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On the accumulation of separatrices by invariant circles

Part of: Qualitative theory Stability theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  17 November 2021

A. KATOK  and
R. KRIKORIAN Open the ORCID record for R. KRIKORIAN [Opens in a new window]
A. KATOK
Affiliation:
Department of Mathematics, CNRS UMR 8088, CY Cergy Paris Université (University of Cergy-Pontoise), 2, av. Adolphe Chauvin, F-95302 Cergy-Pontoise, France
R. KRIKORIAN*
Affiliation:
Department of Mathematics, CNRS UMR 8088, CY Cergy Paris Université (University of Cergy-Pontoise), 2, av. Adolphe Chauvin, F-95302 Cergy-Pontoise, France
*
e-mail: raphael.krikorian@cyu.fr
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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

Primary: 34C27: Almost and pseudo-almost periodic solutions 34D20: Stability 37C75: Stability theory

Information

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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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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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