Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms

TL;DR

This paper demonstrates that for certain symplectic cocycles over specific partially hyperbolic systems, the Lyapunov exponents are generically non-zero, indicating stable chaotic behavior in these dynamical systems.

Contribution

It establishes that non-zero Lyapunov exponents are open and dense for symplectic cocycles over particular classes of partially hyperbolic diffeomorphisms.

Findings

Lyapunov exponents are non-zero in an open and dense set

Results apply to systems with compact center leaves and time-one maps of Anosov flows

Provides generic stability of chaotic behavior in these systems

Abstract

We consider symplectic cocycles over two classes of partially hyperbolic diffeomorphisms: having compact center leaves and time one maps of Anosov flows. We prove that the Lyapunov exponents are non-zero in an open and dense set in the H\"older topology.