Ergodic Theory - M1 - 8EC

Prerequisites
A background in measure theory obtained from any elementary bachelor course in .
measure theory is necessary. In addition some elementary knowledge of undergraduate functional analysis and point set topology. The relevant information can be found in the appendix in the course book. Alternatively, one can check the first four chapters of the book Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07- 054234-1 by Walter Rudin would provide sufficient background for this course.

Aim of Course
At the end of the course, the student is expected to have knowledge in the following concepts:

  1. The notion of measure preservingness (stationarity) with several interpretations and exam ples. The Poincar´e Recurrence Theorem. The notion of ergodicity (which is a weak notion of independence) and its characterizations, the notion of conservativity (for infinite measure preserv ing systems).
  2. Ergodic Theorems (generalizations of the Strong Law of Large Numbers), such as Birkhoff's and Von Neumann's Ergodic Theorems. Some consequences of the Ergodic Theorems and the notions of weakly and strongly mixing.
  3. Isomorphism, factor maps and natural extensions.
  4. Some worked examples with a probabilistic and number theoretic flavor: Bernoulli and Markov shifts, continued fractions, normal numbers.
  5. The notion of entropy, the Shannon-Mcmillan-Breiman Theorem, and Lochs' Theorem.
  6. Construction of invariant and ergodic measures for continuous transformations, unique ergod icity, uniform distribution and Benford's Law.
  7. The Perron-Frobenius operator and the existence of absolutely continuous invariant measures.
  8. Introduction to Infinite ergodic theory, infinite ergodic theorems, induced and jump transfor mations