Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On two noteworthy deformations of negatively curved Riemannian metrics

Pages: 639 - 650, Volume 5, Issue 3, July 1999      doi:10.3934/dcds.1999.5.639

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Gabriel P. Paternain - Centro de Matemática, Facultad de Ciencias, Iguá 4225, 11400 Montevideo, Uruguay (email)

Abstract: Let $M$ be a closed connected $C^\infty$ Riemannian manifold whose geodesic flow $\phi$ is Anosov. Let $\theta$ be a smooth 1-form on $M$. Given $\lambda\in \mathbb R$ small, let $h_{E L}(\lambda)$ be the topological entropy of the Euler-Lagrange flow of the Lagrangian

$L_\lambda (x, v) =\frac{1}{2}|v|^2_x-\lambda\theta_x(v),$

and let $h_F(\lambda)$ be the topological entropy of the geodesic flow of the Finsler metric,

$F_\lambda(x, v) = |v|_x-\lambda\theta_x(v),$

We show that $h_{E L}''(0) + h''_F(0) = h^2$Var$(\theta)$, where Var$(\theta)$ is the variance of $\theta$ with respect to the measure of maximal entropy of $\phi$ and $h$ is the topological entropy of $\phi$. We derive various consequences from this formula.

Keywords:  Entropy, variance, Euler-Lagrange flow, Finsler geodesic flow.
Mathematics Subject Classification:  58F15, 58F17, 58F05.

Received: October 1998;      Revised: February 1999;      Available Online: May 1999.