Kyushu University Academic Staff Educational and Research Activities Database
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Yutaka ISHII Last modified date:2024.04.12

Professor / Department of Mathematics
Division of Analysis
Faculty of Mathematics

Graduate School
Undergraduate School

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Academic Degree
Ph D (Mathematical Sciences), University of Tokyo, March 1998
Country of degree conferring institution (Overseas)
Field of Specialization
dynamical systems
Total Priod of education and research career in the foreign country
Outline Activities
The main theme of my research is a combinatorial study of complex dynamical systems in dimension two, namely the complex Henon maps. I am now working on a criterion of their hyperbolicity, topology and combinatorics of their Julia sets and of their parameter loci, and applications of such complex methods to real dynamics.

I gave a series of lectures on the dynamics of complex Henon maps both in Tokyo Institute of Technology and in Hokkaido University, on symbolic dynamics and data storage in Tokushima University.

I gave a public lecture under the title "Dimension, Fractals and Dynamical Systems" organized by the Department of Mathematics, Kyushu University.
Research Interests
  • Visualization of 4D spaces with VR
    keyword : higher-dimensional spaces, virtual reality, Julia sets
  • Combinatorial study of complex Henon maps
    keyword : Henon maps, complex dynamics, Julia sets, hyperbolicity, parameter loci
Academic Activities
1. Keigo Matsumoto, Nami Ogawa, Hiroyuki Inou, Shizuo Kaji, Yutaka Ishii, Michitaka Hirose, Polyvision
4D space manipulation through multiple projections, SIGGRAPH Asia 2019 Emerging Technologies - International Conference on Computer Graphics and Interactive Techniques, SA 2019 SIGGRAPH Asia 2019 Emerging Technologies, SA 2019, 10.1145/3355049.3360518, 36-37, 2019.11, Seeing is believing. Our novel virtual reality system, Polyvision, applies this old saying to the fourth dimension. Various shadows of an object in a four-dimensional (4D) space are simultaneously projected onto multiple three-dimensional (3D) screens created in a virtual environment to reveal its intricate shape. The understanding of high-dimensional shapes and data can essentially be enhanced when good visualization is complemented by interactive functionality. However, a method to implement an interface for handling complex 4D transformations in a user-friendly manner must be developed. Using our Polyvision system, the user can manipulate each shadow as if it were a 3D object in their hand. The user’s action on each projection is reflected to the original 4D object, and in turn its projections, in real time. While controlling the object’s orientation minutely on one shadow, the user can grasp its global structure from multiple changing projections. Our system has a wide variety of applications in visualization, education, mathematical research, and entertainment, as we demonstrate with a variety of 4D objects that appear in mathematics and data sciences..
2. Zin ARAI, Yutaka ISHII, On parameter loci of the H'enon family., Commun. Math. Phys., 2018.12, We characterize the hyperbolic horseshoe locus and the maximal entropy locus of the Henon family. The proof employs a combination of complex analytic and complex dynamical methods together with rigorous numerics..
3. Yutaka ISHII, Dynamics of polynomial diffeomorphisms of C^2: Combinatorial and topological aspects., Arnold Math. J., 2017.01, The purpose of this paper is to survey some results, questions and problems on the dynamics of polynomial diffeomorphisms of C^2 including complex Henon maps with an emphasis on the combinatorial and topological aspects of their Julia sets..
4. Yutaka ISHII, Hyperbolic polynomial diffeomorphisms of C^2. III: Iterated monodromy groups., Advances in Mathematics, 255, 242-304, 2014.01.
5. Yutaka Ishii, John Smillie, Homotopy shadowing., Amer. J. Math. , 132, 4, 987-1029, 2010.12, Michael Shub proved in 1969 that the topological conjugacy class of an expanding endomorphism on a compact manifold is determined by its homotopy type. In this article we generalize this result in two directions. In one direction we consider certain expanding maps on metric spaces. In a second direction we consider maps which are hyperbolic with respect to product cone fields on a product manifold. A key step in the proof is to establish a shadowing theorem for pseudo-orbits with some additional homotopy information..
Educational Activities
I taught calculus and linear algebra to freshmen and introduction to metric space topology to undergraduate students in math department.