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 $$f.$$