Tese de Doutorado - Edhin Franklin Mamani Castillo

About the measure of maximal entropy and horospherical foliations of geodesic flows of compact manifolds without conjugate points

In this thesis, we study some dynamical and geometrical properties of the geodesic flow of certain compact manifolds without conjugate points. The thesis has two main parts. In the first one, we extend the work of Gelfert and Ruggiero about the existence of a expansive factor for the geodesic flow of compact surfaces without focal points to compact surfaces without conjugate points and genus greater than one. The main idea of their work is to define an equivalence relation that collapses bi-asymptotic orbits of the geodesic flow. The expansive factor is time-preserving semi-conjugate to the geodesic flow through the quotient map. We verify that this expansive model is topologically mixing, and has a local product. These properties imply that the model has the specification property and a unique measure of maximal entropy. We lift this measure to the unit tangent bundle and verify that it is the unique measure of maximal entropy for the geodesic flow. This provides an alternative proof ot the uniqueness of the measure of maximal entropy in the same context by Climegnaga-Knieper-War. In the second part of the thesis, we extend some results of Gelfert and Ruggiero from higher genus compact surfaces without conjugate points having continuous Green bundles to compact n-manifolds without conjugate points with Gromov hyperbolic fundamental group and having continuous Green bundles. In this setting, we see that if there exists a hyperbolic closed orbit for the geodesic flow, then hyperbolic periodic points are dense and the Pesin set agrees with a dense open set a. e. with respect to the Liouville measure. We deduce that Green bundles are the unique n-1 dimensional bundles invariant by the geodesic flow derivative. Moreover, these bundles are tangent to the horospherical foliations. We also show that horospherical foliations are the unique n-1 dimensional foliations of the unit tangent bundle, invariant by the geodesic flow. This fact was only known for compact surfaces without conjugate points by the work of Barbosa-Ruggiero, and in higher dimensions assuming the stronger condition of bounded asymptote by the work of Eschenburg. With respect to the expansive model for this case, we prove that the quotient space is a 2n-1 topological manifold