On global singularities of Sobolev mappings

We define various invariants for Sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong Sobolev topology. We show that these invariants classify various types of ``global singularities" of Sobolev maps. These invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the Sobolev space.     

Form preserving diffeomorphisms on open manifolds

We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D _infw,0 ^supr of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D _infw,0 ^supr inherits a natural Riemannian geometry and we can solve Euler equations in this context.Research supported by NSF grant # DMS-9303215 and Emory-Greifswald Exchange Program     

Une méthode de Γ-convergence pour un modèle de membrane non linéaire

We consider a cylindrical three-dimensional nonlinear hvperelastic body whose stored energy goes to infinity when the jacobian of the deformation gradient goes to zero. We show, under appropriate hypotheses on the applied loads, that the three-dimensional energies Γ-converge to the energy of a nonlinear membrane when the thickness of the cylinder goes to zero. The Γ-convergence takes place in the weak topology of a Sobolev space. As a consequence, we show that the sequence of minimizers also converges toward a minimizer of the limit energy.     

Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices

We study bifurcations of Morse-Smale diffeomorphisms under a change of the embedding of the separatrices of saddle periodic points in the ambient 3-manifold. The results obtained are based on the following statement proved in this paper: for the 3-sphere, the space of diffeomorphisms of North Pole-South Pole type endowed with the C ¹ topology is connected. This statement is shown to be false in dimension 6.     

Nonuniform hyperbolicity for C1-generic diffeomorphisms

We study the ergodic theory of non-conservative C ¹-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C ¹-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C ¹-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.     

Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diff _μ (M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev ${\dot{H}^1}$ -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler–Arnold equation is a completely integrable system in any space dimension whose smooth solutions break down in finite time. We also show that the ${\dot{H}^1}$ -metric induces the Fisher–Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.     

The set of axiom A diffeomorphisms with no cycles

LetM be aC ^∞ closed manifold and Diff¹ (M) be the space of diffeomorphisms ofM endowed with theC ¹ topology. This paper contains an affirmative answer to the following conjecture raised by Mañé, which is an extension of the stability and Ω-stability conjectures of Palis and Smale, as follows: theC ¹ interior of the subset of diffeomorphism such that all the periodic points are hyperbolic is characterized as the set of diffeomorphisms satisfying Axiom A and the no-cycles condition. Moreover, it is showed that theC ¹ interior of the set of all Kupka-Smale diffeomorphisms coincides with the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.     

Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov

We show that partially hyperbolic diffeomorphisms of \(d\) -dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.     

3-manifolds which are orbit spaces of diffeomorphisms

In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either S²×S¹ or irreducible.We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to a finite number of exceptions, which we explicitly describe, all these manifolds are Haken and, by changing the diffeomorphism by a finite power, all the Seifert components of the Jaco-Shalen-Johannson decomposition of these manifolds are made into product circle bundles.     

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

Impedance profile inversion via the first transport equation

The subject of this paper is the inverse reflection problem for a stratified elastic half-space. That is, a linear elastic medium, whose elastic properties depend only on depth from a planar free surface, is stimulated at t = 0 by a plane wave impulsive source. The motion of a typical surface element is recorded for 0 ? t ? 2T. It is shown that this surface trace determines the acoustic impedance of the medium as a function of travel time, to (travel-time) depth T. Moreover, we give a precise characterization of those functions which may appear as surface traces, and show uniqueness, existence, and continuous dependence of the logarithm of the impedance as a function of the surface trace in the Sobolev H¹ topology.     

A Density Result for Homogeneous Sobolev Spaces on Planar Domains

Potential Analysis - We show that in a bounded simply connected planar domain Ω the smooth Sobolev functions Wk,∞(Ω) ∩ C∞(Ω) are dense in the homogeneous Sobolev...     

L'invariant de Calabi pour les homéomorphismes quasiconformes du disqueThe Calabi invariant for quasiconformal mappings of the unit disk

We prove that the Calabi invariant for the symplectic diffeomorphisms of the unit disk with compact support is well defined for quasiconformal maps and depends continuously with respect to these homeomorphisms in the quasiconformal topology. To cite this article: P. Ha??ssinsky, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 635–638.     

Schnyder woods for higher genus triangulated surfaces (abstract)

We study a well known characterization of planar graphs, also called Schnyder wood or Schnyder labelling, which yields a decomposition into vertex spanning trees. The goal is to extend previous algorithms and characterizations designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. We define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and colouration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how to characterize our edge coloration in terms of genus g maps.     

On the symbol of nonlocal operators in Sobolev spaces

We consider nonlocal operators generated by pseudodifferential operators and the operator of shift along the trajectories of an arbitrary diffeomorphism of a smooth closed manifold. We introduce the notion of symbol of such operators acting in Sobolev spaces. As examples, we consider specific diffeomorphisms, namely, isometries and dilations.     

Minimal distortion morphs generated by time-dependent vector fields

A morph between two Riemannian n-manifolds is an isotopy between them together with the set of all intermediate manifolds equipped with Riemannian metrics. We propose measures of the distortion produced by some classes of morphs and diffeomorphisms between two isotopic Riemannian n-manifolds and, with respect to these classes, prove the existence of minimal distortion morphs and diffeomorphisms. In particular, we consider the class of time-dependent vector fields (on an open subset Ω of Rn+1 in which the manifolds are embedded) that generate morphs between two manifolds M and N via an evolution equation, define the bending and the morphing distortion energies for these morphs, and prove the existence of minimizers of the corresponding functionals in the set of time-dependent vector fields that generate morphs between M and N and are L² functions from [0,1] to the Sobolev space .     

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C¹-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C¹-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C¹-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff¹(M).     

Results on infinite-dimensional topology and applications to the structure of the critical set of nonlinear Sturm-Liouville operators

We consider the nonlinear Sturm-Liouville differential operator F(u)=−u″+f(u) for u∈H_D²([0,π]), a Sobolev space of functions satisfying Dirichlet boundary conditions. For a generic nonlinearity we show that there is a diffeomorphism in the domain of F converting the critical set C of F into a union of isolated parallel hyperplanes. For the proof, we show that the homotopy groups of connected components of C are trivial and prove results which permit to replace homotopy equivalences of systems of infinite-dimensional Hilbert manifolds by diffeomorphisms.