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| Von Neumann's original proof of the ergodic theorem for one-parameter families of unitary operators relies on a delicate analysis of the spectral measure of the associated flow operator and the observation that over long times only functions that are invariant under the flow make a contribution to the ergodic integral. In this talk I shall show that for a specific class of generators -- namely vector fields -- the spectral measure is rather simple to understand. For some nicely behaved flows this allows us to obtain a uniform ergodic theorem, while for other flows we show that the spectral measure can be purely singularly continuous. These results are closely related to recent results on the 2D Euler equations, and have potential applications for other conservative flows, such as those governed by the Vlasov equation. |
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