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The set of axiom A diffeomorphisms with no cycles

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Abstract

LetM be aC ∞ closed manifold and Diff1 (M) be the space of diffeomorphisms ofM endowed with theC 1 topology. This paper contains an affirmative answer to the following conjecture raised by Mañé, which is an extension of the stability and Ω-stability conjectures of Palis and Smale, as follows: theC 1 interior of the subset of diffeomorphism such that all the periodic points are hyperbolic is characterized as the set of diffeomorphisms satisfying Axiom A and the no-cycles condition. Moreover, it is showed that theC 1 interior of the set of all Kupka-Smale diffeomorphisms coincides with the set of all diffeomorphisms satisfying Axiom A and the strong transversality condition.

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  1. Department of Mathematics, Tokyo Metropolitan University, Japan

    Nobuo Aoki

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  1. Nobuo Aoki
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Aoki, N. The set of axiom A diffeomorphisms with no cycles. Bol. Soc. Bras. Mat 23, 21–65 (1992). https://doi.org/10.1007/BF02584810

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