General concepts in ergodic theory
Official Course Description
Ergodic theory provides some of the most profound tools of analyzing ''chaotic systems'' with statistical properties. While it is impossible to understand the behavior of the complete system, ergodic theory tells you to make predictions for ''typical'' initial conditions. Ergodic theory might be considered as a mixture of probability theory, measure theory, dynamical system theory, Hamiltonian dynamics or functional analysis. However the very essence is to provide new methods and tools to understand these fields in a better way. In the lectures we will explain and prove the main ergodic theorems due to von Neumann, Birkhoff, Hopf, Kingman, Oseledets and their generalizations and implications in dynamical system theory. To avoid more advanced algebraic knowledge we will restrict to one-dimensional discrete or continuous time dynamical systems.