Convergence of the Krylov–Bogolyubov Procedure in Bowan's Example

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.