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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4 июл 2024
