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 hμ of the geodesic flow on M satisfies the inequality\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{\mu } \geqslant {\int\limits_{SM} {{\text{Tr}}{\sqrt { - K{\left( v \right)}} }d\mu {\left( v \right)}} },$\end{document}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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$K{\left( \upsilon \right)} = {\user1{\mathcal{R}}}{\left( { \cdot ,\upsilon } \right)}\upsilon $\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\user1{\mathcal{R}}}$\end{document} is the Riemannian curvature of M, and μ is the normalized Liouville measure on SM.