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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

37C75 Stability theory for smooth dynamical systems
54H20 Topological dynamics (MSC2010)
Full Text: DOI Numdam EuDML
[1] R. Bowen, On Axiom A diffeomorphisms,Conference Board Math. Sciences,35,Amer. Math. Soc., 1977. · Zbl 0383.58010
[2] M. Hirsch andC. Pugh, Stable manifolds and hyperbolic sets, Global Analysis,Proc. AMS Symp. in Pure Math.,14 (1970), 133–163.
[3] M. Hirsch, J. Palis, C. Pugh andM. Shub, Neighborhoods of hyperbolic sets,Invent. Math.,9 (1970), 121–134. · Zbl 0191.21701 · doi:10.1007/BF01404552
[4] R. Mañé, A proof of the C1 Stability Conjecture,Publ. Math. I.H.E.S.,66 (1987), 161–210.
[5] S. Newhouse, Lectures on dynamical systems,Progress in Math.,8, Birkhäuser, 1980. · Zbl 0444.58001
[6] J. Palis andS. Smale, Structural stability theorems,Proc. AMS Symp. in Pure Math.,14 (1970), 223–231. · Zbl 0214.50702
[7] J. Palis, A note on \(\Omega\)-stability, sameProceedings,14 (1970), 221–222.
[8] S. Smale, The \(\Omega\)-stability theorem, sameProceedings,14 (1970), 289–297.
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