On the relevance of resonances

For a chain transitive flow, the averages produced by the invariant probability measures are approximated by multi-valued averages of an inflated flow. The results are applied to singularly perturbed differential equations with slow and fast trajectories. Here, the multi-valued character of the averaged system is due to the presence of many invariant probability measures, often caused by resonances. Hence, the inflation of a chain transitive flow gives information on the relevance of resonances. The Takens equation modelling strong elastic constraints is treated in detail.