Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set

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1 Simple Proofs for the Derivative Estimates of the Holomorphic Motion near Two Boundary Points of the Mandelbrot Set arxiv: v [math.ds] 5 Aug 08 Yi-Chiuan Chen and Tomoki Kawahira August 8, 08 Abstract For the complex quadratic family q c : z z +c, it is known that every point in the Julia set J(q c ) moves holomorphically on c except at the boundary points of the Mandelbrot set. In this note, we present short proofs of the following derivative estimates of the motions near the boundary points /4 and : for each z = z(c) in the Julia set, the derivative dz(c)/dc is uniformly O(/ /4 c) when real c /4; and is uniformly O(/ c) when real c. These estimates of the derivative imply Hausdorff convergence of the Julia set J(q c ) when c approaches these boundary points. In particular, the Hausdorff distance between J(q c ) with 0 c < /4 and J(q /4 ) is exactly /4 c. Keywords. quadratic map, holomorphic motion, Hausdorff convergence. Introduction For the family of quadratic maps q c of C, z z + c, with c a complex number not locating on the boundary of the Mandelbrot set M, it is well-known that every point in the Julia set J(q c ) moves holomorphically with respect to c, i.e. the holomorphic motion [L, MSS]. Note that J(q c ) is a Cantor set when c M and is connected when c M. A parameter c is called hyperbolic if the orbit of the origin accumulates on an attracting cycle or diverges to infinity. A hyperbolic component is a connected component of C M containing hyperbolic parameters. It is conjectured that the set C M consists of only the hyperbolic parameters. See [DH, Exposé I] for example. 00 Mathematics Subject Classification. Primary 37F45; Secondary 37F99.

2 Let D denote the open disk of radius one centered at the origin. There is a biholomorphic function Φ from C M to C D with which the set R(θ) := {Φ (re iπθ ) < r } is defined and called the parameter ray of angle θ T = R/Z of the Mandelbrot set M. Given θ, if lim r Φ (re iπθ ) exists, then this limit is called the landing point of the parameter ray R(θ). A parameter ĉ in M is called semi-hyperbolic if the critical point is non-recurrent and belongs to the Julia set [CJY]. A typical example is a Misiurewicz parameter, that is, for which the critical point eventually lands on a repelling periodic point. The set consisting of semi-hyperbolic parameters is dense with Hausdorff dimension in M [S]. For each semi-hyperbolic parameter ĉ M, there exists at least one parameter ray R(θ) landing at ĉ. (See [D, Theorem ].) In a recent paper [CK], we proved the following result concerning the estimate for the derivative of the holomorphic motion. Theorem.. Let ĉ M be a semi-hyperbolic parameter that is the landing point of R(θ). Then there exists a constant K > 0 that depends only on ĉ such that for any c R(θ) sufficiently close to ĉ and any z = z(c) J(q c ), the point z(c) moves holomorphically with dz(c) dc K c ĉ. () This result enables us to obtain one-sided Hölder continuity of the holomorphic motion along the parameter ray (i.e. the holomorphic motion lands). More precisely, let ĉ M be a semi-hyperbolic parameter that is a landing point of R(θ), and let c = c(r) := Φ (re iπθ ) with r (, ]. Then for any z(c()) in J(q c() ), the improper integral +δ dz(c) z(ĉ) := z(c()) + lim δ 0 dc dc(r) dr exists in the Julia set J(qĉ). In particular, z(c) is uniformly one-sided Hölder continuous of exponent / at c = ĉ along R(θ): There exists a constant K depending only on ĉ such that for any c = c(r) R(θ) with < r. z(c) z(ĉ) K c ĉ () The primary aims of this paper are two-folds. The first is to show that the same estimate as () holds when c approaches /4, which is a parabolic parameter, along the real line in the interior of the Mandelbrot set, namely dr

