Research of Complex Ergodic Theory

研究了复杂动力系统遍历理论中最基本的对象——表示不变测度的超函数，以及作用于这种超函数空间上的传递算子。到目前为止，以Schwarz的分布作为电流的系数，首席研究员使用Sato的超泛函的广义版本定义了超电流的概念，并构建了复遍历理论。超流标准地定义了具有轴系数的上同调，复动力系统运行的Fredholm行列式与动力zeta函数重合。通过将转移算子改写成积分算子的形式，可以将Fredholm完全连续算子理论应用于利用剩余公式求特征函数。另一方面，他开发了一个用于高维复杂动力系统的Julia集可视化的软件，并获得了第二Julia集、高维Siegel盘和Siegel- reinhardt域的最早图像。研究者Tetsuo Ueda证明了单变量全纯映射的抛物不动点的Fatou坐标可以作为施罗德方程对吸引不动点解的一种极限。此外，他还研究了复射影空间上全纯映射生成的动力系统的法图映射解析延拓的条件。研究者Masashi Kisaka将拟共形手术方法应用于整个超越函数的漫游域，构造了具有双连通漫游域的整个超越函数，以及具有n层连通漫游域的更一般的整个超越函数。关于整个超越函数的半双曲性，他用奇异值轨道刻画了整个超越函数的半双曲性，作为应用，他得到了关于整个超越函数动力系统的测度理论性质的一个结果。少

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