40 research outputs found
Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms
We study the time of th return of orbits to some given (union of)
rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we
prove the existence of a scaled generating function for these returns with
respect to any Gibbs measure (associated to a Holderian potential). As a
by-product, we derive precise large deviation estimates and a central limit
theorem for Birkhoff averages of Poincare cycles. We emphasize that we look at
the limiting behavior in term of number of visits (the size of the visited set
is kept fixed). Our approach relies on the spectral properties of a
one-parameter family of induced transfer operators on unstable leaves crossing
the visited set.Comment: 17 pages; submitte
Poisson smooth structures on stratified symplectic spaces
In this paper we introduce the notion of a smooth structure on a stratified
space, the notion of a Poisson smooth structure and the notion of a weakly
symplectic smooth structure on a stratified symplectic space, refining the
concept of a stratified symplectic Poisson algebra introduced by Sjamaar and
Lerman. We show that these smooth spaces possess several important properties,
e.g. the existence of smooth partitions of unity. Furthermore, under mild
conditions many properties of a symplectic manifold can be extended to a
symplectic stratified space provided with a smooth Poisson structure, e.g. the
existence and uniqueness of a Hamiltonian flow, the isomorphism between the
Brylinski-Poisson homology and the de Rham homology, the existence of a
Leftschetz decomposition on a symplectic stratified space. We give many
examples of stratified symplectic spaces possessing a Poisson smooth structure
which is also weakly symplectic.Comment: 21 page, final version, to appear in the Proceedings of the 6-th
World Conference on 21st Century Mathematic
On the zero-temperature limit of Gibbs states
We exhibit Lipschitz (and hence H\"older) potentials on the full shift
such that the associated Gibbs measures fail to converge
as the temperature goes to zero. Thus there are "exponentially decaying"
interactions on the configuration space for which the
zero-temperature limit of the associated Gibbs measures does not exist. In
higher dimension, namely on the configuration space ,
, we show that this non-convergence behavior can occur for finite-range
interactions, that is, for locally constant potentials.Comment: The statement of Theorem 1.2 is more accurate and some new comment
follow i
Chaotic temperature dependence at zero temperature
We present a class of examples of nearest-neighbour, boubded-spin models, in
which the low-temperature Gibbs measures do not converge as the temperature is
lowered to zero, in any dimension
Invariant manifolds and equilibrium states for non-uniformly hyperbolic horseshoes
In this paper we consider horseshoes containing an orbit of homoclinic
tangency accumulated by periodic points. We prove a version of the Invariant
Manifolds Theorem, construct finite Markov partitions and use them to prove the
existence and uniqueness of equilibrium states associated to H\"older
continuous potentials.Comment: 33 pages, 6 figure
Flatness is a Criterion for Selection of Maximizing Measures
For a full shift with Np+1 symbols and for a non-positive potential, locally
proportional to the distance to one of N disjoint full shifts with p symbols,
we prove that the equilibrium state converges as the temperature goes to 0. The
main result is that the limit is a convex combination of the two ergodic
measures with maximal entropy among maximizing measures and whose supports are
the two shifts where the potential is the flattest. In particular, this is a
hint to solve the open problem of selection, and this indicates that flatness
is probably a/the criterion for selection as it was conjectured by A.O. Lopes.
As a by product we get convergence of the eigenfunction at the log-scale to a
unique calibrated subaction
Duality Theorems in Ergodic Transport
We analyze several problems of Optimal Transport Theory in the setting of
Ergodic Theory. In a certain class of problems we consider questions in Ergodic
Transport which are generalizations of the ones in Ergodic Optimization.
Another class of problems is the following: suppose is the shift
acting on Bernoulli space , and, consider a fixed
continuous cost function . Denote by the set
of all Borel probabilities on , such that, both its and
marginal are -invariant probabilities. We are interested in the
optimal plan which minimizes among the probabilities on
.
We show, among other things, the analogous Kantorovich Duality Theorem. We
also analyze uniqueness of the optimal plan under generic assumptions on .
We investigate the existence of a dual pair of Lipschitz functions which
realizes the present dual Kantorovich problem under the assumption that the
cost is Lipschitz continuous. For continuous costs the corresponding
results in the Classical Transport Theory and in Ergodic Transport Theory can
be, eventually, different.
We also consider the problem of approximating the optimal plan by
convex combinations of plans such that the support projects in periodic orbits