Ergodic Theory and Dynamical Systems

Research Article

On the generic existence of homoclinic points

Fernando Oliveiraa1

a1 Departamento de Matemática, ICEX-UFMG, Caixa Postal 702, 30161-Belo Horizonte-MG, Brazil


This work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.

(Received January 17 1986)

(Revised December 09 1986)