On the Existence of Focal Points near Closed Geodesics on Surfaces

We establish some criteria for the existence or nonexistence of focal points near closed geodesics on surfaces. These criteria are in terms of the curvature of the manifold along the closed geodesic and the average values of the partial derivatives of the curvature in the direction perpendicular to the geodesic. Our criteria lead to a new family of examples of surfaces with no focal points. We also show that if S is a compact surface with no focal points and an inequality relating the curvature of the surface to the curvature of the horocycles holds, then the horocycles (considered as curves in S) are uniformly C ^2+Lipschitz.     

On Sinai–Bowen–Ruelle Measures on Horocycles of 3-D Anosov Flows

Let ^t be a topologically mixing Anosov flow on a 3-D compact manifold M. Every unstable fiber (horocycle) of such a flow is dense in M. Sinai proved in 1992 that the one-dimensional SBR measures on long segments of unstable fibers converge uniformly to the SBR measure of the flow. We establish an explicit bound on the rate of convergence in terms of integrals of Hölder continuous functions on M.