Topology of generic foliations on Stein manifolds: structure of leaves and Kupka-Smale property

On generic properties of Lagrangians on surfaces: the Kupka-Smale theorem

We consider generic properties of Lagrangians. Our main result is the Theorem of Kupka-Smale, in the Lagrangian setting, claiming that, for a convex and superlinear Lagrangian defined in a compact surface, for each $k\in \mathbb{R}$, generically, in Mañé's sense, the energy level, $k$, is regular and all periodic orbits, in this level, are nondegenerate at all orders, that is, the linearized Poincaré map, restricted to this energy level, does not have roots of the unity as eigenvalues. Moreover, all heteroclinic intersections in this level are transversal. All the results that we present here …

Kupka-Smale diffeomorphisms at the boundary of uniform hyperbolicity: a model

We construct an explicit example of family of non-uniformly hyperbolic diffeomorphisms, at the boundary of the set of uniformly hyperbolic systems, with one orbit of cubic heteroclinic tangency. One of the leaves involved in this heteroclinic tangency is periodic, and there is a certain degree of freedom for the choice of the second one. For a non-countable set of choices, this leaf is not periodic and the diffeomorphism is Kupka-Smale: every periodic point is hyperbolic and the intersections of stable and unstable leaves of periodic points are transverse. As a consequence of our construction,…

Existence of Birkhoff sections for Kupka-Smale Reeb flows of closed contact 3-manifolds

A Reeb vector field satisfies the Kupka-Smale condition when all its closed orbits are non-degenerate, and the stable and unstable manifolds of its hyperbolic closed orbits intersect transversely. We show that, on a closed 3-manifold, any Reeb vector field satisfying the Kupka-Smale condition admits a Birkhoff section. In particular, this implies that the Reeb vector field of a $C^\infty$-generic contact form on a closed 3-manifold admits a Birkhoff section, and that the geodesic vector field of a $C^\infty$-generic Riemannian metric on a closed surface admits a Birkhoff section.

Dynamical Properties of Gaussian Thermostats

In this work we show that the set of Kupka-Smale Gaussian thermostats on a compact manifold is generic. A Gaussian thermostat is Kupka-Smale if the closed orbits are hyperbolic and the heteroclinic intersection are transversal. We also show a dichotomy between robust transitivity and existence of arbitrary number of attractors or repellers orbits. The main tools are the concept of transitions adapted to the conformally symplectic context and a perturbative theorem which is a version of the Franks lemma for Gaussian thermostats. Finally we provide some conditions in terms of geometrical invaria…

Generic properties of magnetic flows

We obtain Kupka-Smale theorem and Franks lemma for magnetic flows on manifolds with any dimension. This improves Miranda result on surfaces. However our methods relies on geometric control theory, like in Rifford and Ruggiero articles.

具有体积密度性质的Stein流形上的 generic 保守动力学

Generic conservative dynamics on Stein manifolds with the volume density property

该文研究具有体积密度性质的高维Stein流形上的generic保体积自同构动力学，证明其混沌性、无穷拓扑熵等性质，还给出非保守情形对应结果与保守版Kupka-Smale定理。