About the Ergodic Theorem

00:00 Birkhoff's ergodic theorem and von Neumann's ergodic theorem were born at about the same time. Who was first? Here is a table of content. 00:50 The n-body problem as a motivational problem. Poincare. 04:30 Gas dynamics as a second motivational problem. Bolzmann 05:00 Measure preserving transformations 07:00 Ergodicity 07:30 T is ergodic if and only if every invariant random variable is constant 12:00 The ergodic hypothesis and the quasi ergodic hypothesis 13:15 Birkhoffs ergodic theorem. 14:30 Maximal ergodic theorem, ergodic theorem, mean ergodic theorem 15:00 The paper of von Neumann from 1932 15:30 The paper of Birkhoff from 1931 15:45 Statement of the mean ergodic theorem 16:45 Comparing the two theorems: a.e. convergence, L2 convergence 17:36 About George Birkhoff at Harvard 19:10 About John von Neumann at Harvard 20:00 A letter about the controversy from 1931-1932 20:40 Poincare recurrence announcement 21:40 Ergodicity in finite systems 24:15 The rotation system on the circle 24:30 The doubling map on the circle 25:00 Hamiltonian systems like the double pendulum. 26:00 The use of ergodic theory in number theory