Douady, A. and Hubbard, J.  Etude Dynamique des Polyn^ome Complexes, Publications Mathematiques d'Orsay, v. 84-102.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... A rough picture of the Mandelbrot set rst appeared in [BM] A global interest to this set was sparked by [Ma] Study of the Julia sets and the Mandelbrot set (xx3 2) by means of external rays and equipotentials was originated by Douady and Hubbard in the fundamental but unpublished Orsay Notes [DH1]. This study was further re ned in many papers, see in particular [GM, Ki, M4, Sc2] Connectivity of the Mandelbrot set (x4.3) was proven by Douady Hubbard [DH1] The MLC Conjecture was formulated by the same authors [DH1] Combinatorial models for Julia sets and the Mandelbrot set in terms of ....

.... by means of external rays and equipotentials was originated by Douady and Hubbard in the fundamental but unpublished Orsay Notes [DH1] This study was further re ned in many papers, see in particular [GM, Ki, M4, Sc2] Connectivity of the Mandelbrot set (x4.3) was proven by Douady Hubbard [DH1]. The MLC Conjecture was formulated by the same authors [DH1] Combinatorial models for Julia sets and the Mandelbrot set in terms of geodesic laminations (xx4.6 4.7) was given by Thurston [Th1] Basic introductory sources in the dynamics of real unimodal maps (x5) are the books by Collet ....

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A. Douady & J.H. Hubbard.  Etude dynamique des polyn^omes complexes. Publication Mathematiques d'Orsay, 84-02 and 85-04.

....measure subset of C. In case every critical point eventually lands in A, f(z) belongs to the general class of expanding rational maps, for which the result is proven in [18] The general case can be handled similarly, using orbifolds. This is sketched for polynomials by Douady and Hubbard [3]; the orbifold approach for general critically finite maps is discussed in [19] All examples of generally convergent algorithms we will consider employ critically finite maps. In practical terms, these maps have two benefits: convergence is assured almost everywhere, not just on an open dense ....

A. Douady and J. Hubbard. Etude dynamique des polynomes complexes. Pub. Math. d'Orsay, 1984.

....modi ed, since it is not entire) dynamical determinant, similarly to the rst equality of (1.3) For this, we shall conjugate our complex subhyperbolic quadratic polynomial with an expanding analytic dynamical system. This is related to the Thurston orbifold metric that Douady and Hubbard [DH] construct in a rami ed covering space over a neighbourhood of the Julia set in order to exhibit the hyperbolic properties of subhyperbolic polynomials (see comments after the proof of Theorem B) Here we shall use a more direct construction (as was that of [Og] on the real axis) in order to ....

....prescribed singularities along the postcritical orbit) in a neighbourhood of the Julia set of f c . The de nition of f is given in (3.11) and B is constructed in the proof of Theorem B in Section 4 (see (4. 6) and our comments there linking our construction with the rami ed covering in [DH]) The simplest illustration of Theorem B is given by the quadratic map f 2 (z) 2, where f 2 (0) 2 is a xed point with derivative f = 4. We may compute d (z) for = 2 from Theorem A, and nd 1 =4 = 1 z=2) 1 z=4) Theorem B (here m = d = q = 1, and (3.11) gives 4 = 2) ....

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A. Douady and J.H. Hubbard, Etude dynamique des polyn^omes complexes, Parties I & II, Publ. Math. Orsay 84-2, 85-4 (1984, 1985).

....in CI n IR whose exterior is forward invariant under F: More precisely one can take as such a domain the exterior of the curve y = p R 2 x 2 ; x 2 [ R= p 2; R] y = R= p 2 Cjx R= p 2j 1 1=q for R; C large enough. For the proof one may consult the Expos e 9, paragraphe 3 in [DH]. This region corresponds in the z variable to a petal P j S j . and we denote by P j the petal in the Z variable. One can of course build similar petals for f 1 ; this gives backwards invariant petals in repelling sectors. We now introduce Fatou coordinates: we have so far a coordinate Z = ....

