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Oseledec’s theorem from the generic viewpoint. (English) Zbl 0584.58007
Proc. Int. Congr. Math., Warszawa 1983, Vol. 2, 1269-1276 (1984).
[For the entire collection see Zbl 0553.00001.]
Let M be a compact manifold without boundary and \(f: M\to M\) a \(C^ 1\)- diffeomorphism. A point x in M is called regular if the tangent space \(T_ xM\) at x admits a splitting \(\oplus E_ i(x)\) such that the derivatives \(D_ xf^ n\) can be controlled in a certain way on the \(E_ i(x)\). Oseledec has proved that, for every f-invariant probability measure \(\mu\) on M, the set of regular points has \(\mu\)-measure one. Here it is shown that for a generic set of diffeomorphisms f there are residually many ergodic f-invariant measures \(\mu\) such that the above mentioned splitting is in a sense uniform on the support of \(\mu\). The case of volume respecting diffeomorphisms on symplectic manifolds is also discussed. Residually many of these diffeomorphisms admit a classification in terms of their elliptic, hyperbolic and parabolic regions.
Reviewer: E.Behrends

58C25 Differentiable maps on manifolds