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 and Continuous Dynamical Systems, 2009, 25 (4) : 1349-1366. doi: 10.3934/dcds.2009.25.1349
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