What is a topological dynamical system? The doubling map and other basics.

What is a topological dynamical system? Here we go over the basics of discrete dynamics of metrizable spaces, and we will give a handful of examples including the doubling map. Other examples come from two families of well studied cases, those of circle maps and directed graphs. After giving the definition of a topological dynamical system we go through some basic computations for these systems and then introduce the idea of topological conjugacy — the notion of equivalence we use to distinguish these systems along with an infinite family of conjugacy invariants. At the end of the video we will show that there are infinitely many non-conjugate circle maps.

The video can be broken up into the following chapters:
00:00 Intro
00:25 What is a topological dynamical system?
02:10 Some examples, The doubling map and directed graphs
05:15 Basic computations for topological dynamical systems
09:58 Why is the doubling map the "doubling" map
13:40 Where do we start in mathematics? Topological Conjugacy and Invariants
17:00 Count of periodic points of a certain period is a conjugacy invariant
18:40 There are infinitely many non-conjugate circle maps.

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