set and the Julia sets

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1 Notes on Tan s theorem on similarity between the Mandelbrot set and the Julia sets Tomoi Kawahira Abstract This note gives a simplified proof of the similarity between the Mandelbrot set and the quadratic Julia sets at the Misiurewicz parameters, originally due to Tan Lei [TL]. We also give an alternative proof of the global linearization theorem of repelling fixed points. 1 Simlarity between M and J We first give a proof of Tan s theorem, in a way inspired by Schwic [Sch]. The Julia sets and the Mandelbrot set. Let us consider the quadratic family { fc (z) = z 2 + c : c C }. The Mandelbrot set M is defined by M := {c C : f n c (c) 2 ( n N)}. For each c C, the filled Julia set K c is defined by K c := {z C : f n c (z) max{2, c } ( n N)}. The Julia set J c is the boundary of K c. It is nown that all M, K c, and J c (for any c C) are non-empty compact sets. Misiurewicz parameters. We say c M is a Misiurewicz parameter if the forward orbit of c by f c eventually lands on a repelling periodic point. More precisely, there exist minimal l, p 1 such that a := fc l (c ) satisfies a = fc p (a ) and (fc p ) (a ) > 1. Note that the repelling periodic point a is stable: that is, there exists a neighborhood V of c with the following property: there exists a conformal map a : c a(c) on V such that a(c ) = a ; a(c) = fc p (a(c)); and (fc p ) (a(c)) > 1. In the following we set λ(c) := (fc p ) (a(c)) and λ := λ(c ). A ey lemma. planes: ver This is our ey lemma, which bridges the dynamical and paramter 1

2 Lemma 1 Suppose c M is a Misiurewicz parameter as above. For N, set ρ := (fc l+p ) (c ). Then we have: (1) The function ϕ (w) = fc l+p (c + ρ w) converges to a non-constant entire function ϕ : C C as uniformly on any compact sets. (2) There exists a constant Q such that the function Φ (w) := f l+p c +Qρ w (c + Qρ w) converges to the same function ϕ(w) as uniformly on compact sets of C. Proof. It is well-nown (see Section 2 and [Mi, Cor.8.12]) that the sequence of functions (polynomials, indeed) ( w fc p a + w ) ( N) λ converges uniformly on compact sets of C to an entire function ϕ(w) with ϕ(λ w) = f p c ϕ(w). (We do not use this functional equation later.) Such a ϕ is called a Poincaré function. Note that this function satisfies ϕ() = a, ϕ () = 1. Now let us show (1): set A := (fc l ) (c ), where A since otherwise c is strictly periodic. We also have ρ = 1/(A λ ). For sufficiently small t C, we have the expansion fc l (c + t) = a + A t + o(t). Fix an arbitrarily large compact set E C and tae any w E. Then by setting t = w/(a λ ), ( fc p a + w ) ( f l+p λ c c + w ) + o(λ A λ ) fc l+p (c + ρ w) = ϕ (w) when. (Here by A (w) B (w) we mean A (w) B (w) uniformly on E as.) Hence we have ϕ(w) = lim ϕ (w) on any compact sets. Note that ϕ has no poles, since every ϕ is all entire. Next we show (2): suppose that Q C is a constant and set c = c(w) := c +Qρ w. We also set Φ (w) := fc l+p (c) and b(c) := fc(c). l Recall that a(c) denotes a repelling periodic point (of f c ) of period p with a(c ) = a, and λ(c) denotes its multiplier. Then the theorem by Kœnigs again implies that the sequence of functions ϕ c (w) := fc p (a(c) + w/λ(c) p ) converges to an entire function ϕ c (w) uniformly on compact sets. In particular, by the proof of the theorem in Section 2, it is not difficult to chec that the function c ϕ c (w) is holomorphic near c = c when we fix a w C. As in Tan s original proof, we employ a theorem on transversality by Douady and Hubbard [DH]: There exists a B such that b(c) a(c) = B (c c ) + o(c c ). Hence for c = c + Qρ w (taing w in a compact set), we have b(c) = a(c) + B Qρ w + o(ρ ) = a(c) + B Q A λ(c)p λ p w λ(c) p + o(ρ ). 2

