Research on the dynamics of cubic polynomials (on the topological structure of the parameter space)

三次多项式动力学研究（关于参数空间的拓扑结构）

• 批准号: 11640218
• 负责人: NAKANE Shizuo
• 金额: $ 1.41万
• 依托单位: TOKYO INSTITUTE OF POLYTECHNICS
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640218/
• 关键词: stretching rays; parabolic implosion; Boettcher vector; Fatou vector; radial Julia set; stretching ray; parabolic implosion analysis; porous; Fatou coordinates; parabolic arc; Bottcher vector; biquadratic maps

We have investigated the landing property of the stretching rays for the family of real cubic polynomials. Stretching rays are defined by a quasi-conformal deformation in the escape locus and are generalizations of external rays of the Mandelbrot set for quadratic polynomials. The Boettcher vector is an invariant on each stretching ray in the shift locus.We have proved that the rays above the parabolic arc with non-integral Boettcher vectors do not land, i.e. they have non-trivial accumulation sets. The proof is done by applying the theory of parabolic implosion. In case of rational Boettcher vectors, we also use the notion of radial Julia sets. Those rays with integral Boettcher vectors land at points on the parabolic arc with the same Fatou vectors.Below the parabolic arc, all the stretching rays land and we have characterized their landing points. The rays in the shift locus land at critically prefixed maps. In the proof for rays in the remaining locus, we have to use the theorem on density of hyperbolicity in the real quadratic polynomials.

我们研究了实三次多项式族的拉伸射线的着陆性质。拉伸射线由逃逸轨迹中的准共形变形定义，并且是二次多项式的 Mandelbrot 集的外部射线的推广。 Boettcher向量是位移轨迹中每条拉伸射线上的不变量。我们已经证明，具有非整数Boettcher向量的抛物线上方的射线不会着陆，即它们具有非平凡的累积集。证明是通过应用抛物线内爆理论来完成的。对于有理 Boettcher 向量，我们还使用径向 Julia 集的概念。那些具有积分 Boettcher 矢量的光线以相同的 Fatou 矢量落在抛物线弧上的点上。在抛物线弧下方，所有拉伸光线都着陆，我们已经表征了它们的着陆点。移动轨迹中的光线落在关键前缀的贴图上。在证明剩余轨迹中的射线时，我们必须使用实二次多项式中双曲密度定理。

项目成果

Shizuo Nakane: "Accumulation of stretching rays for real cubic polynomials"京都大学数理解析研究所講究緑. 1220. 26-31 (2001)

Shizuo Nakane：“实三次多项式的拉伸射线的累积”京都大学数学科学研究所 Green 1220. 26-31 (2001)。

S. Nakane and D. Schleicher: "On multicorns and unicorns I"International J. Bifurcation and Chaos. 13. (2003)

S. Nakane 和 D. Schleicher：“论多角兽和独角兽 I”International J. Bifurcation and Chaos。

Y.Komori and S.Nakane: "Non-landing of stretching rays for the family of real cubic polynomia"京都大学数理解析研究所講究録. (予定). (2000)

Y. Komori 和 S. Nakane：“实三次多项式族的拉伸射线的非着陆”京都大学数学科学研究所 Kokyuroku（计划）（2000 年）。