Erogodic Theory
Erogodic Theory
One of the main goals of Ergodic theory is to describe the statistical distribution of orbits. A central problem in the theory of Dynamical Systems is Palis Conjecture, which states that typical systems in finite dimensional Riemannian manifolds possess a finite number of measures (physical measures) which describe the time averages of almost all orbits with respect to the Lebesgue (volume) measure, this increased our desire for physical measures.
In this short course, we will introduce the notions of Physical, invariant, and ergodic measures and the role that they play in describing the statistical distribution of orbits. We will get physical measures for some specific classes of dynamical systems.
Lecture 01: Physical measures and their importance
In this lecture we understand what we actually study in ergodic theory, we realize that the orbit can behave very annoying even in a very simple system. we introduce the sequence of Dirac averages and their short and long-term meanings and how this helps us to understand the dynamics while discussing many examples. We define invariant measures and their existence in a system where the underlying space is compact.
References: Foundation of Ergodic Theory by Marcelo Viana and Krerley Oliveira. For further exploration of physical measures read the first chapter of the book by José F. Alves, Nonuniformly Hyperbolic Attractors.
Here you can find the Video Recording and Slides.
Lecture 02: Poincaré Recurrence Theorem
Invariant measures: Invariant measures, examples of invariant measures and Poincare Recurrence Theorem.
Ergodic measures Ergodic measures, Examples of ergodic measures, and Birkhoff’s Ergodic Theorem (the converges of time average to space average).
Reference" Foundation of Ergodic Theory by Marcelo Viana and Krerley Oliveira.