Handling editor: Meirong Zhang

The Kyoto Dynamical Systems Research Group in Mathematics

by Hiroshi Kokubu, Department of Mathematics, Kyoto University, Japan

The Kyoto dynamical systems research group in Mathematics consists of seven faculty members and about ten graduate students and postdocs of Kyoto University, as well as two faculty members of Ryukoku University, a private university in Kyoto. This is the largest research group of dynamical systems in Japan not only in terms of membership but also in research activity. Regular meetings of our group (Dynamical Systems Seminar) are held every Friday afternoon in the Department of Mathematics at Kyoto University. A wide variety of talks are given by the core members as well as researchers of other Japanese universities and visitors from abroad.

Photograph taken after the Dynamical Systems Seminar, December 24, 2004
front from left: Masashi Kisaka, Masayuki Asaoka, Shigehiro Ushiki, Hiroshi Kokubu, Mitsuhiro Shishikura, Hiroyuki Inou
back from left: Vivien Visaya, Hiroki Takahashi, Teisuke Jin, Toshi Sugiyama, Kazutoshi Maegawa, Zin Arai, Yosuke Okumura, Tien-Cuong Dinh (Orsay, speaker of the day), Mitsuru Shibayama, Eiko Kin, Tetsuo Ueda, Kohei Ueno

The research interests of our group members cover a wide spectrum of topics. The names of the faculty members of Kyoto and Ryukoku University together with their brief research descriptions are as follows:

• Shigehiro Ushiki: complex dynamical systems, complex transfer operator, hyperfunction, sheaf cohomology,
• Mitsuhiro Shishikura: complex dynamics in dimension one and higher,
• Hiroshi Kokubu: bifurcation theory, Conley index, application of dynamical systems,
• Masashi Kisaka: complex dynamics, transcendental entire functions,
• Masayuki Asaoka: low dimensional dynamics and foliation,
• Hiroyuki Inou: polynomial dynamics, renormalization,
• Zin Arai: Conley index, homoclinic bifurcation, computer assisted proof,
• Toshikazu Ito: global properties of holomorphic vector fields or 1-forms,
• Hiroe Oka: bifurcation theory, Conley index, chaos, singular perturbation.

Graduate students and postdocs study complex dynamics, hyperbolic dynamics, topology in dynamical systems, ergodic theory, Hamiltonian systems, time series, and so on. Within the University, we frequently have communication with other research groups in nonlinear dynamics (e.g., Physics and Engineering) through seminars and individual discussions.

Our group leads the dynamical systems research in Japan. In August 2002, an international conference on "New Directions in Dynamical Systems" was held in Kyoto as a satellite conference of ICM 2002 (Beijing, P. R. China). There were 217 participants from 30 countries, 131 of whom gave talks and poster presentations. Much of the organization work was successfully carried out by our research group from both Kyoto and Ryukoku University (Chief organizer: Shigehiro Ushiki).

In 2003, the special project of the Research Institute of Mathematical Sciences (RIMS) of Kyoto University was devoted to Complex Dynamics and was organized by Mitsuhiro Shishikura, a core faculty member of our group.

Aside from these two big activities, there is a workshop series called "Kyoto Dynamics Days" which is organized ocassionally with various special topics and a selection of main speakers. The first workshop was held in May 2001 with the main speakers Adrien Douady and John Hubbard, the second was in February 2003, and the third and the fourth were in 2004. The fifth one will be in January 2005 with the main speaker Pieter Collins. Several workshops have been and will be organized at Ryukoku University as well. The most recent ones are:

• Workshop on complex singularities and holomorphic foliation, November 2002,
• International Conference on Geometry and Foliation, September 2003,
• Seminar on holomorphic foliations and complex dynamical systems, November 2004.

RIMS was founded as a cooperative research institute at the inter-university level, and hence yearly organizes more than 60 workshops on various subjects in Mathematics. One of the advantages of being in Kyoto is being able to attend many workshops easily. Almost every year, there are several workshops devoted to or related to dynamical systems. Each one originates from a different point of view -- from geometric aspects to analytical and to computational, and from purely mathematical to applied (this includes physics, biology, and enginering). Naturally, some of these workshops were organized by members of our research group. Below are some titles of the recent workshops related to our research group:

• Dynamical systems and ergodic theory (2004),
• Developments and applications of dynamical systems theory (2003),
• Complex dynamics and related topics (2002),
• Complex systems and theory of dynamical systems (2001).

The city of Kyoto has a unique historic character with the addition of Kyoto University as one of the leading research universities in Japan. Because of this combination, Kyoto University attracts many mathematicians from all over the world. Every year, we continue to have foreign visitors working on dynamical systems making it easy for us to establish international ties with many reasearchers around the world. Some of our members continue to have long-term collaboration with foreign research groups in the USA, France, Germany, Brazil, China, and other countries.