A. Duady, J. Hubbard,  Etude dynamique des polynomes complexes (premiere et deuxieme partie), Publications Mathematiques d`Orsay, 84-02, 85-04 (1984).

....the shortest spoke is located 3=7 of a turn from the principal spoke. The longest spoke is more difficult to determine. This illustrates why this visual method is not completely accurate. 2 The Mandelbrot set In this section we recall some of the remarkable results of Douady and Hubbard (see [5]) The Mandelbrot set is given by M = fc 2 C j Q n c (0) 6 1g: 3 Figure 3: The 3=7 bulb. See Figure 1. The following facts are well known. The largest cardioidshaped region in M consists of c values for which Q c admits an attracting fixed point. Along the boundary of this cardioid, Q c ....

....this point to the cardioid. This component of the interior of M is called the p=q bulb in M and is denoted by B(p=q) If we remove the root point, then M breaks into two pieces; the component of M Gamma c p=q containing the p=q bulb is called the p=q limb. The main theorem of Douady and Hubbard [5] asserts that there is a unique uniformizing map Phi that takes the exterior of the unit circle in the extended complex plane isomorphically onto the exterior of M, taking 1 to 1 and mapping the positive real axis x 1 onto the line c 1=4. The image under Phi of the straight ray r exp(2i ) ....

A. Douady and J. Hubbard, ' Etude Dynamique des Polynome Complexes, Publications Mathematiques d'Orsay, 1983.

....through angle 2(p=q) As a consequence, for nearby c 2 B p=q , the attracting cycle rotates about the repelling fixed point by jumping approximately 2(p=q) radians at each iteration. For more details see [B] 3 Angle doubling mod 1 In order to use the fundamental results of Douady and Hubbard [DH] regarding the Mandelbrot set we need to digress to recall some facts about the doubling function. The doubling function is defined on the circle considered as the reals modulo one and is given by D( 2 mod 1. We need two facts about D: Fact 1: The angle is periodic under D iff is a ....

....(p=q) where s Gamma (p=q) s (p=q) We call s Gamma (p=q) the lower angle of B p=q and s (p=q) the upper angle. As we will see, s Sigma (p=q) is a string of q digits (0 or 1) and so s Sigma (p=q) denotes the infinite repeating sequence whose basic block is s Sigma . Douady and Hubbard [DH] have a geometric method involving Julia sets to determine these angles. Our method is more combinatorial and resembles algorithms due to Atela [A] LaVaurs [L] and Lau and Schleicher [LS] To describe this algorithm, let R p=q denote rotation of the unit circle through p=q turns, i.e. R p=q ....

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A. Douady and J. Hubbard ' Etude Dynamique des Polynome Complexes, Publications Mathematiques d'Orsay.

....in CI n IR whose exterior is forward invariant under F: More precisely one can take as such a domain the exterior of the curve y = p R 2 x 2 ; x 2 [ R= p 2; R] y = R= p 2 Cjx R= p 2j 1 1=q for R; C large enough. For the proof one may consult the Expos e 9, paragraphe 3 in [DH]. This region corresponds in the z variable to a petal P j S j . and we denote by P j the petal in the Z variable. One can of course build similar petals for f 1 ; this gives backwards invariant petals in repelling sectors. We now introduce Fatou coordinates: we have so far a coordinate Z = ....

A. Duady, J. Hubbard,  Etude dynamique des polynomes complexes (premiere et deuxieme partie), Publications Mathematiques d`Orsay, 84-02, 85-04 (1984).

....of the bifurcations of a quadratic polynomial and many related maps. Secondly, the local connectivity of M implies the density of hyperbolic dynamics ( Axiom A ) for degree two polynomials, another well known conjecture which has eluded proof for many years. For more details see [Dou1] Dou2] [DH1], DH2] Lav] Th] Recent progress. Yoccoz has recently made important progress towards the local connectivity of M . He shows that M is locally connected at all except perhaps those for which z 2 is infinitely renormalizable. The Feigenbaum polynomial is an example of an infinitely ....

