Related papers: Lyapunov exponents of minimizing measures for glob…

Finding special orbits (as periodic orbits) of dynamical systems by variational methods and especially by minimization methods is an old method (just think to the geodesic flow). More recently, new results concerning the existence of…

We consider the irrational Aubry-Mather sets of an exact symplectic monotone twist map and explain what is the link between the Lyapunov exponents and the shape of such a set. The main tools that we use in the proofs are the so-called Green…

We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points which admit some expansion with a positive Lyapunov exponent (in a weak sense) then there exists an invariant…

In this article, we study the minimizing measures of the Tonelli Hamiltonians. More precisely, we study the relationships between the so-called Green bundles and various notions as: - the Lyapunov exponents of minimizing measures; -the weak…

We use numerical and analytical tools to demonstrate arguments in favor of the existence of a family of smooth, symplectic diffeomorphisms of the two-dimensional torus that have both a positive measure set with positive Lyapunov exponent…

We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there…

Let $M$ be a 2$d-$dimensional compact connected Riemannian manifold and $\omega$ be a symplectic form on $M$. In this paper, we prove that a symplectic diffeomorphism, with all Lyapunov exponent zero for almost everywhere, can be $C^1$…

On a one-sided shift of finite type we prove that for a generic Holder continuous function there is a unique maximizing measure. We show that b-Holder continuous functions can be approximated in the a-Holder topology, a<b, by a function…

In this paper we investigate the relation between measure expansiveness and hyperbolicity. We prove that non atomic invariant ergodic measures with all of its Lyapunov exponents positive is positively measure-expansive. We also prove that…

We consider locally minimizing measures for the conservative twist maps of the $d$-dimensional annulus or for the Tonelli Hamiltonian flows defined on a cotangent bundle $T^*M$. For weakly hyperbolic such measures (i.e. measures with no…

We characterize the maximal entropy measures of partially hyperbolic C^2 diffeomorphisms whose center foliations form circle bundles, by means of suitable finite sets of saddle points, that we call skeletons. In the special case of…

We consider a smooth area-preserving Anosov diffeomorphism $f\colon \mathbb T^2\rightarrow \mathbb T^2$ homotopic to an Anosov automorphism $L$ of $\mathbb T^2$. It is known that the positive Lyapunov exponent of $f$ with respect to the…

These are notes for a minicourse given at Regional Norte UdelaR in Salto, Uruguay for the conference CIMPA Research School-Hamiltonian and Lagrangian Dynamics. We will present Birkhoff and Aubry-Mather theory for the conservative twist maps…

We consider random perturbations of a topologically transitive local diffeomorphism of a Riemannian manifold. We show that if an absolutely continuous ergodic stationary measures is expanding (all Lyapunov exponents positive), then there is…

We obtain the following dichotomy for accessible partially hyperbolic diffeomorphisms of 3-dimensional manifolds having compact center leaves: either there is a unique entropy maximizing measure, this measure has the Bernoulli property and…

In this paper we continue our study of polynomial diffeomorphisms of C^2. Let us recall that there is an invariant measure $\mu$, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C. In this…

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has…

We show that the positive mass theorem holds for continuous Riemannian metrics that lie in the Sobolev space $W^{2, n/2}_{loc}$ for manifolds of dimension less than or equal to $7$ or spin-manifolds of any dimension. More generally, we give…

The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds…

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute…