On behaviour of trajectories near hyperbolic sets

Let $F$ and $F'$ be hyperbolic sets of diffeomorphisms $f$ and $f'$ respectively. Suppose that the restrictions $f|_F$ and $f'|_{F'}$ are topologically conjugated by a homeomorphism $h$. Then a restriction of $f$ to an invariant set comprised by all trajectories of $f$ that are close to $F$, and a restriction of $f'$ to a set of all trajectories of $f'$ that are close to $F'$, are conjugated by a homeomorphism $H$, which is an extenstion of $h$.