  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.
