Dichotomies between uniform hyperbolicity and zero Lyapunov exponents for SL(2,R) cocycles

Abstract

We consider the linear cocycle $(T,A)$ induced by a measure preserving dynamical system $T:X\to X$ and a map $A:X\to SL(2,\mathbb{R})$. We address the dependence of the upper Lyapunov exponent of $(T,A)$ on the dynamics $T$ when the map $A$ is kept fixed. We introduce explicit conditions on the cocycle that allow to perturb the dynamics, in the weak and uniform topologies, to make the exponent drop arbitrarily close to zero. In the weak topology we deduce that if $X$ is a compact connected manifold, then for a $C^r$ ($r\ge 1$) open and dense set of maps $A$, either $(T,A)$ is uniformly hyperbolic for every $T$, or the Lyapunov exponents of $(T,A)$ vanish for the generic measurable $T$. For the continuous case, we obtain that if $X$ is of dimension greater than 2, then for a $C^r$ ($r\ge 1$) generic map $A$, there is a residual set of volume-preserving homeomorphisms $T$ for which either $(T,A)$ is uniformly hyperbolic or the Lyapunov exponents of $(T,A)$ vanish.

To appear on the Bulletin of the Brazilian Mathematical Society

References in corpus (2)

• Genericity of zero Lyapunov exponents
• $L^p$-generic cocycles have one-point Lyapunov spectrum

Cited by in corpus (10)

• Continuity properties of the lower spectral radius
• Examples of Discontinuity of Lyapunov Exponent in Smooth Quasi-Periodic Cocycles
• Generic dynamics of 4-dimensional C2 Hamiltonian systems
• $C^1$-Generic Symplectic Diffeomorphisms: Partial Hyperbolicity and Zero Center Lyapunov Exponents
• Shades of Hyperbolicity for Hamiltonians
• Quasi-Periodic Schrödinger Cocycles with Positive Lyapunov Exponent are not Open in the Smooth Topology
• Hamiltonian elliptic dynamics on symplectic 4-manifolds
• Trivial and simple spectrum for SL(2,R) cocycles with free base and fiber dynamics
• Cocycles with Quasi-Conformality I: Stability and abundance
• On the entropy of conservative flows