By Chern S., Osserman R. (eds.)

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1. L=' (V) _ 1 (V) U,U2 Note that 10(0) is independent of V. < o. 2) 50 On Some Aspects of the Theory of Anosov Systems For an arbitrary covering {Ui } of W n , by diam{UJ denote sUPi diam U,. Suppose that {Ui } is a finite covering of wn with open subsets, each of which has the A-property with respect to 1+ 1 and k . Divide Vk into nonintersecting V. such that for any i, V. belongs to Ui and coincides with an intersection of a finite number of open or closed sets. Let V be an open subset of V k , and V n 8V k = 0.

79). 5 is complete. D Theorem 4. If f is a continuous function on wn and t > 0, then . 80) 6 Asymptotics of the Number of Periodic 'frajectories 47 Proof. Let us construct a finite covering of wn with U. , Pi, to). Then f = l: f. with f. supported in U•. , we get the assertion of the theorem. 0 Let II(R) be the number of periodic trajectories of {Tt} with minimal periods smaller than R. Theorem 5. 81) Proof. By theorem 4, n - - 1im H(W, dRR - t, R + t) R-oo (5/+1 -1 t - dt _ (5/+1 d -t t. 83) R1 '5:.

The set U(w, Q, P) is naturally isomorphic to the direct product of Q and P, and the topology on U(w, Q, P) coincides with the direct product of topologies on Q and P. Denote by 71"1 the natural projection U(w, Q, P) ~ Q and by 71"2 the natural projection U(w, Q, P) ~ P. 64) Let "( > 0 be fixed. } be a covering of Q with a finite system of its open subsets so that for any i, diamQ. < ,,(, and let {PJ } be a similar covering of P with diam PJ < "( (diam with respect to Pst+l and PSk). , diamPJ ).