Lecture 4: Metrics, Lipschitz Continuity, and the Contraction Principle

The relevant text sections for this lecture are Section 2.1.3, 2.1.4, and 2.4.1.  Please read these prior to meeting on Thursday.

Last lecture, we saw that understanding discrete dynamical systems will require an understanding of orbits and how they behave.  As discrete orbits are sequences in a space, convergence properties will be important.  One aspect of measuring convergence is via a metric, and metric spaces will play a vital role in this entire course.  In this lecture, we define a metric and discuss some of its properties. 

We also introduce a notion of continuity, Lipschitz continuity, that will facilitate our study.  With that, we can move into our first major topic and theorem of the course:  The Contraction Principle.  While a straightforward concept, there are aspects of it that are a bit tricky, and we will take care to expose some of these thoughtful ideas as we discuss what is a contraction and what information does it convey. 

Here are the lecture notes. 

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