1. title: Quasimultiplicativity of a typical fiber-bunched SL_n(R) cocycle over a subshift of finite type.

Abstract: Following Bonatti-Viana/Avila-Viana, a fiber-bunched SL_n(R) cocycle $$extract_itex$$A$$/extract_itex$$ $$extract_itex$$A$$/extract_itex$$ over a SFT $$extract_itex$$\left(Sigma,f\right)$$/extract_itex$$ $$extract_itex$$(Sigma, f)$$/extract_itex$$ is called typical if there is a fixed point p in which A(p) has simple eigenspectrum and a homoclinic point z of p such that the holonomy loop psi_p^z = H^{s}_{z,p}H^u_{p,z} twists the eigendirections of A(p) into general position. Such are known to be sufficient conditions for the simplicity of the Lyapunov exponents for any ergodic measure with local product structure. Under the same condition, we show that the quasimultiplicativity holds for the cocycle A; suitably interpreted, the quasimultiplicativity can be considered as the submultiplicativity on the norm of the dynamically defined product A^n(x). We will also discuss some applications coming from the quasimultiplicativity.

2. title: Unique equilibrium state for geodesic flow on surfaces with no focal points.

Abstract: Due to work of Bowen, it is well-known that any system with expansivity and the specification property has a unique equilibrium state for any potential with Bowen property (bounded variation). Such pair includes a hyperbolic system with a Holder potential. Recently, Climenhaga and Thompson developed a program to relax the assumptions from Bowen's work to a non-uniformly hyperbolic setting. Using their techniques, Burns-Climenhaga-Fisher-Thompson established the existence and the uniqueness of an equilibrium state for a large class of potentials over the geodesic flow on closed rank one manifolds. We show that their results can be extended to the geodesic flow over surfaces with no focal points, and will discuss properties of the unique equilibrium state coming from the geometry.