The C^r-stability conjecture
Enrique Pujals (IMPA and CUNY)
Abstract: This is a talk in dynamical systems aiming to be as palatable as
possible for an audience working in geometry and topology.
We will try to present the C^r-stability conjecture in dynamics, its
meaning, origins, significance and to provide a glimpse of the proof
in the C^1-case. If time allows (and if there is interest), we will
discuss the state of the general C^r-case.
In few words, given a finite dimensional dynamical systems f (for
instance a diffeomorphisms or a flow acting on a Riemannian
manifold), it is said to be C^r-stable, if it is conjugate to any
dynamic nearby (i.e. for any g C^r-clse to f, there is an
homeomorphisms h such that f\circle h=h\circle g). The C^r-stability
conjecture states that the C^r-stable systems are the hyperbolic ones.
To get an idea about the notion of hyperbolic dynamic, the motivating example is the
geodesic flow on a Riemannian manifold with negative sectional
curvature.