[Paper Review] Positive Lyapunov exponents for symplectic cocycles

Mário Bessa, Paulo Varandas|arXiv (Cornell University)|Jul 1, 2014

Mathematical Dynamics and Fractals9 references3 citations

TL;DR

This paper establishes that for an open and dense set of Hölder continuous symplectic cocycles over non-uniformly hyperbolic diffeomorphisms, all invariant ergodic measures with local product structure exhibit non-zero Lyapunov exponents. The result confirms Viana's conjecture by proving the generic existence of positive Lyapunov exponents in symplectic dynamical systems using perturbation techniques and the properties of local product structure.

ABSTRACT

In the present paper we give a positive answer to a question posed by Viana on the existence of positive Lyapunov exponents for symplectic cocycles. Actually, we prove that for an open and dense set of Holder symplectic cocycles over a non-uniformly hyperbolic diffeomorphism there are non-zero Lyapunov exponents with respect to any invariant ergodic measure with the local product structure.

Motivation & Objective

• To resolve a conjecture by Viana regarding the generic existence of positive Lyapunov exponents in symplectic cocycles.
• To establish that non-zero Lyapunov exponents are prevalent in symplectic systems under mild regularity and hyperbolicity conditions.
• To analyze the dynamical behavior of symplectic cocycles over non-uniformly hyperbolic diffeomorphisms using ergodic measures with local product structure.
• To demonstrate that positivity of Lyapunov exponents is not an exceptional property but holds generically in the Hölder topology.

Proposed method

• Utilizes perturbation theory in the Hölder topology to construct symplectic cocycles with non-zero Lyapunov exponents.
• Applies the theory of local product structure of invariant measures to ensure hyperbolicity and regularity in the Lyapunov spectrum.
• Employs the Oseledets multiplicative ergodic theorem to analyze the growth rates of vector norms under cocycle dynamics.
• Relies on the openness and density of the set of cocycles with positive Lyapunov exponents in the space of Hölder symplectic cocycles.
• Uses the symplectic structure to constrain the Lyapunov spectrum, ensuring that positive exponents must appear in pairs.

Experimental results

Research questions

• RQ1 Does a generic Hölder symplectic cocycle over a non-uniformly hyperbolic diffeomorphism have non-zero Lyapunov exponents for invariant ergodic measures with local product structure?
• RQ2 Is the set of symplectic cocycles with positive Lyapunov exponents open and dense in the Hölder topology?
• RQ3 Can the positivity of Lyapunov exponents be guaranteed for all such invariant measures under mild regularity assumptions?
• RQ4 How does the local product structure of invariant measures influence the spectrum of Lyapunov exponents in symplectic systems?

Key findings

• For an open and dense subset of Hölder symplectic cocycles over non-uniformly hyperbolic diffeomorphisms, all invariant ergodic measures with local product structure have non-zero Lyapunov exponents.
• The existence of positive Lyapunov exponents is generic in the Hölder topology, confirming Viana's conjecture.
• The result holds uniformly across all invariant ergodic measures that possess the local product structure, a key property in non-uniformly hyperbolic systems.
• The symplectic structure ensures that Lyapunov exponents appear in pairs of opposite signs, but the genericity result implies that positive exponents are prevalent.