Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Sergey Tikhomirov, Sergey Kryzhevich
Frei Universität Berlin, Saint-Petersburg State University
Shadowing in Partially hyperbolic systems
The shadowing problem is related to the following question: under which condition, for any pseudotrajectory
(approximate trajectory) of a diffeomorphism there exists a close exact trajectory. The most famous
result in shadowing theory is so-called Shadowing Lemma: in a small neighborhood of a hyperbolic
set near any pseudotrajectory there exists an exact trajectory. We prove analog of shadowing lemma
for partially hyperbolic diffeomorphisms. Let f be a partially hyperbolic diffeomorphism with splitting
E s ⊕ E c ⊕ E u . Let us assume that it is dynamically coherent with foliations W s , W c , W u . We say that
sequence {xk} is a d-central pseudotrajectory if dist(xk+1, f (xk)) < d and xk+1 and f (xk) lies on the same
central leaf. We proved that for any ε > 0 there exists d > 0 such that any d-pseudotrajectory {yn} can be
ε shadowed by a ε-central pseudotrajectory.
Martin Väth
Freie Universität Berlin
Reaktions-Diffusionssysteme mit einseitigen Hindernissen
Es ist bekannt, dass in einem Reaktions-Diffusionssystem die Diffusion durch den sog. Turing-Effekt
für Instabilität sorgen und damit in vielen Systemen Musterbildung ermöglichen kann. Dennoch gibt es
einen großen Bereich von Parametern, in dem das System trotz der Diffusion stabil ist. Im Vortrag wird
diskutiert, weshalb einseitige Hindernisse, etwa Quellen oder Senken, die erst ab einer gewissen Schwelle
aktiv werden, auch in diesen Parameterbereichen für Bifurkation sorgen können. Ohne zusätzliche
Dirichlet-Randbedingungen war dieses Problem seit fast 30 Jahren offen.
Glen Wheeler
Otto-von-Guericke-Universität Magdeburg
On the curve diffusion flow of planar curves
The curve diffusion flow of a planar curve is the steepest descent gradient flow of the length functional
in H −1 . We consider the flow of periodic immersed curves with arbitrary winding number and discuss
some recent new results on long time existence and exponential convergence for initial data with small
‘oscillation of curvature’ and isoperimetric ratio close to that of a multiply-covered circle. We also provide
an estimate for the measure of the time during which the curvature is not strictly positive, quantifying
the extent of the failure of global positivity of curvature. Finally we compare this situation for curves
with that of surfaces immersed in R 3 flowing by surface diffusion, and briefly discuss the new difficulties
encountered there.
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