A Horseshoe with Positive Measure.
A note on LaSalle's problems
In LaSalle's book "The Stability of Dynamical Systems", the author gives four conditions which imply that the origin of a discrete dynamical system defined on ℝ is a global attractor, and proposes to study the natural extensions of these conditions in ℝⁿ. Although some partial results are obtained in previous papers, as far as we know, the problem is not completely settled. In this work we first study the four conditions and prove that just one of them implies that the origin is a global attractor...
A proof existence of whiskered tori with quasi flat homoclinic intersections in a class of almost integrable hamiltonian systems.
A proof of the stability conjecture
A solution to the bidimensional global asymptotic stability conjecture
A sufficient condition for -stability of vector fields on open manifolds
A theorem on the structural stability of non-autonomous differential equations
Anosov endomorphisms
Asymmetric bound states of differential equations in nonlinear optics
Asymptotic behaviors of complex analytic dynamical systems.
Asymptotic properties of the Dulac map near a hyperbolic saddle in dimension three
Asymptotic stability for perturbed hamiltonian systems, II
Asymptotic stability of dynamical systems with multiplicative perturbations
Axiom A Actions.
Bifurcations and global stability of families of gradients
Bifurcations and stability of families of diffeomorphisms
Breaking Cycles on Surfaces.
C¹ stability of endomorphisms on two-dimensional manifolds
A set of necessary conditions for C¹ stability of noninvertible maps is presented. It is proved that the conditions are sufficient for C¹ stability in compact oriented manifolds of dimension two. An example given by F. Przytycki in 1977 is shown to satisfy these conditions. It is the first example known of a C¹ stable map (noninvertible and nonexpanding) in a manifold of dimension two, while a wide class of examples are already known in every other dimension.
C¹ stable maps: examples without saddles
We give here the first examples of C¹ structurally stable maps on manifolds of dimension greater than two that are neither diffeomorphisms nor expanding. It is shown that an Axiom A endomorphism all of whose basic pieces are expanding or attracting is C¹ stable. A necessary condition for the existence of such examples is also given.
C¹-Stably Positively Expansive Maps
The notion of C¹-stably positively expansive differentiable maps on closed manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.