[Paper Review] Lyapunov exponents and rigidity of Anosov automorphisms and skew products

Radu Saghin, Jiagang Yang|arXiv (Cornell University)|Feb 22, 2018

Mathematical Dynamics and Fractals54 references11 citations

TL;DR

This paper establishes local rigidity for linear Anosov automorphisms and skew products over Anosov diffeomorphisms by showing that volume-preserving $ C^1 $-small perturbations with identical Lyapunov exponents are smoothly conjugate to the original system. The key result is that under irreducibility and simple real eigenvalues with distinct absolute values, Lyapunov exponents alone suffice to guarantee smooth conjugacy, extending rigidity results beyond periodic data.

ABSTRACT

In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism $L$ with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to $L$. We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism $f_0$ over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation $f$ of $f_0$ with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

Motivation & Objective

• To determine whether Lyapunov exponents alone can imply smooth conjugacy for Anosov diffeomorphisms, beyond periodic data.
• To extend local rigidity results from periodic data to Lyapunov exponents in higher-dimensional Anosov systems.
• To investigate whether volume-preserving perturbations of skew products over Anosov maps with matching Lyapunov exponents yield smooth center foliations.
• To explore the conditions under which Lyapunov exponents determine the smooth structure of invariant foliations.
• To provide a framework for extending rigidity results to derived-from-Anosov and higher-dimensional skew product systems.

Proposed method

• Uses Lyapunov exponent equality as a sufficient condition for smooth conjugacy in irreducible linear Anosov automorphisms with simple real eigenvalues and distinct absolute values.
• Applies the invariance principle and Journé's regularity theorem to establish $ C^{1+ε} $ regularity of holonomy maps along unstable and stable foliations.
• Employs uniform $ C^1 $-boundedness of holonomy along leaves and projection techniques to show that the conjugacy is uniformly $ C^{1+ε} $.
• Utilizes the fact that $ C^1 $-close perturbations with preserved Lyapunov exponents and volume imply that the center foliation is $ C^{1+ε} $, leveraging uniformity in holonomy regularity.
• Applies inductive arguments on the Lyapunov flag and uses the minimality of foliations to rule out non-trivial holonomy distortions.
• Relies on the structure of the universal cover and the action of the linear map to contradict non-trivial holonomy, proving conjugacy.

Experimental results

Research questions

• RQ1 Can Lyapunov exponents alone determine smooth conjugacy for $ C^1 $-small volume-preserving perturbations of linear Anosov automorphisms?
• RQ2 Under what conditions does the equality of Lyapunov exponents imply $ C^{1+ε} $ regularity of the center foliation in skew products over Anosov diffeomorphisms?
• RQ3 Is the smooth conjugacy result extendable to derived-from-Anosov maps, where the center foliation is not uniformly expanding?
• RQ4 Can the rigidity result be generalized to higher-dimensional Anosov base maps with simple real eigenvalues and distinct absolute values?
• RQ5 Does the preservation of Lyapunov exponents and volume imply that the perturbed system is smoothly conjugate to a true skew product over the base Anosov map?

Key findings

• For an irreducible linear Anosov automorphism $ L $ with simple real eigenvalues and distinct absolute values, any $ C^1 $-small volume-preserving perturbation with identical Lyapunov exponents is smoothly conjugate to $ L $.
• The center foliation of a volume-preserving $ C^1 $-small perturbation of a skew product over an Anosov automorphism is $ C^{1+ε} $ if the average stable and unstable Lyapunov exponents match those of the base map.
• The conjugacy between the perturbed system and the linear model is uniformly $ C^{1+ε} $, established via Journé's regularity theorem applied to holonomy maps.
• The projection of the unstable foliation of the perturbed system to the linear model is a uniform $ C^{1+ε} $ diffeomorphism on leaves, ensuring regularity of the conjugacy.
• The proof relies on contradiction via holonomy distortion: non-trivial holonomy would contradict the invariance under the linear map $ L $, implying that holonomy must be trivial and thus the conjugacy smooth.
• The results extend to skew products over higher-dimensional tori under the same spectral and volume conditions, suggesting a general framework for Lyapunov-based rigidity.