Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

Abstract

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