Research of Complex Ergodic Theory

复遍历理论研究

• 批准号: 15540167
• 负责人: USHIKI Shigehiro
• 金额: $ 1.86万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540167/
• 关键词: complex dynamical system; fractal; ergodic theory; 複素測度; ジュリア集合

The head investigator studied hyperfunctions representing invariant measures, which are the most fundamental object in the ergodic theory of complex dynamical systems, and the transfer operators operating on the space of such hyperfunctions. So far, Schwarz's distributions are taken as the coefficient for currents, the head investigator defined the concept of hyper-currents by using a generalized version of Sato's hyperfunctons, and he constructed the complex ergodic theory. Hypercurrents canonically define a cohomology with sheaf coefficients, and the Fredholm determinant of the operation of the complex dynamical system coincides with the dynamical zeta function. By rewriting the transfer operator in a form of integral operator, the Fredholm theory of complete continuous operator can be applied to find the eigenfunctions utilizing the residue formula. On the other hand, he developed a software for the visualization of Julia sets of higher dimensional complex dynamical systems, and obt … More ained the earliest pictures of the second Julia sets, Siegel disks in higher dimension, and the Siegel-Reinhardt domains.Investigator Tetsuo Ueda proved that the Fatou coordinate of parabolic fixed point of a holomorphic mapping in one variable can be obtained as a kind of limit of the solution to Schroeder's equation for attractive fixed point. Moreover, he studied the conditions concerning the analytic continuation of Fatou's mapping for dynamical systems generated by holomorphic mappings on the complex projective space.Investigator Masashi Kisaka applied the method of quasiconformal surgery to the wandering domain of an entire transcendental function to construct entire transcendental functions having a doubly connected wandering domain, and more generally entire transcendental functions with n-ply connected wandering domains. About the semi-hyperbolicity of entire transcendental functions, he Characterized the semi-hyperbolicity of entire transcendental function by the orbits of the singular values, and, as an application, he obtained a result concerning the measure theoretical properties of dynamical systems of entire transcendental functions. Less

首席研究员研究了代表不变测度的超函数，这是复杂动力系统遍历理论中最基本的对象，以及在这种超函数空间上操作的转移算子。到目前为止，施瓦茨的分布被作为系数的电流，首席研究员定义的概念，超电流使用的一个广义版本的佐藤的hyperfunctons，他构建了复杂的遍历理论。超电流规范地定义了一个具有层系数的上同调，而复杂动力系统的Fredholm行列式与动力学zeta函数相一致。通过将转移算子改写为积分算子的形式，完全连续算子的Fredholm理论可以应用留数公式求本征函数。另一方面，他开发了一个软件，用于高维复杂动力系统的Julia集的可视化，并获得了 ...更多信息 研究者上田哲雄证明了一元全纯映射抛物不动点的Fatou坐标可以作为Schroeder方程吸引不动点解的一种极限。此外，他还研究了复射影空间上由全纯映射生成的动力系统的Fatou映射的解析延拓条件。研究者Masashi Kisaka将拟共形手术的方法应用于整超越函数的游荡域，构造了具有双连通游荡域的整超越函数，更一般的是具有n重连通游荡域的整超越函数。关于整超越函数的半双曲性，他用奇异值的轨道刻画了整超越函数的半双曲性，作为应用，他得到了关于整超越函数动力系统的测度论性质的一个结果。少

项目成果

On mutiply connected wandering domains of entire funtions

关于整个函数的多重连通游走域

• 发表时间: 2006
• 期刊: Trascendental Dynamics and Complex Analysis, Cambridge Univ. Press
• 作者: Masashi Kisaka;M. Shishikura
• 通讯作者: M. Shishikura

Some properties of Julia sets pf semi-hyperbolic entire functions

Julia 集半双曲整函数的一些性质

• 发表时间: 2007
• 期刊: Surikaisekikenkyusho Kokyuroku 1537
• 作者: S.Ushiki;S.Ushiki;Nasashi Kisaka
• 通讯作者: Nasashi Kisaka

Second Julia sets of complex dynamical systems in C^Δ2---computer visualization---

C^Δ2 中的第二组 Julia 复杂动力系统---计算机可视化---

• 发表时间: 2006
• 期刊: RIMS Kokyuroku (発表予定)
• 作者: Masashi Kisaka;M. Shishikura;Masashi Kisaka;S.Usiki
• 通讯作者: S.Usiki

Fixed points of polynomial automorphisms of C^n

C^n 多项式自同构的不动点

• 发表时间: 2004
• 期刊: Adv.Stud.Pure Math. 42
• 作者: Michihiko Fujii;Masaaki Ue;Masaaki Ue;Michihiko Fujii;Masaaki Ue;Masaaki Ue;Masaaki Ue;Masaaki Ue;Michihiko Fujii;Tetsuo Ueda
• 通讯作者: Tetsuo Ueda

Second Julia sets of complex dynamical systems in C^2-computer visualization-

C^2 中的第二组 Julia 复杂动力系统-计算机可视化-

• 发表时间: 2006
• 期刊: RIMS Kokyuroku 1485
• 作者: S.Ushiki;S.Ushiki;Nasashi Kisaka;Shigehiro Ushiki;S.Ushiki
• 通讯作者: S.Ushiki