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.) 
@article{IM2_1974_8_3_a6,
     author = {D. V. Anosov},
     title = {Existence of smooth ergodic flows on smooth manifolds},
     journal = {Izvestiya. Mathematics },
     pages = {525--552},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {1974},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1974_8_3_a6/}
}
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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/