Quasi-periodic motions on symplectic tori. (English) Zbl 1531.37057
The authors study quasi-periodic motions on symplectic tori. In particular, they prove the following: Consider a quasi-periodic motion defined by a real analytic vector field \( X_{\nu} \) on a neighbourhood \( \mathbb{T}^{2d}\times V \) of a symplectic torus \( \mathbb{T}^{2d}\times\{\nu(\omega^0)\}\) such that:
(a) The vector \( \nu(\omega^0) \) satisfies a subquadratic arithmetic condition;
(b) The map \( \nu:V\to \mathbb{R}^{2d} \) is a submersion.
Then, for any real analytic symplectic vector field \( X \) sufficiently close to \( X_{\nu} \), there exists a positive measure-parametrising torus that can be symplectic conjugated to a quasi-periodic one.
Note that the arithmetic condition (a) on the frequency vector \( \nu(\omega^0) \) is weaker than a Diophantine condition but stronger than a Brjuno condition.
(a) The vector \( \nu(\omega^0) \) satisfies a subquadratic arithmetic condition;
(b) The map \( \nu:V\to \mathbb{R}^{2d} \) is a submersion.
Then, for any real analytic symplectic vector field \( X \) sufficiently close to \( X_{\nu} \), there exists a positive measure-parametrising torus that can be symplectic conjugated to a quasi-periodic one.
Note that the arithmetic condition (a) on the frequency vector \( \nu(\omega^0) \) is weaker than a Diophantine condition but stronger than a Brjuno condition.
Reviewer: Qi Zhou (Nankai)
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37C55 | Periodic and quasi-periodic flows and diffeomorphisms |
| 70K43 | Quasi-periodic motions and invariant tori for nonlinear problems in mechanics |
| 70H08 | Nearly integrable Hamiltonian systems, KAM theory |
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