3 Theorem.. For any point z = z(c) in the Julia set J(q c ), we have that z(c) moves holomorphically with derivative dz(c) dc /4 c as c /4 along the real line in the interior of M. (3) Figure : Top: The Julia set J(q c ) for c = k/0 (k = 0,,, 5). analytic motion of the preimages of the repelling fixed point for 0 c < /4. Bottom: Real The second is to present a simple proof of Theorem. for the case ĉ =. (See Remark.4.) As a matter of fact, what we present is a proof of Theorem. for the logistic map f µ : C C, z µz( z), with the semi-hyperbolic (Misiurewicz) parameter µ = 4 case, stated as follows. Theorem.3. For z = z(µ) in the Julia set J(f µ ), the point z(µ) moves holomorphically with derivative as µ 4 along the real line. ( ) dz(µ) dµ = O µ 4 3

4 Figure : Real analytic motion of the Julia set J(f µ ) for µ 4. See Figure of [CK] for the corresponding motion for q c for c. The proof of Theorem. in [CK] relies on the hyperbolic metric, the hyperbolicity of the ω-limit set of a semi-hyperbolic parameter ĉ, the John condition on C J(qĉ), and the asymptotic similarity between J(qĉ) and M at ĉ. To the best of our knowledge, the difficulty of the proof for parameter ĉ = is essentially the same as the general semihyperbolic parameter case. Moreover, the difficulty of proof remains unchanged even if we consider quadratic maps in the form of logistic maps for general semi-hyperbolic parameter case. To our surprise, however, we find that for the logistic map with µ = 4 case, the difficulty of the proof can be substantially reduced. We only need a singular metric, and the proof is very straightforward. This is one of the motivations of this paper. Remark.4. The logistic map f µ is affinely conjugate to q c via the conjugacy G(, µ) : z w = µz + µ (so that G(, µ) f µ G(, µ) = q c ) (4) with µ( µ) c = (µ 0). (5) 4 Fix a point z 0 J(f µ ), and let w 0 = G(z 0, µ) J(q c ). When µ varies, let z(µ) be the holomorphic motion for f µ, and w(c) be the corresponding holomorphic motion for q c via the above relation (5). From the conjugacy (4), the derivative dw(c)/dc can be 4

5 obtained from dz(µ)/dµ: dw(c) dc = G z dz dµ dµ dc + G µ dµ dc ( = µ dz dµ z + ) µ (µ 0 or ). (6) Also, we have dz(µ) dµ = µ µ dw dc + w µ (µ 0 or ). (7) Theorem. for ĉ = can be derived by using (6): The fact that µ 4 means c leads to ( ) dw(c) dc = O (by Theorem.3) µ 4 ( ) = O (µ + )(µ 4) ( ) = O c (by the identity (5)) as c, in which we have used z + / = w/µ and the uniform boundedness of J(q c ). (An estimate of the size of the Julia set is given in Lemma 3..) Remark.5. The estimates in theorems. and.3 are optimal. For example, let z = z(c) be the repelling fixed point of q c for 0 c < /4, then dz(c) dc = /4 c. (This means that the equality in (3) is attained.) Also, let z = z(µ) be a pre-image point of J(f µ ), then dz(µ) dµ = ± µ µ µ 4. Hausdorff convergence and dynamical degeneration In this section, we discuss the convergence of the Julia set J(q c ) and the degeneration of dynamics of q c on J(q c ) when c approaches the boundary points c = /4 or of the Mandelbrot set. We also discuss corresponding properties for the map f µ on its Julia set J(f µ ) by using (4) and (5). The equality (5) shows that the map from real µ to real c in (, /4] is two-fold, therefore µ or µ when c /4, and µ 4 or µ as c (or more precisely, µ = ± 4ɛ when c = /4 ɛ and 5

