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 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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