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On the accumulation of separatrices by invariant circles

Part of: Stability theory Qualitative 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-95302Cergy-Pontoise, France
R. KRIKORIAN*
Affiliation:
Department of Mathematics, CNRS UMR 8088, CY Cergy Paris Université (University of Cergy-Pontoise), 2, av. Adolphe Chauvin, F-95302Cergy-Pontoise, France
*
e-mail: [email protected]
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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
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.

References

Angenent, S.. A remark on the topological entropy and invariant circles of an area preserving twist map. Twist Mappings and Their Applications (The IMA Volumes in Mathematics and its Applications, 44). Eds McGehee, R. and Meyer, K. R.. Springer, New York, 1992, pp. 1–5.Google Scholar
Arnaud, M.-C.. Three results on the regularity of the curves that are invariant by an exact symplectic twist map. Publ. Math. Inst. Hautes Études Sci. 109 (2009), 1–17.CrossRefGoogle Scholar
Arnaud, M.-C.. A non-differentiable essential irrational invariant curve for a ${C}^1$ symplectic twist map. J. Mod. Dyn. 5(3) (2011), 583–591.CrossRefGoogle Scholar
Arnaud, M.-C.. Boundaries of instability zones for symplectic twist maps. J. Inst. Math. Jussieu 13(1) (2013), 19–41.CrossRefGoogle Scholar
Avila, A. and Fayad, B.. Non-differentiable irrational curves for ${C}^1$ twist map. Ergod. Th. & Dynam. Sys. this issue, doi:10.1017/etds.2021.118.CrossRefGoogle Scholar
Avila, A. and Krikorian, R.. Reducibility or non-uniform hyperbolicity for quasi-periodic Schrödinger cocycles. Ann. of Math. (2) 164(3) (2006), 911–940.CrossRefGoogle Scholar
Banyaga, A., de la Llave, R. and Wayne, C. E.. Cohomology equations near hyperbolic points and geometric versions of Sternberg linearization theorem. J. Geom. Anal. 6(4) (1996), 613–649.CrossRefGoogle Scholar
Chaperon, M.. Géométrie differentielle et singularités des systèmes dynamiques. Astérisque 138–139 (1986), 444.Google Scholar
Fayad, B. and Krikorian, R.. Herman’s last geometric theorem. Ann. Sci. Éc. Norm. Supér. (4) 42(2) (2009), 193–219.CrossRefGoogle Scholar
Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.CrossRefGoogle Scholar
Herman, M. R.. Sur les courbes invariantes par les difféomorphismes de l’anneau. (Astérisque 1 (1983), 103–104). With an appendix by Albert Fathi.Google Scholar
Herman, M. R.. Some open problems in dynamical systems. Proceedings of the International Congress of Mathematicians, Volume II (Berlin, 1998). Doc. Math. Extra Volume (1998), 797–808.Google Scholar
Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.CrossRefGoogle Scholar
Krikorian, R.. Global density of reducible quasi-periodic cocycles on ${T}^1\times \mathrm{SU}(2)$ . Ann. of Math. (2) 154 (2001), 269–326.CrossRefGoogle Scholar
Moser, J.. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math-Phys. Kl. II 1962 (1962), 1–20.Google Scholar
Polterovich, L.. The Geometry of the Group of Symplectic Diffeomorphisms (Lectures in Mathematics. ETH Zürich). Birkhäuser Verlag, Basel, 2001.CrossRefGoogle Scholar
Rüssmann, H.. Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nachr. Akad. Wiss. Gottingen, II Math.-Phys. Kl. 5 (1970), 67–105.Google Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. (N.S.) 73 (1967), 747–817.CrossRefGoogle Scholar
Sternberg, S.. On the structure of local homeomorphisms of Euclidean $n$ -space II. Amer. J. Math. 80 (1958), 623–631.CrossRefGoogle Scholar
Sternberg, S.. The structure of local homeomorphism III. Amer. J. Math. 81 (1959), 578–604.CrossRefGoogle Scholar
Treschev, D. and Zubelevich, O.. Introduction to the Perturbation Theory of Hamiltonian Systems (Springer Monographs in Mathematics). Springer-Verlag, Berlin, 2010.CrossRefGoogle Scholar
Yoccoz, J.-C.. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. Éc. Norm. Supér. (4) 17(3) (1984), 333–359.CrossRefGoogle Scholar
Yoccoz, J.-C.. Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Petits diviseurs en dimension 1. Astérisque 231 (1995), 3–88.Google Scholar
Yoccoz, J.-C.. Analytic linearization of circle diffeomorphisms. Dynamical Systems and Small Divisors (Cetraro, 1998) (Lecture Notes in Mathematics/C.I.M.E. Foundation Subseries, 1784). Eds Marmi, S. and Yoccoz, J.-C.. Springer, Berlin, 2002, pp. 125–173.Google Scholar