👾 Welcome to Dynamical Systems II! 👾

I hope you all enjoyed your exams vacation, and that you are all ready to get started with another thrilling semester of Dynamical Systems! 🤓

Our first[1] topic for this semester is hyperbolic differentiable dynamical systems.

We briefly touched upon hyperbolic dynamics under the guise of hyperbolic toral automorphisms all the way back in Lecture 8.

Very roughly speaking, hyperbolicity is the main mechanism used to produce chaos and positive topological entropy in differentiable dynamical systems. A central theme of this course (which we will see time and time again) is that hyperbolicity is persistent: if a given system is hyperbolic, then so are sufficiently “nearby” systems.

  1. and possibly last, if I run out of time... ↩︎

Our eventual aim is to talk about hyperbolic invariant sets for diffeomorphisms on manifolds. We will approach this in three stages:

  • Baby case: linear maps of vector spaces (this is the subject of today's lecture).
  • Local case: local diffeomorphisms of vector spaces (we'll do this on Friday).
  • General case: diffeomorphisms of manifolds (Lecture 35 or thereabouts).

The general case requires us to introduce manifolds and and some of the rudiments of Differential and Riemannian Geometry. If you have not seen this before, don't worry—we will cover everything we need more or less from scratch in this course.

As usual, here are today's notes. Please leave a comment if you spot one of the several million typos that I have deliberately left for you to find.

Comments and questions?