ANNALES SCIENTIFIQUES DE L É.N.S. ADRIEN DOUADY JOHN HAMAL HUBBARD On the dynamics of polynomial-like mappings Annales scientifiques de l É.N.S. 4 e série, tome 18, n o 2 (1985), p

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ANNALES SCIENTIFIQUES DE L É.N.S. ADRIEN DOUADY JOHN HAMAL HUBBARD On the dynamics of polynomial-like mappings Annales scientifiques de l É.N.S. 4 e série, tome 18, n o 2 (1985), p Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L accès aux archives de la revue «Annales scientifiques de l É.N.S.» (http://www., implique l accord avec les conditions générales d utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques Ann. scient. EC. Norm. Sup., 4 6 serie, t. 18, 1985, p. 287 a 343. ON THE DYNAMICS OF POLYNOMIAL-LIKE MAPPINGS BY ADRIEN DOUADY AND JOHN HAMAL HUBBARD TABLE OF CONTENTS Introduction Chapter I. Polynomial-like m a p p i n g s Definitions and statement of the straightening theorem External equivalence The Ahlfors-Bers theorem Mating a hybrid class with an external class Uniqueness of the mating Quasi-conformal equivalence in degree Chapter II. Analytic families of polynomial-like mappings Definitions Tubings Review and statements The first Mane-Sad-Sullivan decomposition Continuity on ^? The locus of hybrid equivalence Continuity of % in degree Chapter III. Negative r e s u l t s Non-analyticity of % Non-continuity of 'k- p^ in degree A family of polynomials Non-continuity of X -» P^ in degree Chapter IV. One parameter families of maps of degree Topological holomorphy The Riemann-Hurwitz formula Topological holomorphy of ^ The case M, compact Further ressemblance of My and M ANNALES SCIENTIFIQUES DE L'fecoLE NORMALE SUP^RIEURE /85/ /S 7.70/ Gauthier-Villars 288 A. DOUADY AND J. H. HUBBARD Fig. 1 4 s SERIE - TOME N 2 Fig. 4 POLYNOMIAL-LIKE MAPPINGS 289 Chapter V. Small copies ofm in M Tunable points of M Construction of a sequence (c,,) Tunability of the c,, Chapter VI. Carrots for dessert A description of Figure Carrots for z^- z Carrots in a Mandelbrot-like family References INTRODUCTION Figure 1 is a picture of the standard Mandelbrot set M. The article [M] by Benoit Mandelbrot, containing the first pictures and analyses of this set aroused great interest. The set M is defined as follows. Let P^ : z \- z 2 + c and for each c let K^ be the set of zec such that the sequence z, Pc(z), P^(P^(z)),... is bounded. A classical Theorem of Fatou [F] and Julia [J] asserts that K^ is connected if OeK^ and a Cantor set otherwise (0 plays a distinguished role because it is the critical point of P^). More recent introductions to the subject are [Br] and [Bl], where this Theorem in particular is proved. The set M is the set of c for which K^ is connected. In addition to the main component and the components which derive from it by successive bifurcations, M consists of a mass of filaments (see Figs. 2 and 3) loaded with small droplets, which ressemble M itself. The combinatorial description of these filaments is complicated; it is sketched in [D-H] and will be the object of a later publication. Figure 4 concerns an apparently unrelated problem. Let f^ be the polynomial (z l)(z+l/2+x,)(z4-l/2 ^), which we will attempt to solve by Newton's method, starting at ZQ = 0 and setting ^+i=n,(z,)=z^-a(z^)//,(z,). As we shall see, 0 is the worst possible initial guess. Color X blue if z^- 1, red if z^- 1/2+X and green ifz^^ 1/2 ^; leave ^ white if the sequence z^ does not converge to any of the roots. Figure 4 represents a small region in the ^--plane; a sequence of zooms leading to this picture is given in chapter VI. This family has been studied independantly by Curry, Garnett and Sullivan [C-G-S] who also obtained similar figures. If two cubic polynomials have roots which form similar triangles, the affine map sending the roots of one to the roots of the other conjugates their Newton's methods. Up to this equivalence, the family f^ contains all cubic polynomials with marked roots except z 3 and z 3 z 2 with 0 as first marked root. The point ZQ=O is a point of inflexion of/^ and hence a critical point of the Newton's method; it is the only critical point besides the roots. ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE 2^ A. DOUADY AND J. H. HUBBARD Fig. 2 Hg SERIE - TOME N 2 POLYNOMIAL-LIKE MAPPINGS Fig. 5 Fig. 6 ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE 292 A. DOUADY AND J. H. HUBBARD If there is an attractive periodic cycle of the Newton's method other than the roots of /, by a Theorem of Fatou it attracts a critical point, which is necessarily 0; in that sense ZQ = 0 is the worst possible initial choice. The white region in Figure 4 looks remarkably like M. Figures 5 and 6 show that the ressemblance extends to the finest detail. Here the coloring corresponds to speed of convergence to the roots. We interpret this fact as follows. For a certain value ^o ^ ^ at tne center of the white region, the sequence z^ is periodic of some period k (in this case 3) for Newton's method. Since 0 is critical for Newton's method, the periodic cycle containing 0 is superattractive, and on a neighborhood of 0 the function N^ will behave like z \ z 2. If V is a small neighborhood of 0, then N^(V) c= V, but for a slightly larger neighborhood U we will have N^(U) = U. For /. close to 'ko we wln stln have N^(U) =) U but now N^ will behave on U like some polynomial z i ^ z 2 + c for some 0=7 (X). Figure 8 represents basins of attraction of the roots for the polynomial f^ \= f (that value of \ is indicated with a cross on Figure 6). The shading corresponds to speed of attaction by the appropriate root; the black region is not attracted by any root. Figure 7 is a picture of K^ for c= (again indicated with a cross on Figure 3). The reader will agree that the words behave like , although still undefined, are not excessive. If ^(^)GM, then z^ev for all n and the sequence cannot converge to one of the roots. The remarkable fact is that 7 induces a homeomorphism of ^-1 (M) onto M. The above phenomenon is very general. If f^ is any family of analytic functions depending analytically on X, then an attempt to classify values of 'k according to the dynamical properties of /,, will often produce copies of M in the ^-plane. The object of this paper is to explain this phenomenon, by giving a precise meaning to the words behave like , which appear in the heuristic description above. The notion of polynomial-like mapping was invented for this purpose. It was motivated by the observation that when studying iteration of polynomials, one uses the theory of analytic functions continually, but the rigidity of polynomials only rarely. In the real case, something analoguous happens. Many results extend from quadratic polynomials to convex functions with negative Schwarzian curvature. But there is a major difference: in the real case, one is mainly interested in functions which send an interval into itself. An analytic function which sends a disc into itself is always contracting for the Poincare metric, and there is not much to say. In the complex case, the appropriate objects of study are maps /: U -^ C where U c C is a simply connected bounded domain. / extends to the boundary of U and sends 9\J into a curve which turns several times around U, staying outside U. Our definition is slightly different from the above, mainly for convenience. Many dynamic properties of polynomials extend to the framework of polynomial-like mappings. In fact, you can often simply copy the proof in the new setting. For the two following statements, however, that procedure does not go through. 