On two noteworthy deformations of negatively curved Riemannian metrics
Gabriel P. Paternain
Discrete & Continuous Dynamical Systems - A 1999, 5(3): 639-650 doi: 10.3934/dcds.1999.5.639
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: Euler-Lagrange flow variance Entropy Finsler geodesic flow.
Transparent connections over negatively curved surfaces
Gabriel P. Paternain
Journal of Modern Dynamics 2009, 3(2): 311-333 doi: 10.3934/jmd.2009.3.311
Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of a certain PDE that arises naturally in the problem. We also show a local uniqueness result for the trivial connection and that there is a transparent $SU(2)$-connection associated to each meromorphic function on $M$.
keywords: geodesic flow. Complex Hermitian vector bundle transparent connection the Livsic theoremfor non-abelian cocycles
On the injectivity of the X-ray transform for Anosov thermostats
Dan Jane Gabriel P. Paternain
Discrete & Continuous Dynamical Systems - A 2009, 24(2): 471-487 doi: 10.3934/dcds.2009.24.471
We consider Anosov thermostats on a closed surface and the X-ray transform on functions which are up to degree two in the velocities. We show that the subspace where the X-ray transform fails to be s-injective is finite dimensional. Furthermore, if the surface is negatively curved and the thermostat is pure Gaussian (i.e. no magnetic field is present), then the X-ray transform is s-injective.
keywords: generalised thermostat X-ray transform Pestov identity.

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