A geometrical point of view for the noncommutative ergodic theorems

Tese de mestrado em Matemática, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, em 2018The strong law of large numbers, surely a classical result in probability theory, says that the more an experiment is repeated the closer the sample mean is to the expected value. The attempts at bringing an analogue of this result to statistical mechanics, although hard to formulate from a mathematical point of view, gave rise to Ergodic Theory. Ergodic theory includes itself in the study of dynamical systems, namely it studies asymptotic behaviours of orbits from a measure theory viewpoint by looking at averages. The first results in ergodic theory, von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theorem as well as the strong law of large numbers all have an important aspect in common - the commutativity of the operation at hand. In the 50’s Furstenberg and Kesten asked themselves how could they extend such results to more general scenarios, specifically the case in which we work with groups whose commutativity may fail. It took until the 60’s for the first answers to such problems to be recorded. These were Furstenberg-Kesten Theorem, Kingman Subadditive Ergodic Theorem and Oseledets Multiplicative Ergodic Theorem. This text aims to present the noncommutative ergodic theorems from a geometrical point of view. The first to notice the relationship between geometry and Oseledets theorem was Kaimanovich by looking at it as a consequence of the action of GL(d;R) on the space of Symmetric Positive Definite Matrices. Later on, Karlsson and Margulis further extended the works of Kaimanovich to semigroups of semicontractions of more general spaces. This allows us to translate the problem into a geometric one on which we can use different machinery. The thesis is comprised of three chapters with the goal of presenting all the results above. The first consists of the classical ergodic theory, the second is about the theory of geodesic metric spaces whilst the proof for the main theorems as well as some of the classic ones are presented in the third