4 6 SERIE - TOME N 2 POLYNOMIAL-LIKE MAPPINGS 293 Fig. 7 Fig. 8 ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE 294 A. DOUADY AND J. H. HUBBARD (a) The density of periodic points in Jy=^Ky, proved by Fatou for polynomials (and rational mappings) using Picard's theorem. (b) The eventual periodicity of components ofkp proved by Sullivan [Sl] using the fact that a polynomial (or a rational function) depends only on a finite number of parameters. The Straightening Theorem (1,1) allows one to deduce various properties of polynomiallike mappings from the analogous properties of polynomials. In particular, this applies to statements (a) and (b) above. Its proof relies on the measurable Riemann mapping Theorem of Morrey-Ahlfors- Bers ([A-B], [L], [A]). A review of this Theorem, as well as a dictionary between the languages of Beltrami forms and of complex structures, will be given in (1,3). Chapter I is centered on the Straightening Theorem. Chapters II to IV are devoted to the introduction of parameters in the situation. Chapters V and VI give two applications. Chapter V explains the appearance of small copies of M in M. The computation is similar to one made by Eckmann and Epstein [E-E] in the real case. They were able, using our characterization of Mandelbrot-like families, to get results similar to those in chapter V in the complex case also. Chapter VI is a study of Figure 4. This work owes a great deal to the ideas of D. Sullivan, who initiated the use of quasiconformal mappings in the study of rational functions. It fits in with Sullivan's papers ([Sl], [S2]), and the article by Mane, Sad and Sullivan [M-S-S]. We thank Pierrette Sentenac for her help with the inequalities, particularly those in IV, 5 and V,3. We also thank Homer Smith. He wrote the programs which drew the pictures in the article, and also provided expert photographic assistance. A preliminary version of this article appeared in French as a preprint in RCP25, I.R.M.A., Strasbourg. CHAPTER I Polynomial-like mappings 1. DEFINITIONS AND STATEMENT OF THE STRAIGHTENING THEOREM. Let P : C - C be a polynomial of degree d and let U=DR be the disc of radius R. If R is large enough then IT = P~ 1 (U) is relatively compact in U and homeomorphic to a disc, and P : IT - U is analytic, proper of degree d. The object above will be our model. DEFINITION. A polynomial-like map of degree d is a triple (U, U', /) where U and U' are open subsets ofc isomorphic to discs, with U' relatively compact in U, andf: U' - U a C-analytic mapping, proper of degree d. We will only be interested in the case d^2. 4 s SERIE - TOME N 2 POLYNOMIAL-LIKE MAPPINGS 295 Examples. (1) The model above. When we refer to a polynomial/as a polynomiallike mapping, a choice of U and IT such that IT =f~ 1 (U) and that /: LT - U is polynomial-like will be understood. (2) Let / (z) = cos (z)-2 and let 17= {z Re(z) 2, lm(z) 3}. Then U=/(IT) is the region represented in Figure 9, and /: U' - U is polynomial-like of degree 2. Fig. 9 (3) Let P be a polynomial of degree d and let U be open in C, such that P'^U) has several connected components U^, U2,...,Ufc with k^d. If \J\ is contained and relatively compact in U, then the restriction of P to U^ is polynomial-like of some degree d^ d. In this situation, one often gets much better information about the behaviour of P in Ui by considering it as a polynomial-like map of degree d^ than as a polynomial of degree d, especially when ^=128 and d^ =2 as will occur in chapter V. (4) A small analytic perturbation of a polynomial-like mapping of degree d is still polynomial-like of degree d. More precisely, let /: U' - U be polynomial-like of degree d. Then/has d \ critical points C0i,...,co^_i, counting with multiplicities. Choose 0, E d (IT, C U) and let Ui be the component of { z\d (z, C U) e } containing U'. Suppose that e is so small that U^ contains the / (0)^). Then if g : U' - C is an analytic function such that \g{z) f{z) for all zeit, the set \J\=g~^{\3^ is homeomorphic to a disc and g : \J\ - U^ is polynomial-like of degree d. If /: U' - U is a polynomial-like mapping of degree d, we will note K f~ n/'^uo, the set of zelt such that/ (z) is defined and belongs to U' for all ne fu The set K^ IS. a compact subset of U', which we will