Passeios e bilhares: Uma incursão em sistemas dinâmicos aleatórios

Abstract

In the first part of the thesis, we work with the random walk in a random environment determined by a partially hyperbolic diffeomorphism. In this work, we found necessary and sufficient conditions for the existence of a stationary measure for this random process, recurrence and we made a study of the dynamics of this process. As a particular case, we studied the time 1 of the geodesic flow in a compact hyperbolic manifold. In addition, we obtained a law of large numbers and a Central Limit Theorem. In the second part of the thesis we defined a random billiard, with a pertur bation in the exit angles and found an invariant measure for this random billiard in general tables. We did a more detailed study in the circle and in this case we found a null Lyapunov exponent, we showed the non-ergodicity of this system and a law called Knudsen’s Strong Law. We show that under certain conditions, almost all (random) trajectory is dense at the edge of the circular table. We also introduced the concept of pseudo caustics.