The aim of the course is the understanding of various phenomena in topological and measurable dynamics via concepts and methods from functional analysis. Recurrence and mixing properties will be examined from a systematic perspective based on combinatorial independence and involving ideas rooted in the geometric theory of Banach spaces. We will touch for example on embeddings of finite-dimensional l_p spaces, Rosenthal's l_1 theorem, and the Radon-Nikodym property. Dynamical topics will include entropy, hereditary nonsensitivity, nullness, tameness, and weak mixing. A recurring theme in both the topological and measurable settings will be the appearance of dichotomies which separate tame systems from those which exhibit chaotic or random behavior. In particular we will study the structure theorem which underlies Furstenberg's dynamical approach to Szemeredi's theorem on the existence of arbitrarily long arithmetic progressions in subsets of natural numbers with positive upper density. Since we will typically work in a general group action framework, certain analytic aspects of the theory of discrete groups such as amenability will also be discussed. The course will be introductory from the perspective of dynamics, but some background in functional analysis will be assumed (e.g., l_p spaces, spectral theory for unitary operators on a Hilbert space). Prerequisite: MATH 608.
The following are basic references for the dynamical concepts in the course.
- Eli Glasner, Ergodic Theory via Joinings (American Mathematical Society, Providence, RI, 2003)
- Karl Petersen, Ergodic Theory (Cambridge University Press, Cambridge, 1989)
- Peter Walters, An Introduction to Ergodic Theory (Springer-Verlag, New York, 2000)