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Groups and Dynamics

Groups describe symmetries, which make them important in Mathematics and Science. In Mathematics they appear as groups of automorphisms of different structures (like metric and topological spaces, vector spaces, algebraic systems, etc.), as fundamental groups of topological spaces, as homology and co-homology, groups, as Galois groups, as renorm-groups and as groups given by generators and relations.

Theory of Dynamical Systems studies the long-term behavior of systems evolving in time. In the heart of dynamical systems is investigation of global and local structures of flows and of mappings under iteration. Ergodic theory, symbolic dynamics, theory of chaos, complex dynamics, entropy theory, differential equations are among the important branches of dynamical systems.

While in the classical situation a dynamical system with discrete time corresponds to iteration of a transformation of a space, modern theory of dynamical systems studies actions of groups on topological or measure spaces. This brings together group theory and dynamics and brings investigation in both fields to a new level.

Topics of interest of our research group are: combinatorial group theory, geometric methods in group theory, asymptotic group theory, amenability, topological groups and invariant means, random walks on groups and graphs, representations, associated C*-algebras and von Neumann algebras, bounded and L²-cohomology, actions on trees, growth, self-similar groups, groups generated by finite automata, groups of homeomorphisms of the real line, the mapping class groups and other groups arising in topology.

The topics related to dynamical systems include theory of billiards, geodesic flows on flat surfaces, symbolic dynamics, substitutional dynamical systems, holomorphic dynamics, analysis on graphs and fractals, entropy, ergodic theorems, low-dimensional dynamics, statistical models on groups and graphs.Many interesting connections between groups and dynamics are provided in theory of iterated monodromy groups, actions on boundaries of trees and hyperbolic spaces, actions on the Cantor set, interval exchange transformations, orbit equivalences.

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Groups, Geometry and Dynamics