Abstract. We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. We prove a similar result for automorphisms of the Heisenberg manifold fibering over the torus. Our methods also give an alternate proof of Brin and Gromov's result on the ergodicity of these frame flows.
Given a Riemannian manifold (M, g) and a geodesic γ, the perpendicular part of the derivative of the geodesic flow φ t g : SM → SM along γ is a linear symplectic map. We give an elementary proof of the following Franks' lemma, originally found in [7] and [6]: this map can be perturbed freely within a neighborhood in Sp(n) by a C 2 -small perturbation of the metric g that keeps γ a geodesic for the new metric. Moreover, the size of these perturbations is uniform over fixed length geodesics on the manifold. When dim M ≥ 3, the original metric must belong to a C 2 -open and dense subset of metrics.
Let γ be an orbit of the billiard flow on a convex planar billiard table; then the perpendicular part of the derivative of the billiard flow along γ is a symplectic linear map DP. This paper contains a proof of the following Franks' lemma for a residual set of convex planar billiard tables: for any closed orbit, the map DP can be perturbed freely within a neighbourhood in Sp(1) by a C 2 -small perturbation in the space of convex planar billiard tables.
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