Symmetric Hyperbolic Trap

An arbitrary $$C^1$$ diffeomorphism $$f$$ of an open subset $$U$$ of a Riemannian $$m$$ -manifold $$M$$ , $$m\ge 2$$ , onto a set $$f(U)\subset M$$ is considered. Sufficient conditions for the domain $$U$$ to be a hyperbolic trap are proposed. This means that any set $$A\subset U$$ satisfying the condition $$f(A)=A$$ is automatically a hyperbolic set of the diffeomorphism $$f$$ . Moreover, this hyperbolic trap is symmetric in the sense that the conditions for its existence do not change under the passage from $$f$$ to the inverse map $$f^{-1}$$ .