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July  1999, 5(3): 639-650. doi: 10.3934/dcds.1999.5.639

On two noteworthy deformations of negatively curved Riemannian metrics

1. 

Centro de Matemática, Facultad de Ciencias, Iguá 4225, 11400 Montevideo, Uruguay

Received  October 1998 Revised  February 1999 Published  May 1999

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.

Citation: Gabriel P. Paternain. On two noteworthy deformations of negatively curved Riemannian metrics. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 639-650. doi: 10.3934/dcds.1999.5.639
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