Geodesic
Convergence of the Krylov--Bogolyubov Procedure in Bowan's Example
Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 678-689

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In this paper, we study the behavior of time averages of a measure in Bowan's example: a vector field on the plane with two saddles joined by two separatrix connections. We present an explicit criterion for the convergence of averaged measures and describe the set of their partial limits. As a consequence, we show that, for a typical initial measure, its time averages do not converge.
Keywords: vector field in the plane, invariant measure, time average of a measure, Poincaré map, Krylov–Bogolyubov procedure, Bowan's example. 
@article{MZM_2007_82_5_a3,
     author = {T. I. Golenishcheva-Kutuzova and V. A. Kleptsyn},
     title = {Convergence of the {Krylov--Bogolyubov} {Procedure} in {Bowan's} {Example}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {678--689},
     publisher = {mathdoc},
     volume = {82},
     number = {5},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2007_82_5_a3/}
}
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T. I. Golenishcheva-Kutuzova; V. A. Kleptsyn. Convergence of the Krylov--Bogolyubov Procedure in Bowan's Example. Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 678-689. http://geodesic.mathdoc.fr/item/MZM_2007_82_5_a3/