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On the accumulation of separatrices by invariant circles

Anatole Katok, Raphaël Krikorian

TL;DR

The paper analyzes accumulation of invariant circles near a non-split separatrix for smooth symplectic diffeomorphisms of ${\mathbb R}^2$. By combining Birkhoff and Symplectic Sternberg normal forms with a careful renormalization via fundamental domains and first-return maps, the authors prove that, for small perturbations that arise as time-1 maps of autonomous Hamiltonians, the separatrix is accumulated by a positive-measure set of $C^r$ invariant circles, which are KAM circles when regularity allows. A central tool is the Translated Curve Theorem, applied to a rescaled renormalized map $\mathring{f}_{\varepsilon,n}$, yielding invariant translated graphs that lift to invariant curves for the original map $f_{\varepsilon}$. The construction also exhibits non-perturbative examples where the separatrix is Lyapunov unstable and not accumulated by invariant circles, illustrating the necessity of smallness in the perturbative setting. Together, these results connect Herman’s geometric intuition with modern renormalization and twist-map techniques, deepening understanding of instability zones and the persistence of invariant structures near separatrices.

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.

On the accumulation of separatrices by invariant circles

TL;DR

The paper analyzes accumulation of invariant circles near a non-split separatrix for smooth symplectic diffeomorphisms of . By combining Birkhoff and Symplectic Sternberg normal forms with a careful renormalization via fundamental domains and first-return maps, the authors prove that, for small perturbations that arise as time-1 maps of autonomous Hamiltonians, the separatrix is accumulated by a positive-measure set of invariant circles, which are KAM circles when regularity allows. A central tool is the Translated Curve Theorem, applied to a rescaled renormalized map , yielding invariant translated graphs that lift to invariant curves for the original map . The construction also exhibits non-perturbative examples where the separatrix is Lyapunov unstable and not accumulated by invariant circles, illustrating the necessity of smallness in the perturbative setting. Together, these results connect Herman’s geometric intuition with modern renormalization and twist-map techniques, deepening understanding of instability zones and the persistence of invariant structures near separatrices.

Abstract

Let be a smooth symplectic diffeomorphism of admitting a (non-split) separatrix associated to a hyperbolic fixed point. We prove that if 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.

Paper Structure

This paper contains 42 sections, 35 theorems, 230 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Normal Forms
  3. Birkhoff Normal Form for the time-periodic vector field $X_{\varepsilon}^t$
  4. Periodically forced vector fields
  5. Birkhoff Normal Form
  6. Symplectic Sternberg theorem for the autonomous vector field $X_{0}^{(1)}$
  7. Symplectic Sternberg Normal Form for the diffeomorphism $f_{\varepsilon}^{(1)}$
  8. Proof of Proposition \ref{['prop:2.5']}
  9. Dynamics in a neighborhood of the origin
  10. Fundamental domains and first return maps
  11. Fundamental domains
  12. First return maps
  13. Estimates on first return maps
  14. Renormalization
  15. Glueing
  16. ...and 27 more sections

Key Result

Theorem A

For any $r\in{\mathbb N}^*$ there exists $\varepsilon_{r}>0$, such that, for any $\varepsilon\in ]-\varepsilon_{r},\varepsilon_{r}[$, there exists a set of $f_{\varepsilon}$-invariant $C^r$ KAM-circles accumulating the separatix $\Sigma$ and which covers a set of positive Lebesgue measure of ${\math

Figures (7)

  • Figure 1: A (non-split) separatrix
  • Figure 2: Fundamental domain $\mathcal{F}_{\varepsilon,y_{*}}$ for $f_{\varepsilon}$ and the first return map $\hat{f}_{\varepsilon}$.
  • Figure 3: Glueing: $(\mathcal{F}_{\varepsilon,y_{*}}\cup L_{y_{*}})/f_{\varepsilon}$
  • Figure 4: The diffeomorphism $\bar{f}_{\varepsilon}$ on ${\mathbb R}/{\mathbb Z}\times [e^{-(n+1)},e^{-n}]$.
  • Figure 5: The perturbed map $f_{pert}$.
  • ...and 2 more figures

Theorems & Definitions (46)

  • Theorem A
  • Remark 1.1
  • Remark 1.2
  • Theorem B
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.1
  • Lemma 2.2
  • Proposition 2.3
  • ...and 36 more