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By D. V. Anosov

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W), v = 1, ···,n. This is proven analogously. , 9l) moves with velocity -d d[(w,x(w)) = (f(w),Drx(w)). Thus an invariant interpretation independent of coordinates can be given not to Dt K, but to the pair (/, Dt K), In the following section, the smooth (class C"') sections ; and >. of the fibrations Hom (I , 9l) and Hom

CHAPTER II §6. Introductory remarks In this chapter we prove some lemmas by means of which Theorems 1-3, 8 and 9 will be proven in the following chapter. Since any (U)-cascade can be embedded in a (U)-flow, as was described in §2B, one can, as a rule, limit oneself to the proof of theorems about (U)-flows, and the theorems about (U)cascades can be derived from the corresponding theorems about (U)-flows in a fairly simple way. Therefore in this chapter we consider only (U)-flows. We shall assume that we have some (U)-flow {T' l given by the system of differential equations ~=f(w).

The theory he constructs allows us to deduce from the metrical transitivity of the foliations ~k and ~ 1 the fact that a cascade is a K-cascade [69). Thus Theorem 5 is a corollary to Theorem 11. Later progress in entropy theory allowed Sinai [69] to give a new and possibly better proof of Theorem 11, but here I give my older proof. This is justified by the fact that it is closely connected with the corresponding argument for (U)-flows, for which I think, so far, there are no essential improvements as in the case of cascades.

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