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6). 7 Mixing can be defined for endomorph isms (Appendix 6) which are not automorphisms (see Appendix 14). 8 Between ergodicity and mixing there is another concept, which is also an invariant of the dynamical systems, the concept of weak mixing (see Halmos [1)). (B)1 o n=O 3 1n contrast, if p. (M) = 00, A. 8. (M) = oc, A and B two measurable sets. [,ptA nB) < E. 22 ERGODIC PROBLEMS OF CLASSICAL MECHANICS in the discrete case, for every pair of measurable sets A, 8. R. V. Chacon (forthcoming paper) proved that if (M, /1, ¢ I) is ergodic, then there exists a measurable change of the modulus of the velocity which makes the system weakly mixing.

Series 2, 31 (1963) pp. 62-84. CUAPTER ~ lI\STABLE SYSTE\IS This chapter contains the study of classical systems with strongly stochastic properties, the so-called C-systems. 1 The orbits of a C-system are highly unstable: two orbits with close initial data are exponentially divergent. This property turns out to imply the asymptotic independence of past and future: C-automorphisms are ergodic, mixing, have Lebesgue spectrum, have positive entropy, and, in general, are K-automorphisms. The set of the C-systems defined on a prescribed manifold M is an open set in the space of the classical systems defined.

Kolmogorov [2) introduced this class under the name of quasi-regular sys terns. 33 ERGODIC PROPERTIES by the: Let ct be the algebra generated by the A Ij,s, i < O. We know that: ¢(Aj) = Aj+l where ¢ is the shift. Hence ¢ct is the algebra generated by the k ~ At's, 1, and ct c ¢ct, proving the property (a). On the other hand, every generator ¢q(A/) = Aj+q , i ~ 0, for q = A! of 1 is a r-i. Hence we get the property (b): 00 V ¢nct Let us now prove the property (c). Let 93 = i. be the subalgebra of 1, each element of which belongs to some subalgebra generated by a finite number of AI- To every A = fl(A) " fl(B) 93 f for any B there corresponds an N f ¢ -nct, n fl(A) "fl(B) holds for every B still holds for any A f i f Z such that fl(A ~ N (exercise).

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