6 µ = ± 9 + 4ɛ when c = ɛ, with ɛ > 0). It is easy to see that the dynamics of f 4ɛ is a trivial copy of that of f + 4ɛ, and the same triviality holds for f 9+4ɛ and f + 9+4ɛ. Hence, we shall restrict our discussion to the cases µ and µ 4 only.. c /4 or µ Theorem. imples one-sided Hölder continuity of the holomorphic motion as c /4 along the real line: The improper integral /4 δ dz(c) z(/4) := z(0) + lim δ 0 0 dc exists in the Julia set J(q /4 ). In particular, for any real c [0, /4). z(c) z(/4) /4 c (8) It is well-known that q c is hyperbolic for c in the hyperbolic components of M. Therefore, (J(q c ), q c ), the restriction of q c to J(q c ), is topologically conjugate to (J(q 0 ), q 0 ) via a conjugacy h c ( ; 0) from J(q 0 ) to J(q c ) for c in the main cardioid of M. Inequalities (3) and (8) lead to a result that the conjugacy h c ( ; 0) converges uniformly to a semiconjugacy h /4 ( ; 0) : J(q 0 ) J(q /4 ), z(0) z(/4), as c increases from 0 to /4. As a matter of fact, h /4 ( ; 0) is a conjugacy (see [K] for example). So, (J(q /4 ), q /4 ) is topologically conjugate to (T, T ), where is the angle-doubling map. T : T T, t t (mod ) dc Hausdorff distance. The estimate (8) and the existence of the conjugacy above between (J(q c ), q c ) and (J(q /4 ), q /4 ) imply that the Hausdorff distance between J(q c ) and J(q /4 ) is at most /4 c. The distance is also at least /4 c, because the distance between the parabolic fixed point / J(q /4 ) and the Julia set J(q c ) is attained by the repelling fixed point ( + 4c)/ = / + /4 c of q c. (Indeed, in the proof of Theorem. we will show that z ( + 4c)/ for any z J(q c ).) Hence we obtain the following: Corollary.. For 0 c < /4, the Hausdorff distance between the Julia sets J(q c ) and J(q /4 ) is exactly /4 c. 6

7 Remark.. From (7) and Theorem., when < µ < we get dz(µ) dµ µ µ c (by Lemma 3.) 4c µ = µ µ µ (by the identity (5)) µ +. This unexpected result implies that the Hausdorff distance between J(f µ ) and J(f ) is at most ( + )(µ )/ as µ along the real axis. Notation. When variables X, Y 0 satisfy X/C Y CX with a uniform constant C >, we denote this by X Y. Remark.3. Let dim H J(q c ) denote the Hausdorff dimension of J(q c ). It has been known from [HZ] that < dim H J(q 4 ) < 3 and that there exists c 0 < /4 such that for all c [c 0, /4) one has ( ) dimh J(q d 4 ) 3 dc dim H J(q c ) 4 c. (9) Therefore, the derivative of the Hausdorff dimension of the Julia set J(q c ) with respect to c tends to infinity from the left of /4, and the graph of dim H J(q c ) versus c has a vertical tangent on the left at /4. The unexpected result in Remark. suggests us to examine whether or not a similar vertical tangency holds for the graph of the Hausdorff dimension dim H J(f µ ) as µ decreases to. What we find is that the graph dim H J(f µ ) versus µ has a horizontal tangent on the right at. (See the proposition below.) Proposition.4. There exists µ 0 > such that for any µ (, µ 0 ]. d dµ dim H J(f µ ) (µ ) dim H J(f ) Proof. For µ 0, the affine map G(, µ) in (4) sends J(f µ ) to J(q µ( µ)/4 ), thus dim H J(f µ ) = dim H J(q µ( µ) ) 4 for µ 0. In particular, dim H J(f ) = dim H J(q ). 4 7

