Hyperbolic limit sets

Newhouse, Sheldon E.

doi:10.1090/S0002-9947-1972-0295388-6

Hyperbolic limit sets
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by Sheldon E. Newhouse

Trans. Amer. Math. Soc. 167 (1972), 125-150

DOI: https://doi.org/10.1090/S0002-9947-1972-0295388-6

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Abstract:

Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set ${L^ - }(f)$ of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in ${L^ - }(f)$. A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if ${L^ - }(f)$ is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is $\Omega$-stable. Examples are given where ${L^ - }(f)$ is hyperbolic, there are cycles, and f fails to satisfy Axiom A.

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