Quasi-morphismes de Calabi et graphe de Reeb sur le tore

We construct homogeneous quasi-morphisms on the identity component of the group of area preserving diffeomorphisms of the two dimensional torus, whose restriction to the subgroup of diffeomorphisms with support in any fixed disc equals Calabi's invariant. To cite this article: P. Py, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Slow area-preserving diffeomorphisms of the torus

We construct area-preserving real analytic diffeomorphisms of the torus with unbounded growth sequences of arbitrarily slow growth.     

Products of quasi-measures

A quasi-state is a positive functional on that is only assumed to be linear on singly-generated subalgebras. We consider the ``iterated integral' of two quasi-states and determine when this gives a quasi-state on the product space. We also provide explicit formulas for the corresponding quasi-measures in case it does. Finally, we show the general failure of Fubini's Theorem for quasi-states.

     

Quasi-morphismes et invariant de Calabi

In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism, denoted by CalS, is defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface S of genus greater than 1. This construction is motivated by a question of M. Entov and L. Polterovich [M. Entov, L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30 (2003) 1635-1676]. If U⊂S is a disk or an annulus, the restriction of CalS to the subgroup of diffeomorphisms which are the time one map of a Hamiltonian isotopy in U equals Calabi's homomorphism. The second quasi-morphism is defined on the universal cover of the group of Hamiltonian diffeomorphisms of a symplectic manifold for which the cohomology class of the symplectic form is a multiple of the first Chern class.     

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

We show that a Py-Calabi quasi-morphism on the group of Hamiltonian diffeomorphisms of surfaces of higher genus gives rise to a quasi-state.     

Transitivity of generic semigroups of area-preserving surface diffeomorphisms

We prove that the action of the semigroup generated by a C ^r generic pair of area-preserving diffeomorphisms of a compact orientable surface is transitive.     

Nonergodicity of Euler fluid dynamics on tori versus positivity of the Arnold-Ricci tensor

Brownian motions above the group G of volume preserving diffeomorphisms of the torus Td, d?2, are constructed. The asymptotic behaviour for large time of those processes shows the nonexistence of a probability measure invariant under the deterministic incompressible fluid dynamics. The energy induces on the group of volume preserving diffeomorphisms of T² a Riemannian structure which has a positive renormalized Ricci tensor.     

Non-uniformly hyperbolic horseshoes in the standard family

We show that the non-uniformly hyperbolic horseshoes of Palis and Yoccoz occur in the standard family of area-preserving diffeomorphisms of the two-torus.     

Difféomorphismes holomorphes Anosov

We classify the holomorphic diffeomorphisms of complex projective varieties with an Anosov dynamics and holomorphic stable and unstable foliations: The variety is finitely covered by a compact complex torus and the diffeomorphism corresponds to a linear transformation of this torus.

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Difféomorphismes holomorphes Anosov

     

Analytic smoothing of geometric maps with applications to KAM theory

We show that finitely differentiable diffeomorphisms which are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving or contact. We prove that the approximating functions are uniformly bounded on some complex domains and that the rate of convergence, in Cr-norms, of the approximation can be estimated in terms of the size of such complex domains and the order of differentiability of the approximated function. As an application to this result, we give a proof of the existence, the local uniqueness and the bootstrap of regularity of KAM tori for finitely differentiable symplectic maps. The symplectic maps considered here are not assumed either to be written in action-angle variables or to be perturbations of integrable systems. Our main assumption is the existence of a finitely differentiable parameterization of a maximal dimensional torus that satisfies a non-degeneracy condition and that is approximately invariant. The symplectic, volume-preserving and contact forms are assumed to be analytic.     

Navier–Stokes equations and forward–backward SDEs on the group of diffeomorphisms of a torus

We establish a connection between the strong solution to the spatially periodic Navier–Stokes equations and a solution to a system of forward–backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier–Stokes equations in terms of diffusion processes.     

On a Version of the Hyperbolic Annulus Principle

A sufficiently general class of diffeomorphisms of the annulus (the direct product of a ball in \(\mathbb{R}^{k}\), k ≥ 2, by an m-dimensional torus) is studied. The so-called annulus principle, i.e., a set of sufficient conditions under which the diffeomorphisms of the class under study have a mixing hyperbolic attractor, is obtained.     

Cremona transformations and diffeomorphisms of surfaces

We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.     

A characterization of annularity for area-preserving toral homeomorphisms

We prove that if an area-preserving homeomorphism of the torus in the homotopy class of the identity has a rotation set which is a nondegenerate vertical segment containing the origin, then there exists an essential invariant annulus. In particular, some lift to the universal covering has uniformly bounded displacement in the horizontal direction.     

A $${C^\infty}$$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces

We prove a \({C^\infty}\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a \({C^\infty}\) closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.     

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T², perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T³. In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.     

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C² volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions. Partially funded by VIGRE grant DMS-9977371 Received: January 2005 Revision: August 2005 Accepted: September 2005     

Motion by weighted mean curvature is affine invariant

Suppose curves are moving by curvature in a plane, but one embeds the plane in R³ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an “elliptical point,” and the surface energy would appear to be anisotropic as would the mobility. The result of this paper is that if one uses the apparent surface energy and the apparent mobility, then the motion by weighted curvature with mobility in the apparent plane is the same as motion by curvature in the original plane but then viewed from the angle. This result applies not only to the isotropic case but to arbitrary surface energy functions and mobilities in the plane, to surfaces in 3-space, and (in the case that the surface energy function is twice differentiable) to the case of motion viewed through distorted lenses (i.e., diffeomorphisms) as well. This result is to be contrasted with an earlier result [4], which states that for area-preserving affine transformations of the plane where the energy and mobility are not also transformed, motion by curvature to the power 1/3 (rather than 1) is invariant.     

Separating diffeomorphisms of the torus

There exists a diffeomorphism on the n-dimensional torus Tⁿ which is conjugate with a hyperbolic linear automorphism, but is not an Anosov diffeomorphism. A diffeomorphismf: Tⁿ→Tⁿ has such a property iff is separating and belongs to the C⁰ closure of the Anosov diffeomorphisms.     

Conservative homoclinic bifurcations and some applications

We study generic unfoldings of homoclinic tangencies of two-dimensional area-preserving diffeomorphisms (conservative New house phenomena) and show that they give rise to invariant hyperbolic sets of arbitrarily large Hausdorff dimension. As applications, we discuss the size of the stochastic layer of a standard map and the Hausdorff dimension of invariant hyperbolic sets for certain restricted three-body problems. We avoid involved technical details and only concentrate on the ideas of the proof of the presented results.