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 Ergodic Theory

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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 following staff members participate in research in this area:

We organise a weekly Seminar (Arbeitsgemeinschaft)
and a monthly Seminar together with the Ergodic Theory group in Budapest.

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 (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.

  • Infinite Ergodic Theory (D. Terhesiu, R. Zweimüller)

    In certain interesting systems, the dynamics is governed by an invariant measure which happens to be infinite. This is naturally the case in systems with non-compact symmetries, like random walks on Z^d or R^d, and their dynamical counterparts (for example, the Lorentz process given by the billiard motion of a particle in a Z^d-periodic array of scatteres). A different dynamical mechanism which often leads to infinite measures is that of weak repellors (indifferent orbits) which cause intermittency. The ergodic and probabilistic theory of such systems still presents us with a large number of open questions. We are interested both in the abstract theory, and in the analysis of specific classes of dynamical systems.  
  • Maps of the interval (H. Bruin, F. Hofbauer, 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.

  • Multi-dimensional time systems (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.

  • Statistical properties of dynamical systems (H. Bruin, O. Butterley, R. Zweimüller)
  • Although fully deterministic, chaotic dynamical systems tend to be so unpredictable that the long-term behaviour is captured est in the language of probability theory, where invariant measures on the space serve as underlying probabilities of stationary (but, typically, not independent) stochastic processes. For instance, Birkhoff's Ergodic Theorem can be seem as a dynamical version of the Law of Large Numbers. Further properties of interest are Decay of Correlations (rate of mixing), Central Limit Theorem, Invariance Principles (convergence on the level of processes), Return Time Statistics and Extreme Value Statistics. We study such properties mostly for non-uniformly hyperbolic systems, both as maps and (semi-)flows, using techniques like inducing and operator approaches (Operator-valued Renewal Theory and Fourier/Laplace Transforms).

  • Thermodynamics formalism (H. Bruin, F. Hofbauer, P. Raith)

  • Motivated by Statistical Physics, this topic involves finding measures (called equilibrium states) that maximize a certain combination of entropy and potential energy. The maximum value is called the pressure of the system. Comprehensive theory was initiated in the 1970 by Bowen, Ruelle and Sinai for subshifts of finite type and Axiom A systems, see lecture notes in English and German. Over the years the theory extended to include many non-uniformly hyperbolic dynamical systems. Topics of interest remain the smoothness of the pressure (as function of parameters such as temperature) and phase transitions, existence and uniqueness of equilibrium states and measure of maximal entropy, ergodic optimization.

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