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 \(f^{-1}\) .