Related papers: Homoclinic Points For Area-Preserving Surface Diff…

In this paper, we prove that if an area-preserving non-degenerate diffeomorphism on the open disk which extend smoothly to the boundary with non-degeneracy has at least 2 interior periodic points, then there are infinitely many positive…

We use symplectic tools to establish a smooth variant of Franks theorem for a closed orientable surface of positive genus $g$; it implies that a symplectic diffeomorphism isotopic to the identity with more than $2g-2$ fixed points, counted…

We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically…

Given a closed smooth four-dimensional manifold, we construct a diffeomorphism that has a homoclinic class whose continuation locally generically satisfies the following condition: it does not admit any kind of dominated splittings whereas…

We develop the theory of spectral invariants in periodic Floer homology (PFH) of area-preserving surface diffeomorphisms. We use this theory to prove $C^\infty$ closing lemmas for certain Hamiltonian isotopy classes of area-preserving…

We prove the existence of infinitely many periodic points of symplectomorphisms isotopic to the identity if they admit at least one (non-contractible) hyperbolic periodic orbit and satisfy some condition on its flux. The obtained periodic…

We prove a criteria for uniform hyperbolicity based on the periodic points of the transformation. More precisely, if a mild (non uniform) hyperbolicity condition holds for the periodic points of any diffeomorphism in a residual subset of a…

We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic…

Inspired by the techniques of Givental and Th\'eret, we provide a proof with generating functions of a recent result of Ginzburg-G\"urel concerning the periodic points of Hamiltonian diffeomorphisms of $\mathbb{C}\text{P}^d$. For instance,…

We obtain some properties of $C^1$ generic surface diffeomorphisms as finiteness of {\em non-trivial} attractors, approximation by diffeomorphisms with only a finite number of {\em hyperbolic} homoclinic classes, equivalence between…

We prove that a "positive probability" subset of the boundary of the set of hyperbolic (Axiom A) surface diffeomorphisms with no cycles $\mathcal{H}$ is constituted by Kupka-Smale diffeomorphisms: all periodic points are hyperbolic and…

We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions…

We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point, or the set of…

Let $M$ be a manifold with a volume form $\omega$ and $f : M \to M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $\omega$. In this paper, we do \textit{not} assume $f$ is $\mathcal{C}^1$-generic. We have two main themes in…

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other…

In this paper we study the effect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of…

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with…

We prove that for any compact toric symplectic manifold, if a Hamiltonian diffeomorphism admits more fixed points, counted homologically, than the total Betti number, then it has infinitely many simple periodic points. This provides a vast…

We prove that for any $C^r$ diffeomorphism, $f$, of a compact manifold of dimension $d>2$, $1\leq r\leq \infty$, admitting a transverse homoclinic intersection, we can find a $C^1$-open neighborhood of $f$ containing a $C^1$-open and…

We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S^1 spaces. Additionally, we show that all these spaces are Kaehler, that every such space is obtained from a…