Article contents
On the accumulation of separatrices by invariant circles
Published online by Cambridge University Press: 17 November 2021
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. However, we provide examples of smooth symplectic diffeomorphisms with a Lyapunov unstable non-split separatrix that are not accumulated by invariant circles.
MSC classification
- Type
- Original Article
- Information
- Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1057 - 1097
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
Footnotes
A preliminary version of this paper was discussed by the authors some months before Anatole Katok passed away in April 2018.
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