The link between the shape of the irrational Aubry-Mather sets and their Lyapunoy exponents

• 成果类型: Article
• 署名作者: Arnaud, Marie-Claude
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.3.4
• 发表日期: 2011
• 页码: 1571-1601
• 关键词: conjugate-points; EXISTENCE; annulus; criterion; systems; THEOREM; PROOF; MAPS

摘要

We consider the irrational Aubry-Mather sets of an exact symplectic monotone C-1 twist map of the two-dimensional annulus, introduce for them a notion of C-1-regularity (related to the notion of Bouligand paratingent cone) and prove that a Mather measure has zero Lyapunov exponents if and only if its support is C-1-regular almost everywhere; a Mather measure has nonzero Lyapunov exponents if and only if its support is C-1-irregular almost everywhere; an Aubry-Mather set is uniformly hyperbolic if and only if it is irregular everywhere; the Aubry-Mather sets which are close to the KAM invariant curves, even if they may be C-1-irregular, are not too irregular (i.e., have small paratingent cones). The main tools that we use in the proofs are the so-called Green bundles.