### Dynamical Systems - 8EC

Prerequisites

Prerequisite is material covered in a standard bachelor program in mathematics, containing in particular a bachelor course on ordinary differential equations, standard concepts from topology and a bit of measure theory. Results for dynamical systems generated by maps or differential equations are usually developed in parallel. Our focus will be on dynamical systems generated by maps. A bachelor course on differential equations treats how a differential equation gives rise to a flow, i.e. a dynamical system, and starts a study of its qualitative properties. An example of a textbook that develops this theory is R.C. Robinson, An Introduction to Dynamical Systems. Notions and techniques from topological dynamical systems are used throughout the course and require knowledge of topology as taught in a bachelor program. See for instance part I (General Topology) from J.R. Munkres, Topology. Familiarity with basic notions of measure theory is used when some results from ergodic theory, i.e. the statistical means to study dynamical systems, are addressed. There are many introductory texts to measure theory and integration, such as R.L. Schilling, Measure, Integrals and Martingales or H. Bauer, Measure and Integration Theory.

Aim of the course

The aim of this course is to introduce students to the concepts, examples, results and techniques for studying smooth dynamical systems generated by ordinary differential equations or maps. The student learns to apply techniques from analysis and topology to study properties of dynamical systems.

We provide a broad introduction to the subject of dynamical systems. In particular we develop theory of topological dynamics, symbolic dynamics and hyperbolic dynamics. Several examples are used to illustrate the theory and clarify the development of the theory. An aim of dynamical systems theory is to describe asymptotic properties of orbits for typical initial points. The strength and beauty of the theory lies herein that techniques to do so work not only for special examples, but for large classes of dynamical systems. The focus of the course will always be on learning techniques to analyse dynamical systems without relaying on explicit formulas for the dynamical system.

As an example, the hyperbolic torus automorphism  $(x,y) \mapsto (2 1 // 1 1) (x,y) \mod 1$ on the torus $R^2/Z^2$ is a topologically transitive dynamical system for which most orbits lie dense in the torus. What makes the example relevant is that small perturbations of it share relevant properties. The automorphism is for instance $C^1$-structurally stable, so that a $C^1$ small perturbation is also topologically transitive. To see this requires much more advanced techniques than needed to study the linear automorphism. These techniques rely on the construction of stable and unstable manifolds. The stable manifold theorem is among the highlights of the course. Another central result we will cover is the structural stability theorem for hyperbolic sets.

A topical description of contents:

• Topological dynamics. Notions to describe attractors, limit sets and chaotic dynamics such as recurrence, topological transitivity, topological mixing.
• Symbolic dynamics and their use to study chaotic dynamics. Full shift. Subshift of finite type. Topological Markov chain.
• Examples of chaotic dynamical systems such as hyperbolic torus automorphisms, the Smale horseshoe map and the solenoid.
• Ergodic theory: basic notions such as the Birkhoff ergodic theorem.
• Hyperbolic dynamics. Stable manifolds. Shadowing (finding real orbits near approximate orbits).
• Structural stability and its relation with hyperbolicity. Shadowing as a technique to study structural stability.