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On the $$C^ 1\Omega$$-stability conjecture. (English) Zbl 0648.58019
Let M be a compact $$C^{\infty}$$ manifold without boundary. Denote by $$\Omega$$ (f) the nonwandering set of a diffeomorphism f of M. A diffeomorphism is said to be $$C^ r\Omega$$-stable if there exists a neighbourhood U of f (in the C r topology) such that for each $$g\in U$$ there exists a homeomorphism $$h: \Omega(f)\to \Omega(g)$$ with $$g\circ h = h\circ f$$ on $$\Omega(f)$$.
The main result of the present paper is: If a $$C^ 1$$ diffeomorphism f is $$\Omega$$-stable then it satisfies axiom A. - As is explained in the papeer, this result constitutes the final step towards an affirmative answer of the $$C^ r\Omega$$-stability conjecture for $$r=1$$. The $$C^ r\Omega$$-stability conjecture asserts: A $$C^ r$$ diffeomorphism f is $$C^ r\Omega$$-stable if and only if f satisfies Axiom A, and there are no cycles on $$\Omega(f)$$.
Reviewer: H.Crauel

##### MSC:
 37C75 Stability theory for smooth dynamical systems 54H20 Topological dynamics (MSC2010)
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