call the filled-in Julia set of / The Julia set J^ of ANNALES SCIENTIFIQUES DE L'ECOLE NORMALE SUPERIEURE 296 A DOUADY AND J. H. HUBBARD /is the boundary of Ky. As a dynamical system, /is mainly interesting near Ky; we will neglect what occurs near the boundary of IT. The following statements are standard for polynomials; they are also valid for polynomial-like mappings. The proofs are simply copies of the classical proofs. PROPOSITION 1. Every attractive cycle has at least one critical point in its immediate basin. PROPOSITION 2. The set Ky is connected if and only if all the critical points of f belong to Ky. // none of the critical points belong to Ky then Ky is a Cantor set. PROPOSITION 3 AND DEFINITION. The following conditions are equivalent: (i) Every critical point of f belonging to Ky is attracted by an attractive cycle. (ii) There exists a continuous Riemannian metric on a neigborhood of Jy and ^ 1 such that for any x e Jy and any tangent vector t at x \ve have KAOII/M^IHI,. // the above conditions are satisfied, f is said to be hyperbolic. Sometimes proofs in the setting of polynomial-like mappings give slightly better results. For instance, by Proposition 1 a polynomial-like mapping of degree d has at most d \ attractive cycles. One can deduce from this that if a polynomial has n attractive periodic cycles and m indifferent ones, then n+m/2^d l, by perturbing P so as to make half the indifferent points attractive. That is as well as one can do if the perturbation is constrained to be among polynomials of degree d. However, if one is allowed to perturb among polynomial-like maps of degree d, it is easy to make all the indifferent points attractive, and to see that n+m^d 1. Let /: IT - U and g : V - V be polynomial-like mappings. We will say that / and g are topologically equivalent (denoted /^iop^) ^ ihere is a homeomorphism (p from a neighborhood of Ky onto a neighborhood of Kg such that cp /=^ (p near Ky. If (p is quasi-conformal (resp. holomorphic) we will say that/and g are quasi-conformally (resp. holomorphically) equivalent (denoted f^qc8 2in(^ f^ho\s)- We will say that /and g are hybrid equivalent (noted f^hbs) ^ ^ey are quasi-conformally equivalent, and cp can be chosen so that 9(p==0 on Ky. We see that f^holg ^ f^hb ^ f^qcs = f^tops- If Jf is of measure 0 (no example is known for which this does not hold) the condition 3cp=0 on Kf just means that (p is analytic on the interior of Ky. THEOREM 1 (the Straightening Theorem). (a) Every polynomial-like mapping f: U' - U of degree d is hybrid equivalent to a polynomial P of degree d. (b) If Ky is connected, P is unique up to conjugation by an affine map. Part (a) follows from Propositions 4 and 5 below and part (b) is Corollary 2 of Proposition 6. 4 SERIE TOME N 2 POLYNOMIAL-LIKE MAPPINGS EXTERNAL EQUIVALENCE. Let /: U' - U and g : V - V be two polynomial-like mappings, with Ky and Kg connected. Then / and g are externally equivalent (noted f^ext8) ^ there exist connected open sets Ui, U^, V^, V^ such that K^. c= Ui c= Ui c U, K, c VI c: Vi c= V, /-^U^Ui, and a complex-analytic isomorphism such that (p f=g (p. ^(V^V^ (p: Ui-K^Vi-K,, We will associate to any polynomial-like map /: U' - U of degree d a real analytic expanding map hf : S 1 - S 1 also of degree d, unique up to conjugation by a rotation. We will call hf the external map of / The construction is simpler if Kf is connected; we will do it in that case first. Let a be an isomorphism of U - K^ onto W+={z l z R} (log R is the modulus of U-K^.) such that a(z) -^l when rf(z,k^)-^0. Set W^o^U'-K^) and ^-^=a / a -l :W+^W+. LetT:z(-»l/z be the reflection with respect to the unit circle, and set W_=r(W+), W'_=T(W+), W=W+ JW_US 1 and w^w+uwc-us 1. By the Schwarz reflection principle [R], h+ extends to an analytic map fc:w'- w; the restriction of h to S 1 is hf. The mapping is strongly expanding. Indeed, K: W - W is an isomorphism, and n~ 1 : W - W' c W is strongly contracting for