Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$, the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

Let M={\tilde M}/\Gamma be a closed negatively curved manifold with universal covering {\tilde M} and fundamental group \Gamma. Every Gibbs equilibrium state \nu of a Hölder continuous function on the unit tangent bundle T^1M of M projects to a \Gamma-invariant ergodic measure class mc(\nu_+) on the ideal boundary \partial{\tilde M} of {\tilde M}. We show that this measure class is also ergodic under the action of any normal subgroup \Gamma^\prime of \Gamma for which the factor group \Gamma/\Gamma^\prime is nilpotent.

Let $f$ be a flip-invariant Hölder continuous functionon theunit tangent bundle $T^1 M$ of a closed negatively curved Riemannian manifold$M$. We show that conditionals on strong unstable manifoldsof the Gibbs equilibrium state defined by $f$can be realized as Hausdorff measures. Moreover,cohomology classes of flip invariant cocycles are inone-to-one correspondence to cross ratios on the space offour pairwise distinct pointsof the ideal boundary of the universal covering $\tilde M$ of $M$.

It is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.

A smooth transitive Anosov flow on a compact manifold N which is uniformly (a, b)-expanding at periodic points for 1 < a < b is uniformly (a − ε, b + ε)-expanding on all of N for all ε > 0.

Let (resp. ) be a smooth contact flow on a compact manifold V1 (resp.) V2 with Anosov splitting of class C1. We show that every time-preserving conjugacy Λ:(V1, )→( V2, ) is necessarily of class C2.

In this note we study Borel-probability measures on the unit tangent bundle ofa compact negatively curved manifold M that are invariant under the geodesic flow. We interpret the entropy of such a measure as a Hausdorff dimension with respect to a natural family of distances on the ideal boundary of the universal covering of M. This in term yields necessary and sufficient conditions for the existence of time preserving conjugacies of geodesic flows.

The Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.