Our aim in this chapter is to examine the role of ergodicity in the method of averaging from a general standpoint. To this end, we shall follow Anosov’s article [Ano], and see how divergence between exact and averaged systems takes place when the fast variables belong to a fiber on which the unperturbed system is not ergodic, or does not permit sufficiently fast “mixing” (mixing is not meant in the technical sense here). In the case of quasiperiodic unperturbed systems (to be discussed in Chapters 3 through 6), these fibers correspond respectively to exact resonances and to the “resonant zones” around them, a distinction to which we will return at the beginning of Chapter 5.
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