the Poincare metric on W. This metric restricts to a metric of the form a dz with a constant on S 1. In the general case (K^. not necessarily connected), we begin by constructing a Riemann surface T, an open subset T of T and an analytic map F: T - T as follows. Let L c: U' be a compact connected subset containing /^(U') and the critical points of/, and such that Xo=U L is connected. Let X^ be a covering space of Xo of degree ^ Pn'- x n+l~ x n and 7i^: X^ - XQ be the projections and let X be the disjoint union of the X^. For each n choose a lifting 7n: ^(U'-L^X,^, of / Then T is the quotient of X by the equivalence relation identifying x to J^ (x) for all xe(u'-l) and all n==0,1,... The open set T is the union of the images of the X^, n = 1,2,..., and F: T - T is induced by the?. If Kf is connected, we have just painfuly reconstructed \J Kr and /: LT Ky - U Ky. In all cases, T is a Riemann surface isomorphic to an annulus of finite modulus, say log R. We can choose an isomorphism a:t-»w={z l z R} and continue the construction as above, to construct an expanding map hf: S 1 - S 1. If L is chosen differently, neither T, T' or F are changed. ANNALES SCIENTIFIQUES DE L'feCOLE NORMALE SUP^RIEURE 298 A. DOUADY AND J. H. HUBBARD Let /i: Ui - Ui be a polynomial-like restriction of / such that Ui =/~ 1 (Ui). The surface T^ constructed from/i is an open subset of T and if ^ is an external map of/, constructed from an isomorphism o^ of T\ onto an annulus, then P=aoal -l gives, using the Schwarz reflection principle again, a real-analytic automorphism of S 1 conjugating /ii to h. Let /: U' - U and g: V - V be two polynomial-like mappings, and ^ and ^ be their external maps. If Kf and Kg are connected, the same argument as above shows that / and g are externally equivalent if and only if hf and hg are real-analytically conjugate. If Kf or Kg is not connected, we take real-analytic conjugation of their external maps as the definition of external equivalence. Remarks. - (1) If there exist open sets Ui, Ui, Vi, Vi and compact sets L and M such that /-^U^LeUlcrUi, ^OOczMcrViCzVi, Ul = /- l (Ul), Vl^-^V,), L (resp. M) containing the critical points of / (resp. g) and an isomorphism (p:ui L-^Vi M such that (po/=^o(p on U^ L, one can easily construct using (p an isomorphism of Ty i onto T^ i and therefore / and g are externally equivalent. Unfortunately, there exist pairs /, g of externally equivalent polynomial-like maps, whose equivalence cannot be realized by such a (p. However, any external equivalence can be obtained from a chain of three such realizable equivalences. (2) Let K: R - R be a lifting of h to the universal covering space. Then h has a unique fixed point a with h' (a) = 8 1 and there exists a diffeomorphism y ^ - ^ such that yo^oy-i is ri-^8.r. The map S^h-^y'^O+l) satisfies the functional equation 3 d (r) = 8 S (r/5), which is formally identical to the Cvitanovic-Feigenbaum equation. The trivial map zi-^ leads to h(z)=z d, S=d and S(Q=r+l. Up to conjugation by an affine map, the class of S uniquely determines the class of h. We do not know how to express the property that h is expanding in terms of S. (3) Let h^ and h^ be two expanding maps S 1 -^ S 1 of degree d. If h^ and h^ are realanalytically. conjugate, then the eigenvalues along corresponding cycles are equal. It is an open question whether the converse holds. Recently, P. Collet [C] found a partial positive answer. (^) PROPOSITION 4. Let f be a polynomial-like mapping of degree d. Then f is holomorphically equivalent to a polynomial if and only iffis externally equivalent to zi ^. Proof. - Let P be a polynomial. Then P is analytically conjugate to zi-^z^ in a neighborhood of infinity. Therefore if U is chosen sufficiently large, and L is chosen sufficiently large in U^P'^U), the Riemann surface T constructed will be isomorphic to the one for zi-^z^, by an isomorphism which conjugates
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