On the Julia set of a typical quadratic polynomial with a Siegel disk

• 成果类型: Article
• 署名作者: Petersen, CL; Zakeri, S
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2004.159.1
• 发表日期: 2004
• 页码: 1-52
• 关键词: mappings; DYNAMICS; circle; mu

摘要

Let 0 < theta < 1 be an irrational number with continued fraction expansion theta = [a(1),a(2),a(3),...], and consider the quadratic polynomial P-theta : z --> e(2piitheta)z + z(2). By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if log a(n) = O(rootn) as n --> infinity, then the Julia set of P-theta is locally connected and has Lebesgue measure zero. In particular, it follows that for almost every 0 < theta < 1, the quadratic P-theta has a Siegel disk whose boundary is a Jordan curve passing through the critical point of P-theta. By standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.