Teoremas ergódicos para sistemas multívocos e aleatórios com aplicações a semifluxos generalizados

Abstract

This work focuses on four major themes: semiflows; generalized semiflows; invariant measure for set-valued maps; and random dynamical systems. The study of semiflows aims to understand the asymptotic behavior of solutions to problems with unique solutions. The generalized semiflows, on the other hand, allow the solution for an initial data to be not unique, thus allowing a wider variety of problems. Both studies, in this work, have the main goal of providing conditions for the existence of an attractor, and properties on it. When dealing with invariant measures for set-valued maps, it is important to say that different proposals have been made over the past few years to define this concept. In this work, four of these definitions are presented, and at the end it is shown that they are all equivalent under certain conditions. Finally, a version of ergodic theorem for set-valued map is presented, with an application to a dynamical system wich generates a generalized semiflow with atractor. Finally, in random (discrete) dynamical systems, the system is not limited to the iterations of a unique map, but to successive applications of maps chosen randomly within a certain family. A version of ergodic theorem is presented for this case, with a new application to a dynamical system wich generates a generalized semiflow with atractor.