Entropy rigidity and harmonic fields

We study the problem of entropy rigidity in the sense of Gromov-Katok for deformations of the geodesic flow by twisting the symplectic form by a closed 2-form Omega in M corresponding to a magnetic term. we show that if M is a hyperbolic manifold, then the origin is a convex point of the function given by the difference of the topological and Liouville entropies and has vanishing second derivative iff Omega is harmonic, giving a dynamical characterization of such forms. In particular, we have strict convexity for Euler-Lagrange deformations. AMS classification scheme numbers: 58F15, 58F17, 58F05.