Teoria ergodica 2020/2021

Syllabus

Ergodic theory is the mathematical discipline dealing with dynamical systems endowed with invariant measures. A (discrete) dynamical system is a transformation \(f \colon M \mapsto M\) in some measurable space \(M\). We think of \(M\) as the space of all possible states of a given system, and of \(f\) as the evolution law of the system. If the element \(x \in M\) is the initial state of the system, then \(f(x)\) is the state of the system after of unit of time. We always assume that \(f\) is measurable and leaves invariant a measure defined on \(M\).

The topics covered in the course are provisionally the following:

Recurrence:

invariant measures, Poincaré recurrence theorem, examples

Existence of invariant measures:

weak* topology, Krylov-Bogoliubov theorem

Ergodic Theorems:

von Neumann ergodic theorem, Birkhoff ergodic theorem, subadditive ergodic theorem

Ergodicity:

ergodic systems, examples, properties of ergodic measures, ergodic decomposition

Unique ergodicity:

unique ergodicity, minimality, Haar measures

Mixing:

mixing systems, Markov shifts

Equivalent systems:

ergodic equivalence, spectral equivalence

Entropy:

metric entropy, Kolmogorov-Sinai theorem, local entropy, examples

Expanding maps:

existence of absolutely continuous invariant measures

References

1. M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016

2. P. Walters, An introduction to ergodic theory, Springer, 1982