# American Institute of Mathematical Sciences

December  2009, 25(4): 1349-1366. doi: 10.3934/dcds.2009.25.1349

## 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  December 2008 Revised  May 2009 Published  September 2009

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.
Citation: Dawei Yang, Shaobo Gan, Lan Wen. Minimal non-hyperbolicity and index-completeness. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1349-1366. doi: 10.3934/dcds.2009.25.1349
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