8 Consequently, for µ 0 or, d dµ dim H J(f µ ) = d dc dim H J(q c ) µ ( ) dimh J(f µ( µ) ) 3 µ 4 4 ( ) dimh J(f µ ) = (µ ) dim H J(f ), (by (9)) as asserted. The value of µ 0 can be obtained by solving c 0 = µ 0 ( µ 0 )/4 from (5) with the c 0 in Remark.3.. c or µ 4 For c in the exterior of M, the restriction of q c to J(q c ) is topologically conjugate to the one-sided left shift with two symbols. Therefore, there exists a conjugacy h c ( ; c 0 ) from J(q c0 ) to J(q c ) for any fixed c 0 and c not belonging to M. The results () and () give rise to a consequence that for any semi-hyperbolic parameter ĉ M, any parameter ray R(θ) landing at ĉ, and any c 0 R(θ), the conjugacy h c ( ; c 0 ) converges uniformly to a semiconjugacy hĉ( ; c 0 ) : J(q c0 ) J(qĉ), z(c) z(ĉ), (0) as c ĉ along R(θ). This further implies that the Hausdorff distance between J(q c ) and J(qĉ) is O( c ĉ ) as c ĉ along R(θ). Let Σ := { } s = {s 0, s, s,...} s n = 0 or for all n 0 be the space consisting of sequences of 0 s and s with the product topology, and σ be the left shift in Σ, σ(s) = s = {s 0, s, s, } with s i = s i+. Fix θ T {0}, the two points θ/ and (θ+)/ divide T into two open semi-circles T θ 0 and T θ with θ T θ 0. Let θ be such an angle that T n (θ) { θ, } θ+ for all n 0. Define the kneading sequence of θ under T as E(θ) = {E(θ) n } n 0 Σ with 0 for T n (θ) T θ 0 E(θ) n = for T n (θ) T θ. Note that the kneading sequence of non-recurrent θ is well-defined. A point e Σ is said to be aperiodic if σ n (e) e for any n 0. Two points a and s in Σ are said to be equivalent with respect to aperiodic e Σ, denoted by a e s, if there is k 0 such that a n = s n for all n k and σ k+ (a) = σ k+ (s) = e. 8

9 In [CK], we proved that the semiconjugacy hĉ( ; c 0 ) described in (0) leads to the following result. Let ĉ be a semi-hyperbolic parameter with an external angle θ and e = E(θ) be the kneading sequence of θ. Then (J(qĉ), qĉ) is topologically conjugate to (Σ/ e, σ), where σ is induced by the shift transformation σ. The dynamical degeneration for q c as c along real axis, namely, along R(θ) with θ = / is the same as the one for the logistic map f µ as µ 4 along the real axis. Since E(/) = {0,,,...}, the dynamical degeneration of q c at c = or f µ at µ = 4 is that both (J(q ), q ) and (J(f 4 ), f 4 ) are topologically conjugate to (Σ/ {0,,,...}, σ). Note that there is another way to interpret the kneading sequence: For the logistic map f µ, the Julia set J(f µ ) is a Cantor set contained in the real interval [0, ] when µ is real and greater than 4. When µ = 4, J(f 4 ) is the whole interval [0, ]. If the critical point / belongs to the Julia set, one can define the kneading sequence I(f µ ) = {I(f µ ) n } n 0 for f µ by I(f µ ) n = if fµ +n (/) [0, /] J(f µ ) and I(f µ ) n = 0 if fµ +n (/) [/, ] J(f µ ). The sequence I(f µ ) is well-defined if fµ +n (/) / for all n 0. Then, it is not difficult to see that E(/) = I(f 4 ). 3 Proof of Theorem. Assume 0 c < /4. Let r = ( + 4c)/, the distance of the repelling fixed point of q c from the critical point 0. Suppose z J(q c ) and z < r. Then, q c (z) = z + c < r + c = r. Thus, q n c (z) < r for all n 0. Since this is an open condition, this means that z belongs to the filled Julia set but not to the Julia set, a contradiction. Therefore, inf z = r, z J(q c) and Dq c (z) = z + 4c for all z J(q c ) (where D = d/dz). For any c not belonging to M or in a hyperbolic component of M, and for any z = z(c) J(q c ), in [CK] we proved the following derivative formula dz(c) dc = n= Dq n c (z(c)). Hence, dz(c) dc Dq n n c (z) ( ) n = n + 4c /4 c. Remark 3.. (i) Compared with our proof of Theorem. in [CK] for c approaching a semi-hyperbolic 9