3 Set Q := A /B. Since λ(c) is a holomorphic function of c and thus λ(c) = λ +O(c c ), we have λ(c)/λ 1 = O(c c ). This implies that log λ(c)p λ p for c = c + Qρ w = c + O(λ p ). Since ( ) = O(c c ) = O ( ) λ p Φ (w) = fc p (b(c)) = fc p ( a(c) + w ) λ(c) + o(ρ ) p and ϕ c (w) ϕ(w) as c c (uniformly when w is on a compact set), we conclude that lim Φ (w) = ϕ(w), where the convergence is uniform on any compact sets. The Hausdorff topology. Let us briefly recall the Hausdorff topology of the set of non-empty compact sets Comp (C) of C. For a sequence {K } N Comp (C), we say K converges to K Comp (C) as if for any ϵ >, there exists N such that if, K N ϵ (K ) and K N ϵ (K), where N ϵ ( ) is the open ϵ neighborhood in C. Set D(r) := { z < r}. For a compact set K in C, let [K] r denote the set (K D(r)) D(r). For a C and b C, let a(k b) denote the set of a(z b) with z K. Similarity. Let c be a Misiurewicz parameter. To simplify the notation we set f := f c and J := J c. Now our main theorem is: Theorem 2 (Similarity between M and J) There exist an entire function ϕ on C, a sequence ρ, and a constant q such that if we set J := ϕ 1 (J) C, then for any large constant r >, we have (a) [ ρ 1 (J c ) ] [J ] r r (b) [ qρ 1 (M c ) ] [J ] r r as in the Hausdorff topology. Actually ϕ, ρ ( N) and 1/q = Q are all given in (the proof of) Lemma 1. Proof of (a). Let ϕ, ϕ, and ρ = 1/(A λ ) be as given in the proof of Lemma 1. Since f n (J) = J, we have [ ρ 1 (J c ) ] = [ ϕ 1 r (J)]. By [J ] r r = [ϕ 1 (J)] r and uniform convergence of ϕ ϕ on D(r), the claim easily follows. 3

4 Figure 1: The central panel shows small peaces of M (in gray) and J c (in blac) centered at a Misiurewicz parameter c in the same coordinate. By zooming out, we will see that they are actually different sets. Proof of (b). Set q := 1/Q (where Q is defined in the proof of Lemma 1) and M := qρ 1 (M c ). Fix any ϵ >. Since the set D(r) N ϵ (J ) is compact, there exists an N = N(ϵ) such that f N ϕ(w) > 2 for any w D(r) N ϵ (J ). By uniform convergence of Φ (w) = f l+p c +Qρ w (c + Qρ w) ϕ(w) on compact sets in C (Lemma 1), we have f (l+p)+n c +Qρ w (c + Qρ w) > 2 for all. This implies that c + Qρ w / M, equivalently, w / M. Hence we have [M ] r N ϵ ([J ] r ). Next we show the opposite inclusions for large enough. Let us approximate [J ] r by a finite subset E of [J ] r such that the ϵ/2 neighborhood of E covers [J ] r. Now it is enough to prove that for any w E, there exists a sequence w [M ] r such that w w < ϵ/2 for. Let be a dis of radius ϵ/2 centered at w. When D(r), we can tae such w in D(r). Hence we may assume that D(r). Since ϕ(w ) J and repelling cycles are dense in J (see [Sch] and [Mi]. See also the remar below), we can choose a w such that ϕ(w ) is a repelling periodic point of some period m and w w < ϵ/4. This implies that the function χ : w f m (ϕ(w)) ϕ(w) has a zero at w = w. Let us consider the function χ : w fc m +Qρ w (Φ (w)) Φ (w), where Φ (w) = f l+p c +Qρ w (c + Qρ w) as in Lemma 1. By uniform convergence of Φ to ϕ on compact sets of C, χ has a zero w in and w w < ϵ/4 for all. In particular, 4

