Research on Complex Dynamics

上田研究了抛物线不动点的法图坐标（Abel方程的解）和一元复解析函数吸引不动点的线性化函数（Schroeder方程的解）。他证明了法头坐标可以作为乘法器趋向于1的映射序列的线性化函数的适当极限。他还研究了复射影空间上的全纯映射，并刻画了法头映射解析延拓的条件，这是法头分量的广义概念。Tsujii用泛函分析方法研究了（部分）双曲动力系统的遍历理论性质。对于某些二维部分双曲系统，他在一般假设下证明了存在有限数量的测度理论吸引子，并且它们的盆与一组Lebesgue测度零的整个相空间模重合。他还研究了双曲动力系统的动力学zeta函数及其解析延拓。在与Shishikura的合作中，Inou研究了一维复杂动力系统的抛物型重整化问题。他们证明了抛物型重整化函数的不变量空间的存在性，这暗示了它的摄动导致了无理性无关不动点的近抛物型重整化的双曲性。作为一个应用，他们得到了围绕无理性无关不动点的小周期环乘子的普遍性质。Buff和Cheritat也将上述结果作为证明具有Julia集正Lebesgue测度的二次多项式存在性的关键步骤。这成为了一个长期存在的问题的反例这个问题类似于Ahlfors关于有理图的猜想。在与Shishikura的联合工作中，Kisaka发展了准共形手术技术，以证明超越整个函数的双连接漫游域的存在性。Ushiki可视化高维Julia集合。少

Ueda studied the Fatou coordinate (solution to Abel's equation) for parabolic fixed points and the linearization function (solution to Schroeder equation) for attracting fixed points for complex analytic functions of one variable. He showed that The Fatou coordinate can be obtained as an appropriate limit of the linearization functions for the sequence of maps whose multiplier tends to 1. He also studied holomorphic mappings on complex projective spaces and characterized the condition for the analytic continuation of Fatou maps, which is a generalized notion of Fatou components.Tsujii studied using functional analytic methods the ergodic theoretical properties of (partially) hyperbolic dynamical systems. For certain two dimensional partially hyperbolic systems, he showed under a genericity assumption that there exist a finite number of measure theoretic attractors and their basins coincide with the entire phase space modulo a set of Lebesgue measure zero. He also studied the dynamical … More zeta function for hyperbolic dynamical systems and its analytic continuation.In a joint work with Shishikura, Inou studied the parabolic renormalization of one dimensional complex dynamical systems. They showed the existence of an invariant space of function for the parabolic renormalization, and this implied that its perturbation leads to the hyperbolicity of the near-parabolic renormalization for irrationally indifferent fixed points. As an application they obtained the universal behavior of the multipliers of the small periodic cycles around irrationally indifferent fixed points. Buff and Cheritat also used the above result as a key step to show the existence of a quadratic polynomial with Julia set of positive Lebesgue measure. This became a counter-example to a long standing problem which is an analogy of Ahlfors conjecture for rational maps.In a joint work with Shishikura, Kisaka developed the technique of quasiconformal surgery to show the existence of doubly connected wandering domains for transcendental entire functions.Ushiki visualized higher dimensional Julia sets. Less