Dynamical Systems and Ergodic Theory

Abstract

Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function.

Objective

By the end of the unit the student: will have developed a good background in the area of dynamical systems; will be familiar with the basic concepts, results, and techniques relevant to the area; will have detailed knowledge of a number of fundamental examples that help clarify and motivate the main concepts in the theory; will understand the proofs of the fundamental theorems in the area; will have mastered the application of dynamical systems techniques for solving a range of standard problems; will have a firm foundation for further study in the area.

Content

Dynamical systems is an exciting and very active field in pure (and applied) mathematics, that involves tools and techniques from many areas such as analysis, geometry and number theory. This introductory course will focus on discrete time dynamical systems, which can be obtained by iterating a function. Even if the rule of evolution is deterministic, the long term behavior of the system is often chaotic. Different branches of dynamical systems, in particular ergodic theory, provide tools to quantify this chaotic behaviour and predict it on average. We will give a strong emphasis on presenting many fundamental examples of dynamical systems. Driven by the examples, we will first introduce some of the phenomena and main concepts which one is interested in studying. We will then formalize these concepts and cover the basic definitions and some fundamental theorems and results in topological dynamics, in symbolic dynamics and in particular in ergodic theory. During the course we will also mention some applications for example to number theory and information theory. Topics which will be covered include: -Basic examples of dynamical systems (e.g. rotations and doubling map; baker’s map, CAT map and hyperbolic toral automorphisms; the Gauss map and continued fractions); -Elements of topological dynamics (minimality; topological conjugacy; topological mixing; topological entropy); -Elements of symbolic dynamics (shifts and subshifts spaces; topological Markov chains and their topological dynamical properties; symbolic coding); -Introduction to ergodic theory: invariant measures; Poincaré recurrence; ergodicity; mixing; the Birkhoff Ergodic Theorem and applications; Markov measures; the ergodic theorem for Markov chains and applications to Internet Search and MCMC methods; measure theoretic entropy; -Selected topics (time permitting): Shannon-McMillan-Breiman theorem; transfer operators, continuous time dynamical systems, dynamics on hyperbolic surfaces.

Resources