# DYNAMICAL SYSTEMS A

Degree course:
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree:
2017/2018
Year:
1
Academic year in which the course will be held:
2017/2018
Course type:
Compulsory subjects, characteristic of the class
Credits:
8
Period:
First Semester
Standard lectures hours:
80
Detail of lecture’s hours:
Lesson (80 hours)
Requirements:

Basic calculus, measure theory.

Final Examination:
Orale

Written take-home exam plus oral examination.

Assessment:
Voto Finale

A first introduction to ergodic theory.

1. Introduction. Metrical framework. Examples.
2. Measure preserving transformations. Invariant measures.
3. Bogoliubov Krylov theorem. Examples. Transformations not possessing any invariant
measure.
4. One dimensional maps. Absolutely continuous invariant measures.
5. Perron Frobenius operator, and related theorems.
6. Symbolic dynamics and its invariant measures. Markov chains.
7. Subshifts of finite type. Singular continuous measures.
8. Iterated function systems and their balanced measures.
9. Poincare' return theorem.
10. Limit theorems. Von Neumann ergodic theorem.
11. Birkhoff averages, Birkhoff ergodic theorem.
12. Metrical transitivity. Characterization theorems.
13. Examples of metrically transitive systems. Weyl theorem.
14. Piecewise liner transformations.
15. Strongly mixing systems.
16. Weakly mixing systems.
17. Characterization theorems.
18. Spectral theory of dynamical systems. Koopman Von Neumann operator.
19. Return times. Kac theorem.
20. Examples. Von Neumann Kakutani map, Pomeau-Manneville and Gaspard Wang map.
21. Multifractal statistics and Kac Theorem.
22. Shannon entropy and coding. Shannon Kincin Theorem.
23. Metrical entropy. Definition and properties.
24. Kolmogorov theorem on generating partitions. Examples. Arnol'd cat map.
25. Ruelle theorem and Pesin formula.
26. Algorithmic complexity, Brudno's theorem, dynamical chaos

Online material, lecture notes.

Standard lectures.

n.a.