548 research outputs found
Chain recurrence, chain transitivity, Lyapunov functions and rigidity of Lagrangian submanifolds of optical hypersurfaces
The aim of this paper is twofold. On the one hand, we discuss the notions of
strong chain recurrence and strong chain transitivity for flows on metric
spaces, together with their characterizations in terms of rigidity properties
of Lipschitz Lyapunov functions. This part extends to flows some recent results
for homeomorphisms of Fathi and Pageault. On the other hand, we use these
characterisations to revisit the proof of a theorem of Paternain, Polterovich
and Siburg concerning the inner rigidity of a Lagrangian submanifold
contained in an optical hypersurface of a cotangent bundle, under the
assumption that the dynamics on is strongly chain recurrent. We also
prove an outer rigidity result for such a Lagrangian submanifold ,
under the stronger assumption that the dynamics on is strongly chain
transitive.Comment: 26 pages, 2 figure
Mixing-like properties for some generic and robust dynamics
We show that the set of Bernoulli measures of an isolated topologically
mixing homoclinic class of a generic diffeomorphism is a dense subset of the
set of invariant measures supported on the class. For this, we introduce the
large periods property and show that this is a robust property for these
classes. We also show that the whole manifold is a homoclinic class for an open
and dense subset of the set of robustly transitive diffeomorphisms far away
from homoclinic tangencies. In particular, using results from Abdenur and
Crovisier, we obtain that every diffeomorphism in this subset is robustly
topologically mixing
Attractors in spacetimes and time functions
We develop a new approach to the existence of time functions on Lorentzian
manifolds, based on Conley's work regarding Lyapunov functions for dynamical
systems. We recover Hawking's result that a stably causal admits a time
function through a more general result giving the existence of a continuous
function that is non decreasing along all future directed causal curves, and
increasing along such curves that lie outside a special region of the
spacetime, called the chain recurrent set, which is empty for stably causal
spacetimes. The construction is based on a notion of attractive sets in
spacetimes.Comment: 26 pages, 2 figure
Partial hyperbolicity far from homoclinic bifurcations
We prove that any diffeomorphism of a compact manifold can be
C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a
homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which
is partially hyperbolic (its chain-recurrent set splits into partially
hyperbolic pieces whose centre bundles have dimensions less or equal to two).
We also study in a more systematic way the central models introduced in
arXiv:math/0605387
Rate of Converrgence for ergodic continuous Markov processes : Lyapunov versus Poincare
We study the relationship between two classical approaches for quantitative
ergodic properties : the first one based on Lyapunov type controls and
popularized by Meyn and Tweedie, the second one based on functional
inequalities (of Poincar\'e type). We show that they can be linked through new
inequalities (Lyapunov-Poincar\'e inequalities). Explicit examples for
diffusion processes are studied, improving some results in the literature. The
example of the kinetic Fokker-Planck equation recently studied by H\'erau-Nier,
Helffer-Nier and Villani is in particular discussed in the final section
- …