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 Wojtkovski in [4] 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 μ of the geodesic flow on M satisfies the inequality $h_{\mu } \geqslant {\int\limits_{SM} {{\text{Tr}}{\sqrt { - K{\left( v \right)}} }d\mu {\left( v \right)}} },$ where v is a unit vector in T p M if p is a point in M, SM is the unit tangent bundle on M, K(v) is defined as $K{\left( \upsilon \right)} = {\user1{\mathcal{R}}}{\left( { \cdot ,\upsilon } \right)}\upsilon $ , where ${\user1{\mathcal{R}}}$ is the Riemannian curvature of M, and μ is the normalized Liouville measure on SM. We consider a symplectic manifold M of dimension 2n, and a compact submanifold 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 $\hat{R}_z^h$ of the Hamiltonian vector field $\vec{h}$ with respect to the distribution is non-positive. Then we prove that under these assumptions, the dynamical entropy h μ of the Hamiltonian flow with respect to the normalized Liouville measure on N satisfies $h_{\mu } \geqslant {\int\limits_N {{\text{Tr}}{\sqrt { - \ifmmode\expandafter\hat\else\expandafter\^\fi{R}^{h}_{z} } }d\mu } }.$