Résumé
Orateurs/programme :
Thomas Waters 10h30
Title: Focusing and integrability of geodesic flows
Abstract:
A family of geodesics emanating from a point \(p\) on a manifold \(M\)
may come to focus due to the curvature of \(M\). The focal set is
sometimes referred to as the conjugate locus, and it can be very
complex. Nonetheless we can show a simple relationship between the
topological and geometrical properties of the conjugate locus on convex
surfaces, and continue to describe the conjugate locus in manifolds of
dimension 3. We also consider the integrability of geodesic flows on
certain simple surfaces, presenting examples of surfaces with
non-integrable geodesic flow as well as constructing new examples with
integrable geodesic flow.
Samuel Tapie 11h40
Growth gap, amenability and coverings
Let
(M,g) = (X/\Gamma, g) be a negatively curved manifold with universal
cover X. The critical exponent \delta_\Gamma(g) is a number which
measures the topological and geometrical complexity of (M,g). For
locally symmetric manifolds, it is strongly related to the spectrum of
the Laplacian. If N = M/ \Gamma' is a covering of M, the critical
exponent of the subgroup \Gamma'<\Gamma is hence smaller than the
critical exponent of \Gamma. When does equality occurs ?
It was
shown in the 1980's by Brooks that if M is a convex co-compact
hyperbolic manifold and \Gamma' is a normal subgroup of \Gamma, then
equality occurs if and only if \Gamma/\Gamma' is amenable. At the same
time, Cohen and Grigorchuk showed an analogous result when \Gamma is a
free group acting on its Cayley graph.
When the action of M
/\Gamma is not convex cocompact, showing that equality of critical
exponents is equivalent to the amenability of \Gamma/\Gamma' requires an
additional assumption: a "critical gap at infinity". I will explain how
under this (optimal) assumption, we can generalize the result of Brooks
to all non-compact negatively curved manifolds.
Joint work with R. Coulon, R. Dougall and B. Schapira
Anders Karlsson 14h45
Noncommuting random products and metric functionals
I
will discuss a general ergodic theorem for compositions of randomly
selected transformations, which is a setting that generalizes the
classical law of large numbers and the iteration of just one single map.
The theorem will be given in terms of metric functionals, which is an
extension of the concept of horofunction of Busemann-Gromov. The metric
functionals provide a weak topology and compactness to metric spaces,
and as tool allows for some very general statements that can be viewed a
bit in parallel to linear functional analysis. The list of applications
is rather long. The proof of the main theorem is based on a substantial
refinement of Kingman's subadditive ergodic theorem. Joint work with S.
Gouezel.
Sébastien Gouezel 15h55
Titre : Ruelle resonances
Résumé
: The Ruelle resonances of a dynamical system are spectral
characteristics of the system, describing the precise asymptotics of
correlations. While one can usually show their existence by abstract
spectral arguments, they are most of the time not computable. I will
explain that, in the case of linear pseudo-Anosov maps, one can
describe them explicitly in terms of the action of the
pseudo-Anosov on cohomology. Joint with Frédéric Faure and Erwan
Lanneau.