What we do
In the simplest setting a Dynamical System consists of a measure space, a topological space, or a smooth manifold, and of one or more transformations of this space which preserve its structure (i.e. measure-preserving or nonsingular transformations, homeomorphisms, or diffeomorphisms). The mathematical theory of dynamical systems concentrates on asymptotic properties, stability under perturbations, and complexity of such systems (in applied dynamics, the term chaos is used to denote a certain degree of complexity of the system). Depending on whether one is in a measure-theoretic, topological or smooth setting one speaks of ergodic theory, topological dynamics, or differentiable dynamics, although this division is often somewhat artificial.
The mathematical theory of dynamical systems opens up many intriguing connections with other branches of mathematics (e.g. Differential Equations, Probability Theory, Operator Algebras, Number Theory and Commutative Algebra), as well as with areas of Physics (e.g. Statistical Mechanics), or with certain models in Mathematical Biology (cellular automata).
The dynamical systems described in the first paragraph are now sometimes called Commutative Dynamical Systems. A Noncommutative
Dynamical System consists of a group of automorphisms of an operator algebra, which can be either a C*-algebra, a von Neumann algebra, or something more exotic. In this setting the terms commutative and noncommutative do not refer to the group of automorphisms (or semigroup of endomorphisms), but to the algebras they act on: in commutative dynamics these are algebras of functions and hence commutative, whereas in noncommutative dynamics these algebras are nonabelian. Apart from their interest for mathematicians, Noncommutative Dynamical Systems are also very important in many branches of Mathematical Physics. On the one hand, the classical theory of (commutative) dynamical systems has led to some of the most interesting examples of noncommutative systems; on the other hand, methods and problems from noncommutative dynamics have been a major driving force in the development of commutative dynamics.
The following staff members participate in research in this area:
PhD-Students in the research group:
The following list is not comprehensive, but should help to illustrate some of the directions of dynamical research at this department:
Cohomology of dynamical systems (G. Greschonig and K. Schmidt)
The cohomology of ergodic transformations and transformation groups is used in many constructions in dynamical systems, as well as in applications in probability theory (for example, every stationary random walk can be viewed as a cocycle of an appropriate measure-preserving transformation). The systematic analysis of cohomology and of the properties of individual cocycles not only leads to new constructions and examples in ergodic theory, but also to new results on recurrence and exchangeable events in classical probability theory.
Multi-dimensional dynamical systems (M. Einsiedler and K. Schmidt)
The classical theory of dynamical systems deals mainly with the asymptotic behaviour of single transformations and flows (i.e. with actions of N, Z, R+ or R) which are usually interpreted as time evolutions of a given system. Many questions arising from applications lead, however, to the study of spatially extended systems with multi-dimensional symmetry groups which correspond to
a kind of multi-dimensional time evolution. In recent years much attention has focused Z^d-actions by commuting automorphisms of compact, abelian groups. In spite of their very special nature these algebraic Z^d-actions reveal a variety of the new
phenomena occurring in the transition from Z-actions to Z^d-actions which can be analyzed in this case with tools from commutative algebra and harmonic analysis. This class also intersects the family of hyperbolic Z^d-actions
and contains certain higher-dimensional shifts of finite type, which makes it possible to extend some of the methods and results from algebraic Z^d-actions to wider classes of multi-dimensional dynamical systems.
Maps of the interval (F. Hofbauer and P. Raith)
Piecewise monotonic maps on the interval are a thoroughly investigated class of dynamical systems showing chaotic behaviour. One is interested in the structure of the nonwandering set, where the chaotic behaviour takes place. The nonwandering set can be decomposed into finitely or countably many invariant topologically transitive subsets, sometimes called basic sets. Basic sets are either single periodic orbits or sets of positive entropy with periodic orbits dense or sets of entropy zero without periodic orbits. A basic set of positive entropy is either a finite union of intervals or a Cantor set. There are formulae connecting the Hausdorff dimension of these sets with topological pressure. The behaviour of piecewise monotonic maps under small perturbation is also investigated.
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