On the accumulation of separatrices by invariant circles

TL;DR

This paper investigates how invariant circles accumulate around separatrices in smooth symplectic maps, showing conditions under which they do or do not accumulate, with implications for stability analysis.

Contribution

It proves that perturbations of autonomous Hamiltonian flows cause invariant circles to accumulate on certain separatrices, and provides examples where they do not.

Findings

Invariant circles can accumulate on separatrices under specific perturbations.

Examples show non-accumulation of invariant circles on some unstable separatrices.

The results clarify the relationship between stability and invariant circle accumulation.

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.