Prerequisites. The courses of the Laurea Triennale.
Teaching Methods. Assignment of exercises, and correction by the
teacher.
Verification of Learning. Oral exam.
More Information. If required, the course may be taught in English.
Objectives
The student will learn the basic aspects of the theory of dynamical systems. He will learn to formalize and treat problems arising in the theory, and from other areas of mathematics. He will have to present some course topic. He will have to study in an autonomous and creative way.
Contents
Monoid and group actions. Measures and their push-forwards. Topological dynamical systems and ergodic systems. Poincaré recurrence theorem. Haar measure. Characterization of ergodic rotations and of ergodic endomorphisms. The Borel theorem on normal numbers. Koopman operator and ergodic theorems. Mixing and weak-mixing systems. Shift spaces and Markov chains. Perron-Frobenius theory. Theorems of Kriloff-Bogoliuboff. Fixed-point theorems. Induced systems and first-return map. Kakutani skyscraper. Kac theorem. Natural extensions. Ergodic measures as extremal measures. Ergodic decomposition theorem.
Texts
Walters. An introduction to ergodic theory. Springer.
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