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Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians

Santiago Barbieri, Gerard Farré

Abstract

We prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius $ρ>0$ around a Lagrangian, $(γ,τ)-$Diophantine, quasi-periodic torus, are stable over a time $t^{\text{stab}}\simeq 1/(|ρ|^{1+\frac{\ell-1}{τ+1}}|\ln ρ|^{\ell-1})$, where $\ell>2d+1, \ell \in \mathbb R$, is the regularity, and $d$ is the number of degrees of freedom. In the finitely differentiable case (for integer $\ell$), this result improves the previously known effective stability bounds around Diophantine tori. Moreover, by a previous work based on the Anosov-Katok construction, it is known that for any $\varepsilon>0$ there exists a $C^\ell$-Hamiltonian, with $ \ell\ge 3$, admitting a sequence of solutions starting at distance $ρ_n \to 0$ from a $(γ,τ)$-Diophantine torus that diffuse in a time of order $t^{\text{diff}}_n\simeq 1/(|ρ_n|^{1+\frac{\ell-1}{τ+1}+\varepsilon})$. Therefore the stability estimates that we show are optimal up to an arbitrarily small polynomial correction.

Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians

Abstract

We prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius around a Lagrangian, Diophantine, quasi-periodic torus, are stable over a time , where , is the regularity, and is the number of degrees of freedom. In the finitely differentiable case (for integer ), this result improves the previously known effective stability bounds around Diophantine tori. Moreover, by a previous work based on the Anosov-Katok construction, it is known that for any there exists a -Hamiltonian, with , admitting a sequence of solutions starting at distance from a -Diophantine torus that diffuse in a time of order . Therefore the stability estimates that we show are optimal up to an arbitrarily small polynomial correction.
Paper Structure (2 sections, 4 theorems, 13 equations)

This paper contains 2 sections, 4 theorems, 13 equations.

Table of Contents

  1. Introduction, setting, and main result
  2. Proof of Theorem \ref{['thm_main2']}

Key Result

Theorem 1

For any Hamiltonian $H\in C^\ell(\mathbb{T}^d\times B_R)$ as in eq_main_ham, with $\ell> 2d+1$ and $\omega \in \Omega^{d}_{\tau,\gamma}$, there exist constants $\rho^*, C_1 >0$ such that, for any $0<\rho<\rho^*$, and for any $(\theta_0, I_0)\in \mathbb{T}^d \times B_{\rho}$, the flow $\Phi_H^t$ veri

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 1
  • Remark 1
  • Lemma 1: Analytic smoothing
  • Remark 2
  • Lemma 2: Normal form Lemma