Prerequisites. The courses of the Laurea Triennale.
Teaching Methods. Assignment of exercises, and correction by the
Verification of Learning. Oral exam.
More Information. If required, the course may be taught in English.
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.
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.
Walters. An introduction to ergodic theory. Springer.
Università degli Studi di Udine Dipartimento di Scienze Matematiche, Informatiche e Fisiche (DMIF) via delle Scienze 206, 33100 Udine, Italy Tel: +39 0432 558400 Fax: +39 0432 558499 PEC: firstname.lastname@example.org p.iva 01071600306 | c.f. 80014550307
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