Abstract
The idea of studying the global behavior of dynamical systems using their local properties seems to be very attractive. The theory of hyperbolic dynamical systems may be considered as a partial realization of this idea.
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We shall also call it uniformely hyperbolic if there is no misunderstanding.
The indices “s”, “m” mean “stable” and “unstable” respectively.
One uses also a different system of notation and different terminology for continuous time systems. GSM is denoted by Wss(s) and is called the strong stable manifold, and the global weakly stable manifold is denoted by Ws(x) and is called the stable manifold. A similar system of notation and terminology is used with respect to unstable manifolds.
There is an interesting question: If any trajectory of a system is uniformely completely hyperbolic, is this an Anosov system? [Pe2] gives a positive answer to this question for C2-systems which conserve a measure equivalent to the Riemannian volume. 14 For the definition of biregular points and of characteristic Lyapunov exponents, see Sect. 2 of Chap. 1.
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© 1989 Springer-Verlag Berlin Heidelberg
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Pesin, Y.B. (1989). General Theory of Smooth Hyperbolic Dynamical Systems. In: Sinai, Y.G. (eds) Dynamical Systems II. Encyclopaedia of Mathematical Sciences, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06788-8_7
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DOI: https://doi.org/10.1007/978-3-662-06788-8_7
Publisher Name: Springer, Berlin, Heidelberg
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