Research on Complex Dynamics

上田研究了抛物型不动点的Fatou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Fatou坐标可以作为乘子趋于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Fatou映射解析连续的条件，Fatou映射是Fatou分量的推广概念。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型系统，他证明了在一个一般性的假设下，存在有限数量的测度论吸引子，并且它们的盆地与整个相空间模为一组勒贝格测度零时重合。他还研究了动态…双曲动力系统的更多Zeta函数及其解析连续性。在与Shishikura的合作中，Inou研究了一维复杂动力系统的抛物线重正化。它们证明了抛物线重整化存在一个函数不变空间，这意味着它的扰动导致了近抛物线重整化对于无理无关不动点的双曲性。作为应用，他们得到了在无理无关不动点附近的小周期循环乘子的普遍行为。Buff和Cheritat也利用上述结果作为关键的一步，证明了具有Julia集的二次多项式的正勒贝格测度的存在性。这是一个长期存在的问题的反例，这个问题类似于有理映射的Ahlfors猜想。在与Shishikura的合作中，Kisaka发展了准共形运算的技术，证明了超越整函数的双连通游荡域的存在性。Ushiki可视化了高维Julia集。较少

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