Geodesic
Existence of smooth ergodic flows on smooth manifolds
Izvestiya. Mathematics , Tome 8 (1974) no. 3, pp. 525-552.

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It is proved that on any compact, connected, smooth manifold of dimension greater than two there exist smooth flows preserving a given measure with smooth positive density and ergodic with respect to it. (The smoothness is everywhere of infinite order.) 
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D. V. Anosov. Existence of smooth ergodic flows on smooth manifolds. Izvestiya. Mathematics , Tome 8 (1974) no. 3, pp. 525-552. http://geodesic.mathdoc.fr/item/IM2_1974_8_3_a6/

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