5 c := c + Qρ w satisfies fc m+(l+p) (c ) = fc l+p (c ) and thus c M. Hence we have a desired w M with w w < ϵ/2. Remars. Note that ϕ(w ) need not be repelling. Thus we only need the density of periodic points in the Julia set, which is an easy consequence of Montel s theorem. See [Mi, p.157]. The proof can be extended to semi-hyperblic parameters. See my forthcoming paper [Ka]. 2 A proof of the global linearization theorem ) Here we give a proof for the fact used in the proof of Lemma 1 that w fc p (a + w λ converges as. Our proof is based on the normal family argument and the univalent function theory (see [Du] for example), which is different from the proof given in [Mi, Cor.8.12]. Proposition 3 Let g : C C be an entire function with g() =, g () = λ, and λ > 1. Then the sequence ϕ n (w) = g n (w/λ n ) converges uniformly on compact sets in C. Moreover, the limit function ϕ : C C satisfies g ϕ(w) = ϕ(λw) and ϕ () = 1. Proof. Since g(z) = λz + O(z 2 ) near z =, there exists a dis = D(δ) = { z < δ} such that g is univalent and g( ). Hence we have a univalent branch g 1 of g that maps into itself. First we show that ϕ n is univalent on D(δ/4): Since the map ϕ 1 n : w λ n g n (w) is well-defined on = D(δ) and univalent, its image contains D(δ/4) by the Koebe 1/4 theorem. Hence ϕ n is univalent on D(δ/4), and by the Koebe distortion theorem, the family {ϕ n } n is locally uniformly bounded on D(δ/4) and thus equicontinuous. Next we show that ϕ n has a limit on D(δ/4): Fix an arbitrarily large r > and an integer N such that r < δ λ N /4. By using the Koebe 1/4 theorem as above, the function G N, (w) := λ N g (w/λ N+ ) ( N) satisfying ϕ N+ = ϕ N G N, is univalent on the dis D(δ λ N /4). By the Koebe distortion theorem, there exists a constant C > independent of N and such that for any w D(r) and sufficiently large N we have G N, (w) 1 C w / λ N. By integration we have G N, (w) w Cr 2 /(2 λ N ) on D(r). In particular, G N, id uniformly on D(δ/4) as N. Since the family {ϕ n } is equicontinuous on D(δ/4), the relation ϕ N+ = ϕ N G N, implies that {ϕ n } n is Cauchy and has a unique limit ϕ on any compact sets in D(δ/4). Let us chec that the convergence extends to C: (We will not use the functional equation g n ϕ(w) = ϕ(λ n w). Compare [Mi, Cor.8.12].) Since ϕ N+ (w) ϕ N (w) = ϕ N (G N, (w)) ϕ N (w) and G N, (w) w = Cr 2 /(2 λ N ) on D(r), it follows that the family {ϕ N+ } (with fixed N) is uniformly bounded on D(r). Hence {ϕ n } n is normal on any compact set in C and any sequential limit coincides with the local limit ϕ on D(δ/4). 5

6 The equation g ϕ(w) = ϕ(λw) and ϕ () = 1 are immediate from g ϕ n (w) = ϕ n+1 (λw) and ϕ n() = 1. Remar. One can easily extend this proof to the case of meromorphic g by using the spherical metric. References [DH] A. Douady and J. Hubbard. On the dynamics of polynomial-lie mappings. Ann. Sci. Éc. Norm. Sup. 18(1985), [Du] P.L. Duren. Univalent Functions. Springer-Verlag, [Ka] T. Kawahira. Quatre applications du lemme de Zalcman à la dynamique complexe. Preprint, 21. [Mi] J. Milnor. Dynamics in one complex variable. (3rd edition). Ann. of Math. Studies 16, Princeton University Press, 26. [Sch] N. Schwic. Repelling periodic points in the Julia set. Bull. London Math. Soc. 29(1997), no. 3, [TL] Tan L. Similarity between the Mandelbrot set and Julia sets. Comm. Math. Phys. 134(199), [Za] L. Zalcman. A heuristic principle in function theory. Amer. Math. Monthly, 82(1975),