Preprint C162/2013
Markov and Lagrange Dynamical Spectra for Geodesic Flows in Surfaces with Negative Curvature.
Sergio Augusto Romaña Ibarra
Keywords: Lagrange and Markov dynamical spectrum | Horseshoe | Hausdorff dimension | pinched curvature negative | Poincaré map | persistently non-empty interior | the Marstrand's theorem.

The current work consists in two parts, both of them related to the study of the fractal geometry.

The first part focuses on showing that the Lagrange and the Markov dynamical spectrum has a non-empty stable interior. First, we study the Lagrange and the Markov dynamical spectrum for diffeomorphisms in surface that have a horseshoe with Hausdorff dimension greater than 1 and the property $V$. We show that for a large" set of real functions on the surface and for a diffeomorphism with a horseshoe associated and Hausdorff dimension  greater than 1, with the property $V$, both, the Lagrange and the Markov dynamical spectrum have persistently non-empty interior. Then, we find hyperbolic sets for the geodesic flow of surfaces of pinched negative curvature and finite volume, with Hausdorff dimension close to $3$. Associated with this hyperbolic set, we find a horseshoe of Hausdorff dimension close to $2$ for Poincaré map. Finally, we prove that the Lagrange and the Markov dynamical  spectrum (associated to geodesic flow) have persistently non-empty interior.

The second part focuses on showing that the Marstrand's thoerem is true in surfaces simply connected and non-positive curvature.