Description:
Ergodic theory is concerned with the behavior of dynamic systems when these are running for a long time. Vaguely speaking, the long-term statistical behavior of an ergodic dynamical system is not going to depend on its initial condition. This course discusses the mathematical characterization of this property. A central role is going to be played by the so-called transfer operator, which describes the action of the dynamics on a distribution of states. We are also going to higlight its importance in applications, when it comes to the numerical approximation of quantities of interest. If time permits, we will introduce entropy, as a notion of complicatedness (or "non-predictability") of a dynamical system.

Planned content

1. Basic Constructions:
   measure preserving transformations, ergodicity and mixing, the Poincaré Recurrence Theorem, stochastic processes as deterministic dynamical systems, ergodicity and mixing, Markov chains, skew products
2. Ergodic Theorems:
   Perron-Frobenius theory, von Neumann's Mean Ergodic Theorem, Birkhoff's Pointwise Ergodic Theorem, Kingman's Subadditive Ergodic Theorem (measure disintegration, ergodic decomposition)
3. Spectral Theory:
   the Koopman Operator and the spectral approach to ergodic theory
4. Transfer Operators:
   the Perron-Frobenius Operator, quasi-compactness, absolutely continuous invariant measures, Ulam's method
5. Entropy: (content covered here depending on time left) information content, entropy of a partition, metric entropy, topological entropy.
6. Applications:
   Oceanography, atmospheric and molecular dynamics, internet search.