Partial hyperbolicity and homoclinic tangencies
We show that any diffeomorphism of a compact manifold can be approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.
Partial hyperbolicity for symplectic diffeomorphisms
Poincaré-Hopf index and partial hyperbolicity
We use the theory of partially hyperbolic systems [HPS] in order to find singularities of index for vector fields with isolated zeroes in a -ball. Indeed, we prove that such zeroes exists provided the maximal invariant set in the ball is partially hyperbolic, with volume expanding central subbundle, and the strong stable manifolds of the singularities are unknotted in the ball.
Porcupine-like horseshoes: Transitivity, Lyapunov spectrum, and phase transitions
We study a partially hyperbolic and topologically transitive local diffeomorphism F that is a skew-product over a horseshoe map. This system is derived from a homoclinic class and contains infinitely many hyperbolic periodic points of different indices and hence is not hyperbolic. The associated transitive invariant set Λ possesses a very rich fiber structure, it contains uncountably many trivial and uncountably many non-trivial fibers. Moreover, the spectrum of the central Lyapunov exponents of...