Symmetric hyperbolic trap

An arbitrary $C^1$ diffeomorphism $f$ from an open subset $U$ of a Riemannian $m$-manifold $M$, $m\geqslant 2$, to 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 $f^{-1}$.