THREE DIMENSIONAL PARTITION AND INFINITE RENORMALIZATION Yunping Jiang March 13, 2002 Abstract. We provide a detailed construction of the three dimensional partition for the Julia set of an infinitely renormalizable quadratic polynomial. We show that for an unbranched infinitely renormalizable quadratic polynomial having complex bounds, the three-dimensional partition determines points in the Julia set dynamically. Furthermore, we construct a partition in the parameter space about a subset consisting of infinitely renormalizabe points. We prove that this subset is dense on the boundary of the Mandel- brot set. We show that every point in this subset is an unbranched quadratic polynomial having complex bounds. So its Julia set is locally connected. Moreover, we prove that the Mandelbrot set is locally connected at every point in this subset. Contents x1 Introduction. x2 Some basic facts about the dynamics of a quadratic polynomial. x3 Yoccoz partition and renormalizability. x4 Construction of a three-dimensional partition following renormalization. x4.1 Infinite renormalization. x4.2 Three-dimensional partition. x4.3 Local connectivity. x5 A partition in the parameter space. x5.1 Some basic facts about the Mandelbrot set from Douady-Hubbard. x5.2 On partitions around infinitely renormalizable points Key words and phrases. Yoccoz partition, three dimensional partition, infinite renormalization, quadratic polynomial, the Mandelbrot set, local connectivity. 2000 Mathematics Subject Classification: Primary 37F25, 37F45 and Secondary 37Cxx, 37Exx This research is supported in part by NSF grants and by PSC-CUNY awards Typeset by AMS-TEX 1 x1. Introduction. Organizing information provided by a dynamical system is important in the study of this dynamical system. A very useful method is called Markov partitions, which was introduced by Sinai [Si1,Si2] into dynamical systems from probability theory and which was generalized by Bowen [Bo1,Bo2]. For example, the dynamical system generated by the famous horseshoe map constructed by Smale [Sm] has a Markov partition. Actually, Bowen showed that one can always construct a (finite) Markov partition for an Axiom A dynamical system (see [Bo3]). So many interesting questions, in particular, those relate to chaos, of the dynamical system are well understood by using symbolic dynamical systems. When things come to a non Axiom A dynamical system, the situation becomes more complicate. For example, when we study the dynamics of a quadratic polynomial whose critical orbit is recurrent, we can not really find a (finite) Markov partition. This cre- ates a major difficulty in the study of dynamics of a quadratic polynomial. Recently, Yoccoz constructed a partition for a non-renormalizable quadratic polynomial. Using this partition and a puzzle rule on the partition, he proved that the Julia set of a non-renormalizable quadratic polynomial is locally connected. Another similar parti- tion for a cubic polynomial whose one critical orbit is bounded and the other critical orbit in the complex plane tends to infinity is constructed by Branner and Hubbard in [BH1,BH2]. Using this partition and a puzzle rule on the partition, they proved that the Julia set of such a cubic polynomial is a Cantor set whose Lebesgue measure is zero. Moreover, using the partition constructed by Yoccoz for a non-renormalizable quadratic polynomial, Shishikura [Sk] and Lyubich [Ly1] proved that the Lebesgue mea- sure of the Julia set of this polynomial is zero. We have further developed this kind of partitions for an infinitely renormalizable quadratic polynomial in [Ji3] and [Ji2]. Using this development, we proved in [Ji3] that any two Feigenbaum-like folding maps are qua- sisymmetrically conjugate. And in [Ji2], we proved that the Julia set of a unbranched infinitely renormalizable quadratic polynomial having complex bounds is locally con- nected. By combining the result in [Ji2] and Sullivan’s result [Su] on the existence of complex bounds for the Feigenbaum quadratic polynomial, we concluded in [HJ] that the Julia set of the Feigenbaum quadratic polynomial is locally connected. Levins and van Strien [LS] and Lyubich and Yamploski [LY] further proved that a real infinitely renormalizable quadratic polynomial has complex bounds. Thus its Julia set is locally connected. Although the main result in [Ji2] is about the local connectivity of the Julia set of an infinitely renormalizable quadratic polynomial, the construction of a partition in the study is also interesting. We provide a detailed exposition in x4 about this partition which we call the three dimensional partition for the Julia set of an infin- itely renormalizable quadratic polynomial. In the same section, we show that for an unbranched infinitely renormalizable quadratic polynomial having complex bounds, the three-dimensional partition determines points in the Julia set dynamically (Theorem 6). 