A. Douady and J. Hubbard. ' Etude dynamique des polynomes complexes. Pub. Math. d'Orsay, 1984.

.... and Hubbard formulated these theorems somewhat differently (without quasiconformal conjugacy) but proved them using our quasiconformal deformation technique introduced in [44] together with their explicit description of quadratic maps which amplifies greatly on the count of Theorem 10.3 [13] [14]. The papers of this series probably would not have been written without the inspiration and insight of these bonshommes . ....

A. Douady and J. Hubbard. ' Etude dynamique des polynomes complexes. Pub. Math. d'Orsay 84-2, 85-4, 1985.

....measure subset of b C. In case every critical point eventually lands in A, f(z) belongs to the general class of expanding rational maps, for which the result is proven in [Sul] The general case can be handled similarly, using orbifolds. This is sketched for polynomials by Douady and Hubbard [DH]; the orbifold approach for general critically finite maps is discussed in [Th] All examples of generally convergent algorithms we will consider employ critically finite maps. In practical terms, these maps have two benefits: convergence is assured almost everywhere, not just on an open dense ....

A. Douady and J. Hubbard. ' Etude dynamique des polynomes complexes. Pub. Math. d'Orsay, 1984.

....H c , c 2 M 0 , complex analytic (see [L4, x4.3] The fibers Z g , g 2 E , of turn out to be complex analytic curves in QL [L4, Theorem 4.18] They are called vertical fibers. The map (2. 2) provides a smooth extension (actually, real analytic) of the Riemann mapping C r M 0 C r D (see [DH1]) to the complement of the connectedness locus. Moreover, this map is vertically holomorphic, i.e. it holomorphic on the vertical fibers Z g [L4, Lemma 4.9] Note finally that the Green function G = log j j : QL r C R provides us with a dynamically natural way to measure the distance from ....

A. Douady & J.H. Hubbard. ' Etude dynamique des polynomes complexes. Publication Mathematiques d'Orsay, 84-02 and 85-04.

....and an example which is not ergodic on its Julia set appears in [vS] Positive measure sets of ergodic maps are constructed in [Rees1] For local connectivity of geometrically finite Julia sets, see [TY] x6 Renormalization. External angles for Julia sets and the Mandelbrot set are studied in [DH1], which includes a proof that local connectivity of M implies the density of expanding polynomials. A more recent treatment appears in [Dou2] See [Mil1] for the tuning operators and self similarity of the Mandelbrot set. Background on complex renormalization can be found in [Mc4] for a ....

A. Douady and J. Hubbard. ' Etude dynamique des polynomes complexes. Pub. Math. d'Orsay 84-2, 85-4, 1985.

....8.1 was obtained by Camacho in [6] 2. See e.g. 46] for the analytic classification of parabolic fixed points. 3. Parabolic bifurcations can be analyzed in great detail by the technique of Ecalle cylinders, implicit in the renormalization construction above. For more details and examples see [14], 13] and [37] 4. The renormalization construction is applied to small denominators in [47] 9 Continuity of Julia sets The Julia set J(f) determines a map J : Rat d Cl( b C ) from the space of all rational maps of degree d to the space of compact subsets of the sphere. Here Rat d is given ....

A. Douady and J. Hubbard. ' Etude dynamique des polynomes complexes. Pub. Math. d'Orsay 84--2, 85--4, 1985.

....of this theorem especially that the method employed there has turned out to be applicable also in other settings. Proof of Theorem 2.9. First, using the concept of hyperbolic zooms, the distortion property on the balls B(x; r j (x) and Lebesgue s density theorem, one demonstrates as in [Ly1] and [DH1] (cf. also [DU3] that the (2 dimensional) Lebesgue measure of the Julia set J(f) is equal to 0. We now follow the proof of Theorem 8.8 in [ADU] Suppose on the contrary that HD(J(f) 2 and let m be a 2 conformal measure existing by Theorem 2.5. For every x 2 J c (f) let fr j (x)g 1 j=1 be the ....

