We show that any diffeomorphism of a compact manifold can be $C^1$ approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.

This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study to Gevrey classes of functions.

The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x+A_1{x}^2+B_{1}xy+C{y}^2){\partial }_x+(-2y+D{x}^2+A_{2}xy+B_2{y}^2){\partial }_y$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic...

A new concept of stability, closely related to that of structural stability, is introduced and applied to the study of C¹ endomorphisms with singularities. A map that is stable in this sense is conjugate to each perturbation that is equivalent to it in a geometric sense. It is shown that this kind of stability implies Axiom A and Ω-stability, and that every critical point is wandering. A partial converse is also shown, providing new examples of C³ structurally stable maps.

We prove that every $C^1$ diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive.

For n ≥ 1, given an n-dimensional locally (n-1)-connected compact space X and a finite Borel measure μ without atoms at isolated points, we prove that for a generic (in the uniform metric) continuous map f:X → X, the set of points which are chain recurrent under f has μ-measure zero. The same is true for n = 0 (skipping the local connectedness assumption).

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of...

Let $L:TM\to ℝ$ be a Tonelli Lagrangian function (with $M$ compact and connected and $dimM\ge 2$). The tiered Aubry set (resp. Mañé set) ${𝒜}^T\left(L\right)$ (resp. ${𝒩}^T\left(L\right)$) is the union of the Aubry sets (resp. Mañé sets) $𝒜(L+\lambda )$ (resp. $𝒩(L+\lambda )$) for $\lambda$ closed 1-form. Then1.the set ${𝒩}^T\left(L\right)$ is closed, connected and if $dim{H}^1\left(M\right)\ge 2$, its intersection with any energy level is connected and chain transitive;2.for $L$ generic in the Mañé sense, the sets $\overline{{𝒜}^T\left(L\right)}$ and $\overline{{𝒩}^T\left(L\right)}$ have no interior;3.if the interior of $\overline{{𝒜}^T\left(L\right)}$ is non empty, it contains a dense subset of periodic points.We then give an example...

Dans cet article, nous généralisons les résultats de Fusco et Oliva [8], qui ont montré la transversalité de l’intersection des variétés stable et instable associées à des orbites périodiques hyperboliques, pour un système dynamique de la forme $\dot{x}=f\left(x\right)$ (sur un ouvert de ${ℝ}^n$) où $f^{\text{'}}\left(x\right)$ est une matrice de Jacobi cyclique. Dans [8], cette propriété est obtenue en utilisant le nombre de changements de signe de $\dot{x}\left(t\right)$ qui est une fonctionnelle monotone le long des orbites. Tout d’abord, nous étendons ce résultat de transversalité...