HYPERBOLICITY OF HOMOCLINIC CLASSES OF $C^{1}$VECTOR FIELDS

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.