A. Douady, J.H. Hubbard, Etude dynamique des polynomes complexes I,II, Publications mathematique d'Orsay 84-2, 85-4, 1985.

....around in C nM, see [1] In this paper we will restrict to parameter values c 2 RM ( along an external ray RM ( of the Mandelbrot set of angle , with strictly preperiodic. RM ( lands on some c 0 2 M. The Julia set J c 0 is, in this case, locally connected. We refer to Douady and Hubbard [2] as a general reference for external rays of both the Mandelbrot set and Julia sets. In the dynamic plane, for c 2 RM ( and t 2 S 1 = R=Z, the external rays R c (t) of the disconnected Julia set J c (dynamic rays) are of two types (for more details see [1] and next section) Branched rays: ....

.... 2 T, RM ( will denote the corresponding external ray of M. It is well known that c 2 RM ( c 2 R c ( In this paper we consider values which are strictly preperiodic. It is also well known that RM ( lands on a well defined point c 0 2 M which we will also denote by LM ( see [2]) that J c 0 = K c 0 , and that J c 0 is connected and locally connected. Since the Julia set J c 0 is connected, the extension of Psi c 0 turns out to be the Riemann map of the complement of K c 0 , which is the basin of attraction of infinity A(1; c 0 ) onto the complement C nD of the ....

Adrien Douady and John H. Hubbard. Etude Dynamique des Polynomes Complexes. Publ. Math. d'Orsay, 1984.

....plane and then harvest in the parameter plane . There are different ways to pick this harvest. Historically the first one was based on a relation between the Riemann mapping OE M : C n M C n D of the complement of the Mandelbrot set and the corresponding mappings in the dynamical plane [DH1]. Any quadratic polynomial f = P c : z 7 z 2 c near infinity looks like z 2 . The precise statement is that it is analytically conjugate to z 7 z 2 , namely there is a conformal map OE = OE c near 1, fixing 1, tangent to id at 1, and such that OE(f z) fz) 2 . There is a classical ....

....of the orbit of the critical point 0 7 c 7 c c 2 7 (c c 2 ) 2 c 7 : Namely OE M (c) lim n 1 (Q n z) 1=2 n (3. 2) This yields connectivity of the Mandelbrot set together with the following remarkable formula for the corresponding Riemann mapping (Douady Hubbard, Sibony [DH1]) OE M (c) OE c (c) c 2 C n M: 3.3) So a point c 2 C n M has a double personality: as a parameter value and as the critical value for the corresponding polynomial P c . Both personalities can be identified by their uniformizing coordinates: the external angles and equipotential levels. ....

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A. Douady & J.H. Hubbard. ' Etude dynamique des polynomes complexes. Publication Mathematiques d'Orsay, 84-02 and 85-04.

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Douady, A. and Hubbard, J.  Etude Dynamique des Polyn^ome Complexes, Publications Mathematiques d'Orsay, v. 84-102.

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A. Douady and J. H. Hubbard.  Etude dynamique des polyn^omes complexes. Parties I et II. Publications Mathematiques d'Orsay, 84-2 & 85-4.

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Douady, A. and Hubbard, J. ' Etude Dynamique des Polynome Complexes, Publications Mathematiques d'Orsay.

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Douady, A. and Hubbard, J. H. [1986], ' Etude dynamique des polynomes complexes, Publications Math'ematiques d'Orsay 84-02, Universit'e de Paris-Sud. (In French. Two parts).

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A. Douady and J. Hubbard. ' Etude dynamique des polynomes complexes. Pub. Math. d'Orsay 84-2, 85-4, 1985.

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