In this thesis we obtain an abstract continuity theorem for the drift associated with a product of isometries in both Gromov hyperbolic spaces and symmetric spaces as well as the Lyapunov exponents for a product of linear operators over some Hilbert space. We obtain these results by following a recipe of having large deviations estimates and an avalanche principle; a result which allows us to take conclusion of global nature from local hypothesis.
As a main example, we apply the results to cocycles over Markov systems, where we prove the aforementioned large deviations estimates hold, thus providing a large class of examples. Upon presenting the linear setting we also mention the case of quasi-periodic linear cocycles. Whilst exploring Markov systems we also obtain a Fürstenberg type formula.
From the perspective of Gromov hyperbolic spaces, we prove their group of isometries is a topological group and how random products of isometries follow a multiplicative ergodic theorem for the drift, thus describing the behaviour of typical orbits.Financiado pela Universidade de Lisboa ao abrigo do programa de bolsas de doutoramento BD201
