Banca examinadora: Marcelo Viana - Orientador - IMPA Carlos Gustavo Moreira – Coorientador – IMPA Fernando Lenarduzzi – IMPA Karina Marin – UFMG Krerley Irraciel - UFAL Mauricio Poletti - UFC.

Abstract: This thesis is a contribution to the widely studied theory of hyperbolic dynamics. The work is structured in two independent parts which are intrinsically related: Lyapunov exponents for linear cocycles (model for non uniformly hyperbolic dynamics) and geometric properties of horseshoes (uniformly hyperbolic dynamics). In the first part we address the problem of continuity and simplicity of the Lyapunov spectrum for random product k-tuples of quasi periodic cocycles. In dimension two we prove that for any r = 1 there exists a C 0 open and C r dense set of k-tuples of C r quasi periodic cocycles whose random product is a continuity point of the Lyapunov exponents with positive value. Restricting to the k-tuples of Schrödinger cocycles the same results holds. In higher dimensions we prove that among the C r , r = 1, k-tuples with one of the elements diagonal, there exists a C r dense and C 1 open set such that the random product defined by cocycles in this set has simple Lyapunov spectrum and is a C 0 continuity of the Lyapunov exponents. In the second part, using the Erdös probabilistic method, we prove that for typical C r horseshoes with s-splitting, one dimensional weak stable bundle and upper stable dimension smaller than one there exists a sub horseshoe with almost the same upper stable dimension which is contained in a C 1+ locally invariant submanifold tangent to the center unstable direction. .