56 research outputs found
Symmetric random walks on Homeo+(R)
We study symmetric random walks on finitely generated groups of
orientation-preserving homeomorphisms of the real line. We establish an
oscillation property for the induced Markov chain on the line that implies a
weak form of recurrence. Except for a few special cases, which can be treated
separately, we prove a property of "global stability at a finite distance":
roughly speaking, there exists a compact interval such that any two
trajectories get closer and closer whenever one of them returns to the compact
interval. The probabilistic techniques employed here lead to interesting
results for the study of group actions on the line. For instance, we show that
under a suitable change of the coordinates, the drift of every point becomes
zero provided that the action is minimal. As a byproduct, we recover the fact
that every finitely generated group of homeomorphisms of the real line is
topologically conjugate to a group of (globally) Lipschitz homeomorphisms.
Moreover, we show that such a conjugacy may be chosen in such a way that the
displacement of each element is uniformly bounded
Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems
We consider a simple class of fast-slow partially hyperbolic dynamical
systems and show that the (properly rescaled) behaviour of the slow variable is
very close to a Friedlin--Wentzell type random system for times that are rather
long, but much shorter than the metastability scale. Also, we show the
possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon
that turns out to be related to the lack of absolutely continuity of the
central foliation.Comment: To appear in Journal of Statistical Physic
Non-hyperbolic ergodic measures with large support
We prove that there is a residual subset in
such that, for every , any homoclinic class of with
invariant one dimensional central bundle containing saddles of different
indices (i.e. with different dimensions of the stable invariant manifold)
coincides with the support of some invariant ergodic non-hyperbolic (one of the
Lyapunov exponents is equal to zero) measure of
An example of non-coincidence of minimal and statistical attractors
International audienceIn this work we construct an example of a smooth dynamical system for which the minimal and statistical attractors do not coincide. In the language of SRB measures, this is an example for which there exists a unique natural measure, but there is no observable one. Also, as the statistical attractor describes the mean behaviour of individual points, and the minimal one the mean behaviour of the Lebesgue measure, this example provides us with a dynamical realization of the Riesz example
One-end finitely presented groups acting on the circle
International audienceWe study possible one-end finitely presented subgroups of Diff(+)(omega)(S-1), acting without finite orbits. Our main result, theorem 1, establishes that any such action possesses the so-called property (star), that allows one to make distortion-controlled expansion and is thus sufficient to conclude that the action is Lebesgue-ergodic. We also propose a path towards full characterization of such actions (conjectures 3-5)
Proof of the WARM whisker conjecture for neuronal connections
International audienc
Recommended from our members
Parametric Furstenberg Theorem on random products of SL(2,R) matrices
We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice
- …