An estimate for the entropy of Hamiltonian flows
| dc.creator | Chittaro, Francesca C. | |
| dc.date | 2006-02-28 | |
| dc.date.accessioned | 2026-07-07T07:03:54Z | |
| dc.date.available | 2026-07-07T07:03:54Z | |
| dc.description | In the paper we present a generalization to Hamiltonian flows on symplectic manifolds of the estimate proved by Ballmann and Wojtkovski in \cite{BaWoEnGeo} 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 following inequality: $$ h_μ \geq \int_{SM} \traccia \sqrt{-K(v)} dμ(v), $$ \noindent where $v$ is a unit vector in $T_pM$, if $p$ is a point in $M$, $SM$ is the unit tangent bundle on $M,$ $K(v)$ is defined as $K(v) = \mathcal{R}(\cdot,v)v$, with $\mathcal{R}$ 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 of rank $n-1$ on $N,$ and we assume that the reduced curvature $\hat{R}_z^h$ of the Hamiltonian vector field $\vec h$ is nonpositive. Then we prove that under these assumptions the dynamical entropy $h_μ$ of the Hamiltonian flow w.r.t. the normalized Liouville measure on $N$ satisfies: h_μ \geq \int_N \traccia \sqrt{-\hat{R}_z^h} dμ. | |
| dc.description | 10 pages | |
| dc.identifier | https://arxiv.org/abs/math/0602674 | |
| dc.identifier | http://arxiv.org/abs/math/0602674 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/109157 | |
| dc.subject | Dynamical Systems | |
| dc.subject | Symplectic Geometry | |
| dc.subject | 37C10, 53D12, 58F11 | |
| dc.title | An estimate for the entropy of Hamiltonian flows | |
| dc.type | text |