Aspects of Spectral Theory
15-19 Jan 2006
Erwin Schrödinger Institute
Boltzmanngasse 9, A-1090 Wien
Titles and (some) Abstracts:
Wolf-Jürgen Beyn: Homoclinic tangencies, bifurcations, and
symbolic dynamics
It is well known that transversal homoclinic points in finite dimensional
dynamical systems (with continuous as well as discrete time) lead to
Cantor like invariant subsets on which chaotic dynamics prevails.
Famous theorems by Smale, Shilnikov and Conley characterize the
dynamics in terms of a specific subshift of finite type from symbolic
dynamics.
In this talk we discuss parametrized dynamical systems that exhibit a
so called homoclinic tangency, i.e. a first order contact of stable and
unstable manifolds of an equilibrium. Such systems mark the onset
of two branches of transversal homoclinic points and hence two sets
with symbolic dynamics. Despite a series of geometrical results
by Takens and coworkers the problem of a complete
characterization of this onset in terms of symbolic dynamics remains
open. We report about a partial result that allows to reduce
the dynamics mear the homoclinic tangency to a set of bifurcation equations
that are indexed symbolically. For certain examples this local theory allows
to interpret the results of numerical calculations obtained by global
continuation.
Friedrich Götze: Quadratische Formen und Spektraltheorie
Gerhard Knieper: Geschlossene Geodätische auf Riemannschen
Mannigfaltigkeiten nicht positiver Krümmung
Thomas Kriecherbauer: Über Kombinatorik und
Zufallsmatrizen
Bernd Metzger: Tauber-Theorie für das parabolische
Andersonmodell
Peter Müller: On Mott's formula for the ac-conductivity in the
Anderson model
We study the ac-conductivity in linear response theory in the
general framework of ergodic magnetic Schroedinger operators.
For the Anderson model, if the Fermi energy lies in the localization regime,
we prove that the ac-conductivity is bounded by
$ C \nu2 (\log \frac 1 \nu)^{d+2}$ at small frequencies $\nu$.
This is to be compared to Mott's formula, which predicts the
leading term to be $ C \nu2 (\log \frac 1 \nu)^{d+1}$.
This is joint work with Abel Klein and Olivier Leneoble.
Christoph Richard: Entropie der Rautenparkettierung mit periodischen
Randbedingungen
Klaus Schmidt: Mahler measure and entropies of certain ergodic
Z^d-actions
Michael Stolz: Random matrices, symmetric spaces, and mesoscopic
physics
Ivan Veselić: Spectral properties of Anderson-percolation
Hamiltonians on graphs
Peter Zeiner: Mehrfache Koinzidenzen in 2 und 3 Dimensionen
last modified on 11 Jan 2006