2 Moreover, we transfer the three dimensional partitions into the parameter space and prove in x5.2 that there is a subset which is dense on the boundary of the Mandelbrot set such that every point in this subset is an unbranched quadratic polynomial having complex bounds (Theorems 7 and 8). Furthermore, by applying the Rouch´eTheorem in classical complex analysis, we prove in x5.2 that the Mandelbrot set is locally connected at every point in this subset (Theorems 7 and 8). Another interesting subset on the boundary of the Mandelbrot set is constructed by Douady and Hubbard. Douady and Hubbard proved that this subset is generic and the Julia set of every point in this subset is not locally connected. We provide some information about Douady and Hubbard’s construction in x5.1. The reader who is interested in more details could refer to Milnor’s expository article [Mi2]. We review some basic facts about dynamics of quadratic polynomial in x2. There are two text books [Mi1,CG] available in complex dynamics. The reader, who is interested in these basic facts and who would like to learn more about complex dynamics, could refer to them. In x3, we give an outline of the partition constructed by Yoccoz for a quadratic polynomial and show its relation with the renormalizability of the polynomial. There are two expository articles [Hu,Mi2] written by Hubbard and Milnor for the partition constructed by Yoccoz and a puzzle rule on this partition. The reader who would like to learn more may go to there. He may also go to [Ji1, pp. 206-219]. For the reader who is interested in the renormalization theory of quadratic polynomials and related topics there are two well written books [Mc1,Mc2] by McMullen. x2. Some basic facts about the dynamics of quadratic polynomials. Let C be the complex plane and let C be the extended complex plane (the Riemann sphere). Suppose f is a holomorphic map from a domain Ω ½ C into itself. A point z in Ω is called a periodic point of period k ¸ 1 if f ±k(z) = z but f ±i(z) 6= z for all 1 · i < k. The number ¸ = (f ±k)0(z) is called the multiplier of f at z. A periodic point of period 1 is also called a fixed point. A periodic point z of f is said to be super-attracting, attracting, neutral, or repelling if j¸j = 0, 0 < j¸j < 1, j¸j = 1, or j¸j > 1. The proof of the following theorem can be found in Milnor’s book [Mi1, pp. 86-88]. Theorem A (Theorem of B¨ottcher). Suppose p is a super-attracting periodic point of period k of a holomorphic map f from a domain Ω into itself. There is a neighborhood W of p, a holomorphic diffeomorphism h : W ! h(W ) with h(p) = 0, and a unique integer n > 1 such that h ± f ±k ± h¡1(w) = wn for w 2 h(W ). Furthermore, h is unique up to multiplication by an (n ¡ 1)-st root of unity. 2 Denote Dr = fz 2 C j jzj < rg the disk of radius r centered 0. Let qc(z) = z +c be a ¡1 quadratic polynomial. For r large enough, U = qc (Dr) is a simply connected domain 3 and its closure is relatively compact in Dr. Then qc : U ! V = Dr is a holomorphic, proper, degree two branched covering map. This is a model of quadratic-like maps defined by Douady and Hubbard [DH1] as follows: A quadratic-like map is a triple (f; U; V ) such that U and V are simply connected domains isomorphic to a disc with U ½ V and such that f : U ! V is a holomorphic, proper, degree two branched covering mapping. For a quadratic-like map (f; U; V ), 1 ¡n Kf = \n=0f (U) is called the filled-in Julia set. The Julia set Jf is the boundary of Kf . Both Kf and Jf are compact. A quadratic-like map (f; U; V ) has only one critical point which we always denote as 0. Refer to [Mi1, pp. 91-92] for a proof of the following theorem. Theorem B. The set Kf (as well as Jf ) is connected if and only if 0 is in Kf . More- over, if 0 is not in Kf , Kf = Jf is a Cantor set. 2 Consider a quadratic polynomial qc(z) = z + c. By the definition, the filled-in Julia set of qc is the set of all points not going to infinity under forward iterates of qc and the Julia set is the boundary of the filled-in Julia set. We use Kc and Jc to mean its filled-in Julia set and Julia set. For c = 0 and every r > 1, (q0;Dr;Dr2 ) is a quadratic- like map whose filled-in Julia set is K0 = D1. For any c, 1 is a super-attracting fixed point of qc, applying Theorem A, there is a unique holomorphic diffeomorphism hc (called the B¨ottcher change of coordinate) defined on a neighborhood B0 about 1 with 0 hc(1) = 1, hc(1) = 1 such that ¡1 2 hc ± qc ± hc (z) = z ; z 2 B0: ¡n Assume hc(B0) = C n Dr (for a fixed large number r > 1).