What is a measure preserving dynamical system?

Dynamics and Measure Theory are very closely intertwined fields of modern mathematics and they are connected by the study of measure preserving dynamical systems, or those dynamical systems that preserve a measure under backwards iteration. Measure preservation is a very important concept that can be talked about with even introductory examples like with the doubling map and I give a bit of argument as to why the Lebesgue measure is preserved by the doubling map. Here I also go through the Krylov–Bogolyubov theorem and the Poincaré Recurrence theorem, two standard results in the field, the former a justification for the study of the area in some sense, and the latter of historical significance to the field.

00:00 Intro
00:38 What is a measure preserving dynamical system?
01:28 The Lebesgue measure is preserved by the doubling map.
03:30 Preserved Dirac Measure example
04:22 When do you have a preserved measure? The Krylov–Bogolyubov theorem
10:41 The Poincaré Recurrence theorem

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