On a Method for Verifying Hyperbolicity

An arbitrary diffeomorphism $f$ of class $C^1$ acting from an open subset $U$ of Riemannian manifold $M$ of dimension $m,$ $m\geqslant 2,$ into $f(U)\subset M$ is considered. Let $A$ be a compact subset of $U$ invariant for $f,$ i. e., $f(A)=A.$ Various sufficient conditions are proposed under which $A$ is a hyperbolic set of the diffeomorphism $f.$