Iterated monodromy groups were introduced in 2001 as groups naturally associated with (partial) self-coverings of topological spaces, for instance arising from the action of complex rational functions on a punctured Riemann sphere. These groups are used as algebraic encoding of combinatorial information about the dynamical system. If the covering is expanding, then all the essential information (for instance the Julia set) can be recovered from the iterated monodromy groups. The project is devoted to the study of iterated monodromy groups and their application to topology, dynamics and group theory. Iterated monodromy groups are naturally defined for more general structures than partial self-covering. This generalized definition can be used to construct simplicial approximations of Julia sets of multi-dimensional dynamical systems. There are very few examples of rational functions of several variables for which there is a satisfactory understanding of the topology of their Julia set. The project will provide a general method of constructing approximations of such Julia sets and will lead to new homological invariants of dynamical systems.
The aim of the project is to study new connections between algebra and geometric group theory on one side and dynamical systems and the associated fractal Julia sets on the other. The iterated monodromy groups provide a bridge between these two branches of mathematics. Geometric group theory studies large-scale properties of groups of symmetries. Dynamical systems study chaotic dynamics of iterations of maps, which are models of chaotic systems in Science. Iterated monodromy groups encode in a computationally effective way combinatorial information about the dynamical systems. In particular, they give a method to construct approximations by polyhedra of complicated fractals associated with the dynamical systems. This approach can be used in the study of multi-dimensional dynamical systems, where usual computer visualization can not be applied. Iterated monodromy groups are also very interesting and exotic examples of groups. This way dynamical systems can be used to reach better understanding of group theory.
This project is jointly funded by the Topology Program and the Analysis Program.
The project was devoted to the study of connections between chaotic systems and algebraic structures. An example of such a chaotic system is iteration of a polynomial f(x), i.e., the sequence f(x), f(f(x)), f(f(f(x))), ... This sequence can behave tamely (approach a finite set of points or run away to infinity) or chaotically. The set of points x for which it behaves chaotically is called the Julia set, which is usually a complicated set (a fractal). We have discovered that the structure of the Julia set and the transformation defined by the polynomial on this set can be described using an algebraic tool---a self-similar group of symmetries of the set of all words over a finite alphabet. The project studied structure of these groups and connection between algebraic properties of the group and geometric properties of the Julia set. We showed how one can use the self-similar groups to construct "combinatorial models" of the Julia set and their multi-dimensional analogs. These combinatorial modesl are simple geometric structures (polyhedra pasted together along their faces) that successivly approximate the fractal Julia set. We also showed that some chaotic dynamical systems can be completely described using "finitely presented groups" - a very classical object of modern algebra. In another direction of our investigations we studied complexity of self-similar groups and other groups associated with dynamical systems. We showed that a large class of these groups satisfy "amenability condition", which means that they are small in some sense. Our results show deep connections between different branches of Mathematics and provide new tools for the study of complex dynamical systems.
