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==Stable Manifolds==

Many results and techniques from the theory of uniformly hyperbolic dynamical systems extend to  nonuniformly partially hyperbolic dynamical systems (in the broad sense), such as stable or unstable manifolds, cone techniques and <math>u</math>-measures. For nonuniformly completely hyperbolic dynamical systems, one obtains both stable and unstable manifolds. These are not known to be as uniform in size and alignment as in the uniform case. The problem is that although the hyperbolicity conditions may ensure contraction of stable separations in positive time and that of unstable directions in negative time, there is no mechanism to ensure that the reversal of time brings about any expansion. Accordingly, the stable and unstable manifolds, which in the uniformly hyperbolic case can be extended indefinitely, are not known to extend indefinitely. The ''local'' leaves are usually envisioned like a "fence" consisting of moderate-size posts between which there is a finer fence of somewhat crooked and smaller posts, between which in turn there is a yet finer fence of posts that are yet smaller and rather more crooked, and so on ad infinitum. The (noninvariant) sets of points for which the stable and unstable leaves have a certain minimal size and mutual angle, are called <span id="PesinSet"></span>''Pesin sets''. Numerous arguments for the uniformly hyperbolic context that use invariant foliations can be extended to the present context, but they tend to be far more subtle and often have less uniform or global conclusions.

Lyapunov introduced Lyapunov exponents, and the notion of regularity appears in his work and that of Perron in connection with the study of stability of solutions of linear ordinary differential equation with nonconstant coefficients. Oseledec published the Multiplicative Ergodic Theorem in 1968, and the relation between Lyapunov exponents and nonuniform hyperbolicity was established by Pesin in 1977.