Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

Abstract

Let mathfrak {X}^{0,1}_nu (M) be the subset of divergence-free Lipschitz vector fields defined on a closed Riemannian manifold M endowed with the Lipschitz topology Vert ,{cdot },Vert _{0,1} where nu is the volume measure. Let mathfrak {X}^{0,1}_{nu ,ell }(M)subset mathfrak {X}^{0,1}_nu (M) be the subset of vector fields satisfying the ell -property, a property that implies C^1-regularity nu -almost everywhere. We prove that there exists a residual subset with respect to Vert ,{cdot },Vert _{0,1} such that Pesin’s entropy formula holds, i.e. for any the metric entropy equals the integral, with respect to nu , of the sum of the positive Lyapunov exponents.