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HYPERBOLICITY OF HOMOCLINIC CLASSES OF
$C^{1}$ VECTOR FIELDS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 98 / Issue 3 / June 2015
- Published online by Cambridge University Press:
- 21 November 2014, pp. 375-389
- Print publication:
- June 2015
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- Article
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- You have access
- Export citation
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Let
${\it\gamma}$ be a hyperbolic closed orbit of a 
$C^{1}$ vector field 
$X$ on a compact 
$C^{\infty }$ manifold 
$M$ and let 
$H_{X}({\it\gamma})$ be the homoclinic class of 
$X$ containing 
${\it\gamma}$. In this paper, we prove that if a 
$C^{1}$-persistently expansive homoclinic class 
$H_{X}({\it\gamma})$ has the shadowing property, then 
$H_{X}({\it\gamma})$ is hyperbolic.
