SUBJECT

Ergodic theory

lecture

master

Faculty of Science

Mathematics MSc

3

Semesters 1-4

Autumn/Spring semester

Examples. Constructions. Von Neumann L2 ergodic theorem. Birkhoff-Khinchin pointwise ergodic theorem. Poincaré recurrence theorem and Ehrenfest’s example. Khinchin’s theorem about recurrence of sets. Halmos’s theorem about equivalent properties to recurrence. Properties equivalent to ergodicity. Measure preserving property and ergodicity of induced maps. Katz’s lemma. Kakutani-Rokhlin lemma. Ergodicity of the Bernoulli shift, rotations of the circle and translations of the torus. Mixing (definitions). The theorem of Rényi about strongly mixing transformations.  The Bernoulli shift is strongly mixing. The Koopman von Neumann lemma. Properties equivalent to weak mixing. Banach’s principle. The proof of the Ergodic Theorem by using Banach’s principle. Differentiation of integrals. Wiener’s local ergodic theorem. Lebesgue spaces and properties of the conditional expectation. Entropy in Physics and in information theory. Definition of the metric entropy of a partition and of a transformation. Conditional information and entropy. ``Entropy metrics”. The conditional expectation as a projection in L2. The theorem of Kolmogorv and Sinai about generators. Krieger’s theorem about generators (without proof).

• K. Petersen, Ergodic Theory,Cambridge Studies in Advanced Mathematics 2, Cambridge University Press, (1981).
• I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer Verlag, New York, (1981).