International School on Dynamical Systems & Applications - DSA2020.2

Título: A Brief Introduction to Ergodic Theory

Palestrante: Jaqueline Siqueira
Data: 17, 19, 24 e 26 de Agosto 
Horário: 14:00 - 15:30h 

Abstract: Describing the behaviour of the orbits of a dynamical system can be a challenging task, especially for systems that have a complicated topological and geometrical structure. A very useful way to obtain features of such systems is via invariant probability measures. For instance, by Birkhoff’s Ergodic Theorem, almost every initial condition in each ergodic component of an invariant measure has the same statistical distribution in space. Moreover, in recent years, Ergodic Theory has proven to be a very effective tool in solving problems in other fields such as Topology, Differential Geometry and Number Theory (e.g. the celebrated Green-Tao Theorem).
The purpose of this mini course is to introduce some of the fundamental features of Ergodic Theory, such as invariant and ergodic probability measures, the Poincaré Recurrence Theorem and Birkhoff’s Ergodic Theorem, thus providing a starting point in this great field.

The course is divided in 4 lectures:

Lecture 1 - The course will begin with a quick review of Measure Theory, providing the needed requisites. I will then introduce the concept of invariant measure and derive certain dynamical properties via the Poincaré Recurrence Theorem.

Lecture 2 - In this lecture we will show that under general conditions one can guarantee the existence of invariant measures. We will then discuss a few examples of dynamical systems with invariant measures.

Lecture 3 - In this lecture we introduce the concept of ergodicity and discuss examples of ergodic systems.

Lecture 4 - In the last lecture we will formulate Birkhoff’s Ergodic Theorem and present a few nice consequences of this powerful result.