[Paper Review] The link between the shape of the Aubry-Mather sets and their Lyapunov exponents

Marie-Claude Arnaud|ArXiv.org|Feb 19, 2009

Mathematical Dynamics and Fractals26 references14 citations

TL;DR

This paper establishes a direct geometric-dynamical link between the $C^1$-regularity of Aubry-Mather sets in exact symplectic twist maps and their Lyapunov exponents. Using Green bundles, it proves that zero Lyapunov exponents occur iff the support is almost everywhere $C^1$-regular, while non-zero exponents correspond to $C^1$-irregularity; uniformly hyperbolic sets are everywhere non-regular, and sets near KAM curves are only mildly irregular.

ABSTRACT

We consider the irrational Aubry-Mather sets of an exact symplectic monotone twist map and explain what is the link between the Lyapunov exponents and the shape of such a set. The main tools that we use in the proofs are the so-called Green bundles.

Motivation & Objective

• To establish a geometric criterion for the Lyapunov exponents of Mather measures supported on Aubry-Mather sets.
• To define and analyze $C^1$-regularity of Aubry-Mather sets using Bouligand paratingent cones.
• To clarify the relationship between hyperbolicity and geometric regularity in invariant sets of symplectic twist maps.
• To show that sets near KAM curves, though possibly non-$C^1$-regular, are not highly irregular.
• To prove that uniform hyperbolicity is equivalent to everywhere non-regularity of the support.

Proposed method

• Introduces a notion of $C^1$-regularity for Aubry-Mather sets based on Bouligand paratingent cones.
• Constructs Green bundles $G_+$ and $G_-$ along irrational Aubry-Mather sets using variational and symplectic geometry.
• Uses the transversality of Green bundles to characterize hyperbolicity and non-uniform hyperbolicity.
• Applies a dynamical criterion linking the angle between Green bundles to the sign of Lyapunov exponents.
• Employs the continuity and monotonicity properties of Green bundles to analyze regularity and hyperbolicity.
• Uses the fact that $G_-$ and $G_+$ bound the tangent cone $P_M(x)$ to deduce geometric constraints on the support.

Experimental results

Research questions

• RQ1 Can the Lyapunov exponents of a Mather measure be determined from the $C^1$-regularity of its support?
• RQ2 What is the geometric condition on an Aubry-Mather set that implies uniform hyperbolicity?
• RQ3 How does the regularity of the support relate to non-uniform hyperbolicity?
• RQ4 Are Aubry-Mather sets near KAM curves geometrically constrained in their irregularity?
• RQ5 Is there a precise link between the transversality of Green bundles and the non-vanishing of Lyapunov exponents?

Key findings

• A Mather measure has zero Lyapunov exponents if and only if its support is $C^1$-regular at almost every point.
• A Mather measure has non-zero Lyapunov exponents if and only if its support is $C^1$-irregular at almost every point.
• An Aubry-Mather set is uniformly hyperbolic if and only if it is nowhere $C^1$-regular.
• For non-uniformly hyperbolic Mather measures, the set of points where $G_-(x) = G_+(x)$ is a dense $G_ u$-set, and such points are $C^1$-regular.
• Aubry-Mather sets near KAM curves are not uniformly hyperbolic and have small paratingent cones, indicating only mild irregularity.
• The Green bundles $G_+$ and $G_-$ are transverse at every point of a uniformly hyperbolic set, and this transversality characterizes uniform hyperbolicity.