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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

##### MSC:
 58C25 Differentiable maps on manifolds