On Some Sufficient Hyperbolicity Conditions

We consider an arbitrary $C^1$ diffeomorphism $f$ that acts from an open subset $U$ of a Riemannian manifold $M$ of dimension $m$, $m\ge 2$, to $f(U)\subset M$. Let $A$ be a compact $f$-invariant (i.e., $f(A)=A$) subset in $U$. We propose various sufficient conditions under which $A$ is a hyperbolic set of $f$.