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Propriétés ergodiques des mesures de Sinaï. (French) Zbl 0561.58037
The author considers a compact Riemannian manifold M without boundary and a \(C^ 1\)-diffeomorphism \(f: M\to M\) with Hölder-continuous tangential maps. The main result of the paper shows that for an invariant measure m for which the subspace \(E^ 0(x)\) corresponding to the characteristic exponent 0 reduces to \(\{\) \(0\}^ a.\)s., the absolute continuity of m with respect to the unstable foliation is equivalent to the Pesin formula: \(h(m)=\int \sum \lambda^+_ i(x)\dim E^ i(x)dm(x)\). Such a measure the author calls a Sinai measure. Partial results are due to Ya. B. Pesin [Usp. Mat. Nauk 32, No.4(196), 55-112 (1977; Zbl 0359.58010)] and D. Ruelle [Bol. Soc. Bras. Mat. 9, 83-87 (1978; Zbl 0432.58013)]. Various applications and related results for Sinai measures m are given. Among them it is shown that the set of generic points for m has positive Lebesgue measure if m is ergodic and that (M,f,m) is Bernoulli if the system is totally ergodic.
Reviewer: M.Denker

MSC:
37D99 Dynamical systems with hyperbolic behavior
28D05 Measure-preserving transformations
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