An estimate for the entropy of Hamiltonian Flows

In this paper, we present a generalization to Hamiltonian flows on symplectic manifolds of the estimate proved by Ballmann and Woitkovski in [41 for the dynamical entropy of the geodesic flow on a compact Riemannian manifold of nonpositive sectional curvature. Given such a Riemannian manifold M, Ballmann and Wojtkovski proved that the dynamical entropy h. of the geodesic flow on M satisfies the inequalityh(mu) >= integral(SM) Tr root-K(v)d mu(v)where v is a unit vector in TpM if p is a point in M, SM is the unit tangent bundle on M, K(v) is defined as K(v) = R(., v)v, where R is the Riemannian curvature of M, and p is the normalized Liouville measure on SM. We consider a symplectic manifold M of dimension 2n, and a compact submanifeld N of M, given by the regular level set of a Hamiltonian function on M; moreover, we consider a smooth Lagrangian distribution on N, and we assume that the reduced curvature R-Z(h) of the Hamiltonian vector field (h) over right arrow with respect to the di...