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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.

##### MSC:
 37D99 Dynamical systems with hyperbolic behavior
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