Minimal non-hyperbolicity and index-completeness

Dawei Yang; Shaobo Gan; Lan Wen

doi:10.3934/dcds.2009.25.1349

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2009, Volume 25, Issue 4: 1349-1366. Doi: 10.3934/dcds.2009.25.1349

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Minimal non-hyperbolicity and index-completeness

1.
School of Mathematical Sciences, Peking University, Beijing 100871
2.
School of Mathematical Science, Peking University, Beijing 100871
3.
School of Mathematic Sciences, Peking University, Beijing, 100871

Received: November 30, 2008

Revised: April 30, 2009

Published: October 2009

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Abstract

We study a problem raised by Abdenur et. al. [3] that asks, for any chain transitive set $\Lambda$ of a generic diffeomorphism $f$, whether the set $I(\Lambda)$ of indices of hyperbolic periodic orbits that approach $\Lambda$ in the Hausdorff metric must be an "interval", i.e., whether $\alpha\in I(\Lambda)$ and $\beta\in I(\Lambda)$, $\alpha<\beta$, must imply $\gamma\in I(\Lambda)$ for every $\alpha<\gamma<\beta$. We prove this is indeed the case if, in addition, $f$ is $C^1$ away from homoclinic tangencies and if $\Lambda$ is a minimally non-hyperbolic set.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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