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Название: Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities
Авторы: Katok A., Strelcyn J., Ledrappier F.
During the past twenty-five years the hyperbolic properties of smooth dynamical systems (i.e. of diffeomorphisms and flows) were studied in the ergodic theory of such systems in a more and more general framework (see [AnO]l,2, [Sma], [Nit], ri, [Kat] I, [PeS]l, 3, [Rue]2,3). The detailed historical survey of the hyperbolicity and its role in the ergodic theory up to 1967 is given in [Ano]2, Chapter ].
One of the most important features of smooth dynamical systems showing behavior of hyperbolic type is the existence of invariant families of stable and unstable manifolds and their so called "absolute continuity". The most general theorem concerning the existence and the absolute continuity of such families has been proved by Ya. B. Pesin ([PeS]l,2).
The final results of this theory give a partial description of the ergodic properties of a smooth dynamical system with respect to an absolutely continuous invariant measure in terms of the Lyapunov characteristic exponents. One of the most striking of the many important consequences of these results described in [pes] is the 2,3 so called Pesin entropy formula which expresses the entropy of a smooth dynamical system through its Lyapunov characteristic exponents.