We consider the behavior of a passive scalar being transported by a 2-dimensional diffusively-perturbed pseudoperiodic flow. Pseudoperiodic
flows, introduced by Arnold in 1991, have several physically-relevant types of behavior. They are ergodic in parts of the phase space, and they have periodic orbits in other parts (and homoclinic orbits in between). We are interested in small diffusive perturbations of such dynamics.
In particular, we are interested in transitions between the periodic orbits and the ergodic orbits. We adapt the theory of stochastic averaging (in particular, some work of Freidlin) to study these questions when we look at long periods of time. We find that, in an appropriate sense, a Markov process on a graph asymptotically gives the important statistics.
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