Positive Measure of Effective Quasi-Periodic Motion Near a Diophantine Torus

IF 1.4 3区 物理与天体物理 Q2 PHYSICS, MATHEMATICAL Annales Henri Poincaré Pub Date : 2023-04-25 DOI:10.1007/s00023-023-01302-4
Abed Bounemoura, Gerard Farré
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Abstract

It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus \({\mathcal {T}}_0\), invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian Diophantine quasi-periodic invariant tori. While the conjecture is still open, we will prove the following weaker statement: there exists an open set of positive measure (in fact, the relative measure of the complement is exponentially small) around \({\mathcal {T}}_0\) such that the motion of all initial conditions in this set is “effectively” quasi-periodic in the sense that they are close to being quasi-periodic for an interval of time, which is doubly exponentially long with respect to the inverse of the distance to \({\mathcal {T}}_0\). This open set can be thought of as a neighborhood of a hypothetical invariant set of Lagrangian Diophantine quasi-periodic tori, which may or may not exist.

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丢番图环附近有效准周期运动的正测度
Herman推测,一个解析拉格朗日丢番图拟周期环面\({\mathcal {T}}_0\),被实解析哈密顿系统不变量,总是被其他拉格朗日丢番图拟周期不变量环面的一组正勒贝格测度累加。虽然这个猜想仍然是开放的,但我们将证明以下较弱的陈述:在\({\mathcal {T}}_0\)周围存在一个正测度的开放集(事实上,补的相对测度是指数小的),使得该集合中所有初始条件的运动在一段时间间隔内是“有效的”准周期的,即它们接近于准周期,这段时间间隔相对于到\({\mathcal {T}}_0\)的距离的倒数是双指数长的。这个开集可以看作是拉格朗日丢芬图拟周期环面假设不变集的一个邻域,它可能存在也可能不存在。
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来源期刊
Annales Henri Poincaré
Annales Henri Poincaré 物理-物理:粒子与场物理
CiteScore
3.00
自引率
6.70%
发文量
108
审稿时长
6-12 weeks
期刊介绍: The two journals Annales de l''Institut Henri Poincaré, physique théorique and Helvetica Physical Acta merged into a single new journal under the name Annales Henri Poincaré - A Journal of Theoretical and Mathematical Physics edited jointly by the Institut Henri Poincaré and by the Swiss Physical Society. The goal of the journal is to serve the international scientific community in theoretical and mathematical physics by collecting and publishing original research papers meeting the highest professional standards in the field. The emphasis will be on analytical theoretical and mathematical physics in a broad sense.
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