Real analysis, basic notions of measure theory and functional analysis

At least one half of the course will be lectured in english

Ergodic theory is the mathematical discipline dealing with dynamical systems endowed with invariant measures. A (discrete) dynamical system is a transformation f : M → 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 the following:

1. Recurrence: invariant measures, Poincaré recurrence theorem, examples
2. Existence of invariant measures: weak* topology, Krylov-Bogoliubov theorem 
3. Ergodic Theorems: von Neumann ergodic theorem, Birkhoff ergodic theorem, subadditive ergodic theorem
4. Ergodicity: ergodic systems, examples, properties of ergodic measures, ergodic decomposition
5. Unique ergodicity: unique ergodicity, minimality, Haar measures
6. Mixing: mixing systems, Markov shifts
7. Equivalent systems: ergodic equivalence, spectral equivalence
8. Entropy: metric entropy, Kolmogorov-Sinai theorem, local entropy, examples
9. Expanding maps: existence of absolutely continuous invariant measures

1. M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge University Press, 2016
2. P. Walters, An introduction to ergodic theory, Springer, 1982

1. Homework 
2. Final written essay on a topic chosen by student and approved by lecturers
3. Oral presentation of the final essay on Google Meet

The student may choose to write the homework, the final essay and deliver the oral presentation in italian or english.