On the accumulation of separatrices by invariant circles

Abstract

Let $f$ be a smooth symplectic diffeomorphism of ${\mathbb{R}}^2$ admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if $f$ is a perturbation of the time-1 map of a symplectic autonomous vector field, this separatrix is accumulated by a positive measure set of invariant circles. On the other hand, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.

44 pages, 7 figures. To be appear in Ergodic Theory and Dynamical Systems (Katok Memorial Special Issue)

References in corpus (2)

• A non-differentiable essential irrational invariant curve for a $C^1$ symplectic twist map
• On the accumulation of separatrices by invariant circles

Cited by in corpus (1)

• On the accumulation of separatrices by invariant circles