TAILIEUCHUNG - Đề tài " On the Julia set of a typical quadratic polynomial with a Siegel disk "

Let 0 | Annals of Mathematics On the Julia set of a typical quadratic polynomial with a Siegel disk By C. L. Petersen and S. Zakeri Annals of Mathematics 159 2004 1 52 On the Julia set of a typical quadratic polynomial with a Siegel disk By C. L. Petersen and S. Zakeri To the memory of Michael R. Herman 1942-2000 Abstract Let 0 0 1 be an irrational number with continued fraction expansion 0 a1 a2 a3 . and consider the quadratic polynomial Pg z e2nigz z2. By performing a trans-quasiconformal surgery on an associated Blaschke product model we prove that if log an O ffn as n to then the Julia set of Pg is locally connected and has Lebesgue measure zero. In particular it follows that for almost every 0 0 1 the quadratic Pg has a Siegel disk whose boundary is a Jordan curve passing through the critical point of Pg . By standard renormalization theory these results generalize to the quadratics which have Siegel disks of higher periods. Contents 1. Introduction 2. Preliminaries 3. A Blaschke model 4. Puzzle pieces and a priori area estimates 5. Proofs of Theorems A and B 6. Appendix A proof of Theorem C References 1. Introduction Consider the quadratic polynomial Pg z e2nig z z2 where 0 0 1 is an irrational number. It has an indifferent fixed point at 0 with multiplier Pg 0 e2nig and a unique finite critical point located at -e2nig 2. Let Ag to be the basin of attraction of infinity Kg C Ag to be the filled Julia set 2 C. L. PETERSEN AND S. ZAKERI and Jq dK q be the Julia set of Pq. The behavior of the sequence of iterates PQ ra n 0 near Jq is intricate and highly nontrivial. For a comprehensive account of iteration theory of rational maps we refer to CG or M . The quadratic polynomial Pq is said to be stable near the indifferent fixed point 0 if the family of iterates P n 0 restricted to a neighborhood of 0 is normal in the sense of Montel. In this case the largest neighborhood of 0 with this property is a simply connected domain Aq called the maximal Siegel disk of Pq. The .

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