10 parameter, or even with that of Theorem.3, to come in the next section, we find that the proof of Theorem. is surprisingly simple. (ii) In fact, by combining with the lemma below, the Julia set J(q c ) locates inside the annulus {z ( + 4c)/ z ( + + 4c)/} for 0 c < /4. Lemma 3.. z ( c )/ for any z J(q c ) and c C. Proof. Let M = M(c) = ( c )/ and notice that M M c = 0. If z is such a point that z = M + R for some R > 0, then q c (z) z = z + c z z z c = (M )R + R. This implies that the orbit of z tends to infinity thus z J(q c ). 4 Proof of Theorem.3 Note that f µ () = 0 and the origin z = 0 is a repelling fixed point of multiplier µ for any real µ 4. Hence there exists a linearizing coordinate φ µ : Ũ0 C defined on a fixed neighborhood Ũ0 of 0 such that. for any z f µ (Ũ0) Ũ0, φ µ (f µ (z)) = µφ µ (z).. φ µ (z) z when µ is sufficiently close to 4. Fix a point z = z(µ) in the Julia set J(f µ ) for µ > 4. Set f := f µ and z n = z n (µ) := f n µ (z) (n 0). As in [CKLY], we can derive dz n+ dµ = Df(z n) dz n dµ + µ z n+ (where D = d/dz) and thus we have a formal expansion dz dµ = z n µ Df n (z). () n Now suppose that µ is sufficiently close to 4. We will show that the formal expansion above converges absolutely. The main idea is to consider a fixed singular metric of the following explicit form γ(z) dz := dz z z on C {0, } (inspired by [Ka]; see also [R]). Let us fix a z J(f) (0, ). Then it is easy to see γ(z). Moreover, γ(f(z)) Df(z) γ(z) = µ z / f(z). 0

11 Note that f(z) J(f) [0, ], hence f(z) 0. By f(z) = µ/4 µ(z /), we have γ(f(z)) Df(z) γ(z) The right hand side takes its infimum as f(z) 0: = µ/4 f(z) f(z). γ(f(z)) Df(z) γ(z) µ 4 = µ =: A (= + O(µ 4)). Hence for any fixed z = z 0 J(f) whose forward orbit never lands on the fixed point 0, with the help of the identity we obtain n γ(z n ) γ(z) Df n (z) = k=0 γ(z k+ ) γ(z k ) Df(z k), Df n (z) γ(z n) A n γ(z) γ(z n) A γ(z n) n A. () n Now we have dz dµ z n µ Df n (z) µ n n z n γ(z n ) A n n z n γ(z n ) A n (3) for such a z. This implies that if the forward orbit {z n } n 0 is a certain distance away from 0 (and ), then z n γ(z n ) and the derivative dz is uniformly bounded by a dµ constant. More precisely, we have: Proposition 4.. For any real µ 4 and any forward orbit {z n } n 0, if there exists some δ > 0 such that δ z n δ for all n, then dz dµ 8δ. Proof. By assumption we have z n and γ(z n ) /δ. Thus (3) implies dz dµ z n γ(z n ) µ µ A n δ A A. n Since µ = A 4 we have the desired estimate. Non-pre-fixed case. Next we suppose that {z n } n 0 accumulates on 0 but never lands on 0. We may assume that f (Ũ0) is the union of disjoint neighborhoods U 0 of 0 and U of. Now there exist N and m such that

12 z N U J(f), z N, z N+,..., z N+m U 0 J(f), z N+m (Ũ0 U 0 ) J(f). By the linearizing coordinate φ = φ µ : Ũ0 C, we have φ(z N+m ) = φ(f m i (z N+i )) = µ m i φ(z N+i ) for 0 i m. Since φ(z) z, we have φ(z N+m ) z N+m and thus z N+i µ m+i. Since γ(z) / z when 0 < z < /, we have m i=0 z N+i γ(z N+i ) A N+i m µ m+i = m + A N+i A. N+m i=0 µ m i Hence the sum for near zero orbit points z N,..., z N+m Ũ0 are bounded by (m + )/A N+m. Since (m+)/a m 0 as m, there exits a constant K > 0 independent of m with (m + )/A N+m K/A N. z N Let us give an estimate of Df N (z) for z N U. When N = this term is not counted in the formal sum expansion of dz/dµ, so we may assume that N. One can show that z N µ m. Since Df(z) = µ z = µ µ 4 f(z), we have Df(z N ) = µ 4 µ + z N µ µ 4. 4 Moreover, by using () and z N /, we have ( ) Df N (z) = O γ(zn ) A N Hence ( ) = O. A N ( z N Df N (z) = z N Df N (z) Df(z N ) = O A ). N µ 4 By assumption, we have infinitely many near singular orbit points z N,..., z N +m, z N,..., z N +m,... Ũ0 U By calculating the function z (z + )/A z, we will find K = A/(e log A) = /(e log ) + O(µ 4).

