This project involves a exploration of all aspects of low dimensional dynamics: combinatorial and geometric issues of holomorphic dynamics, its interplay with one-dimensional real dynamics (particularly in the rigidity and renormalization theories), interaction with three dimensional hyperbolic geometry, and analytic and geometric aspects of two dimensional complex and real dynamics. Particular emphasis is given to the key problems of describing the typical asymptotical behavior of a typical system and understanding of the mechanisms behind the universality phenomenon.
The theory of dynamical systems was founded in the works of Poincare' about 100 years ago. It began as a branch of celestial mechanics and the study of differential equations, but then developed into an important field in its own right. It has developed in a collaboration between mathematicians and workers in many other sciences, and has preserved strong mutually beneficial connections with natural science including celestial mechanics, statistical physics, fluid dynamics, meteorology, engineering, chemistry and many other fields. Since its founding in 1989, IMS has carried on an active program of research and education in dynamical systems, and related areas of geometry, ergodic theory, and complex analysis, with particular emphasis on low dimensional dynamics over the real and complex numbers. It has supported a number of junior and senior visitors, organised research and learning seminars, mini-courses and conferences, worked with graduate and undergraduate students, incorporated computers into research and education, and helped promote communication in the field by production of the preprint series and maintaining a dynamics web site.
