Grigory Margulis used group actions with this property to prove the normal subgroup theorem for semi-simple Lie groups. For 10 points each:
[10m] Name this mathematical property originally motivated by statistical mechanics. A dynamical system has this property if its map T is measure-preserving, and all T-invariant sets either have measure zero or are of full measure.
ANSWER: ergodic [accept word forms, like ergodicity]
[10h] Leo Furstenburg proved this theorem by showing that it follows from a multiple recurrence property satisfied by ergodic maps. This theorem states that any set of integers with positive upper density contains arithmetic progressions of arbitrary length.
ANSWER: Szemeredi’s theorem
[10e] Results about ergodic averages are used to prove Carleson’s theorem, which establishes pointwise convergence of these series. These sums of trigonometric functions were developed by their French namesake to solve the heat equation.
ANSWER: Fourier series
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