1.1. INTRODUCTION 3 continuity of B implies already that ¯(x) is also Lipshitz continuous in x, and so there exists a unique solution ¯ = ¯ x of the averaged equation (1.1.4) d ¯ ε (t) dt = ε ¯( ¯ ε (t)), ¯ ε (0) = x. In this case the standard averaging principle says (see [73]) that for µ-almost all y, (1.1.5) lim ε→0 sup 0≤t≤T/ε |Xε x,y (t) ¯ ε x (t)| = 0. As the main motivation for the study of averaging is the setup of perturbations described above we have to deal in real problems with the fully coupled system (1.1.1) which only in very special situations can be reduced by some change of variables to a much easier uncoupled case where the fast motion does not depend on the slow one. Observe that in the general case (1.1.1) the averaged vector field ¯(x) in (1.1.3) may even not be continuous in x, let alone Lipschitz, and so (1.1.4) may have many solutions or none at all. Moreover, there may exist no natural well dependent on x Rd family of invariant measures µx since dynamical systems Fxt may have rather different properties for different x’s. Even when all measures µx are the same the averaging principle often does not hold true in the form (1.1.5), for instance, in the presence of resonances (see [63] and [56]). Thus even basic results on approximation of the slow motion by the averaged one in the fully coupled case cannot be taken for granted and they should be formulated in a different way requiring usually stronger and more specific assumptions. If convergence in (1.1.3) is uniform in x and y then (see, for instance, [52]) any limit point ¯(t) = ¯ x (t) as ε 0 of x,y (t) = x,y (t/ε) is a solution of the averaged equation (1.1.6) d ¯(t) dt = ¯( ¯(t)), ¯(0) = x. It is known that the limit in (1.1.3) is uniform in y if and only if the flow Fx t on M is uniquely ergodic, i.e. it possesses a unique invariant measure, which occurs rather rarely. Thus, the uniform convergence in (1.1.3) assumption is too restrictive and excludes many interesting cases. Probably, the first relatively general result on fully coupled averaging is due to Anosov [1] (see also [63] and [52]). Relying on the Liouville theorem he showed that if each flow Fx t preserves a probability measure µx on M having a C1 dependent on x density with respect to the Riemannian volume m on M and µx is ergodic for Lebesgue almost all x then for any δ 0, (1.1.7) mes{(x, y) : sup 0≤t≤T/ε |Xx,y(t) ε ¯ ε x (t)| δ} 0 as ε 0, where mes is the product of m and the Lebesgue measure in a relatively compact domain X Rd. An example in Appendix to [56] shows that, in general, this convergence in measure cannot be strengthened to the convergence for almost all initial conditions and, moreover, in this example the convergence (1.1.5) does not hold true for any initial condition from a large open domain. Such examples exist due to the presence of resonances , more specifically to the ”capture into resonance” phenomenon, which is rather well understood in perturbations of integrable Hamil- tonian systems. Resonances lead there to the wealth of ergodic invariant measures and to different time and space averaging. It turns out (see [11]) that wealth of ergodic invariant measures with nice properties (such as Gibbs measures) for Ax- iom A and expanding dynamical systems also yields in the fully coupled averaging
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