Research of Complex Ergodic Theory

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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