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On the \(C^ 1\) stability conjecture for flows. (English) Zbl 0866.58050
The paper is devoted to the proof of the \(C^1\) stability conjecture of Palais and Smale for flows:
Theorem A: \(C^1\) structurally stable flows of compact manifolds without boundary satisfy Axiom A.
This proof allows to extend the classical results of Peixoto for dimension 2 and those of Hu and Liao for dimension 3. The \(C^1\) connecting lemma of Hayashi is the right key to this work.
As a corollary of Theorem A, the equivalence of two apparently different definitions of structural stability (Peixoto’s and Andronov-Pontryagin’s) is established.

37D99 Dynamical systems with hyperbolic behavior
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