Araceli Bonifant

Cubic polynomial maps with periodic critical orbit, Part II: Escape regions

The parameter space for cubic polynomial maps has complex dimension 2. Its non-hyperbolic subset is a complicated fractal locus which is difficult to visualize or study. One helpful way of exploring this space is by means of complex 1-dimensional slices. This note will pursue such an exploration by studying maps belonging to the complex curve Sp consisting of all cubic maps with a superattracting orbit of period p . Here p can be any positive integer. A preliminary draft of this paper, based on conversations with Branner, Douady and Hubbard, was circulated in 1991 but not published. The present version tries to stay close to the original; however, there has been a great deal of progress in the intervening years. (See especially (Faught 92), (Branner and Hubbard 92), (Branner 93), (Roesch 99, 06), and (Kiwi 06).) In particular, a number of conjectures in the original have since been proved; and new ideas have made sharper statements possible. We begin with the period 1 case. Section 2 studies the dynamics of a cubic polynomial map F which has a superattracting fixed point, and whose Julia set J(F ) is connected. The filled Julia set of any such map consists of a central Fatou component bounded bounded by a Jordan curve, together with various limbs sprouting off at internal angles which are explicitly described.

Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnors work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing. This collection will be useful to students and researchers for decades to come. The contributors are Marco Abate, Marco Arizzi, Alexander Blokh, Thierry Bousch, Xavier Buff, Serge Cantat, Tao Chen, Robert Devaney, Alexandre Dezotti, Tien-Cuong Dinh, Romain Dujardin, Hugo Garcia-Compean, William Goldman, Rotislav Grigorchuk, John Hubbard, Yunping Jiang, Linda Keen, Jan Kiwi, Genadi Levin, Daniel Meyer, John Milnor, Carlos Moreira, Vincente Munoz, Viet-Anh Nguyen, Lex Oversteegen, Ricardo Perez-Marco, Ross Ptacek, Jasmin Raissy, Pascale Roesch, Roberto Santos-Silva, Dierk Schleicher, Nessim Sibony, Daniel Smania, Tan Lei, William Thurston, Vladlen Timorin, Sebastian van Strien, and Alberto Verjovsky.

Errata for “Cubic polynomial maps with periodic critical orbit, Part II: Escape regions”

In this note we fill in some essential details which were missing from our paper. In the case of an escape region Eh with non-trivial kneading sequence, we prove that the canonical parameter t can be expressed as a holomorphic function of the local parameter η = a−1/μ (where a is the periodic critical point). Furthermore, we prove that for any escape region Eh of grid period n ≥ 2, the winding number ν of Eh over the t-plane is greater or equal than the multiplicity μ of Eh. A result which can be stated as follows is claimed in §6 of the paper Cubic Polynomial Maps with Periodic Critical Orbit, Part II: Escape Regions, Conformal Geometry and Dynamics 14 (2010), 68–112 (referred to below as [BKM]). Assertion A. For any escape region Eh, the residue ∮ dt/2πi at the ideal point ∞h is zero. Furthermore, whenever the kneading sequence of Eh is non-trivial, the indefinite integral t = ∫ dt can be expressed as a holomorphic function of the local parameter η = ξ = a−1/μ. This assertion is true; however, there is a gap in our proof when the kneading sequence is non-trivial. In this case, [BKM, Lemma 5.19 and Theorem 6.2] do show that the quotient dt/da can be expressed as a locally holomorphic function of η, vanishing at η = 0. However, this is not enough to prove the assertion. Since a = η−μ, we have dt dη = dt da da dη = −μ dt da η−μ−1 . Thus we must show that dt/da is divisible by η in order to complete the proof. In fact, we will prove a slightly sharper statement. The necessary details follow. Lemma B. Consider a Branner-Hubbard marked grid of period n ≥ 2, denoting its finite column heights by L1, . . . , Ln−1. If Ln−1 > 0, then Lj = Ln−1 − j for 1 ≤ j ≤ Ln−1 . Received by the editors April 2, 2010. 2010 Mathematics Subject Classification. Primary 37F10, 30C10, and 30D05. The first author was partially supported by the Simons Foundation. The second author was supported by Research Network on Low Dimensional Dynamics PBCT/CONICYT, Chile. 1Our mistake was to ignore the ξ2 in the denominator of [BKM, Equation (6.3)]. 2The period p of the critical orbit can be any multiple of the grid period n; but we will work only with the grid. Note that n ≥ 2 if and only if the kneading sequence is non-trivial. c ©2010 American Mathematical Society Reverts to public domain 28 years from publication