13 with strictly increasing N j. Hence the original sum () can be estimated as follows: dz dµ z n γ(z n ) + z n γ(z n ) + z n γ(z n ) (4) A n A n A n z n / Ũ0 U z n Ũ0 z n U A + m j + n A N j+m j + ( ) O A N j z n / Ũ0 U j j µ 4 = O() + ( ) K A + O N j j µ 4 A N j j ( ) = O (µ 4). µ 4 Since the diameter of Ũ0 is fixed for µ 4, one can check that the estimate above is independent of both µ and z = z(µ) J(f µ ). Pre-fixed case. Next we consider the case where the orbit of z = z(µ) eventually lands on 0. Assume that z n = 0 if and only if n N 0. Then the derivative is written as a finite sum dz dµ = µ n<n z n Df n (z) and the sum is divided into n<n = z n / Ũ0 U + + z n Ũ0 z n U as (4) in the previous case. Hence we obtain the same estimate dz/dµ = O(/ µ 4) without extra effort. Acknowledgments Chen was partly supported by MOST 06-5-M Kawahira was partly supported by JSPS KAKENHI Grant Number 6K0593. They thank the hospitality of Academia Sinica, Nagoya University, RIMS in Kyoto University, and Tokyo Institute of Technology where parts of this research were carried out. References [CJY] L. Carleson, P.W. Jones and J.-C. Yoccoz. Julia and John. Bol. Soc. Bras. Mat. 5 (994), 30. 3

14 [CK] Y.-C. Chen and T. Kawahira. From Cantor to semi-hyperbolic parameters along external rays. arxiv: [CKLY] Y.-C. Chen, T. Kawahira, H.-L. Li and J.-M. Yuan. Family of invariant Cantor sets as orbits of differential equations. II: Julia sets. Interna. J. Bifur. Chaos. (0), [D] A. Douady. Conjectures about the Branner-Hubbard motion of Cantor sets in C. Dynamics on the Riemann Sphere: A Bodil Branner Festschrift, 09, Eur. Math. Soc., 006. [DH] A. Douady and J.H. Hubbard. Étude dynamique des polynômes complexes I & II, Publ. Math. d Orsay. 984 & 985. [HZ] G. Havard and M. Zinsmeister. Thermodynamic formalism and variations of the Hausdorff dimension of quadratic Julia sets. Comm. Math. Phys. 0 (000), [K] T. Kawahira. Semiconjugacies between the Julia sets of geometrically finite rational maps. Ergodic Theory Dynam. Systems 3 (003), 5 5. [Ka] C. Kawan. On expanding maps and topological conjugacy. J. Diff. Eq. Appl. 3 (007), [L] M.Yu. Lyubich. Some typical properties of the dynamics of rational mappings. Russian Math. Surveys 38 (983), [MSS] R. Mañé, P. Sad and D. Sullivan. On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 6 (983), [R] C. Robinson. Dynamical Systems. Stability, Symbolic Dynamics, and Chaos, CRC Press, 995. [S] M. Shishikura. The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets. Ann. Math. 47 (998), no., Yi-Chiuan Chen Institute of Mathematics Academia Sinica Taipei 067, Taiwan YCChen@math.sinica.edu.tw 4

15 Tomoki Kawahira Department of Mathematics Tokyo Institute of Technology Tokyo 5-855, Japan Mathematical Science Team RIKEN Center for Advanced Intelligence Project (AIP) -4- Nihonbashi, Chuo-ku Tokyo , Japan 5