Saddle fixed point of a map ($|{\lambda }_s|<1<|{\lambda }_u|$)

A planar map has a hyperbolic saddle fixed point when one Jacobian multiplier has modulus less than one and the other has modulus greater than one. In diagonal coordinates the corresponding linear iterates contract and expand geometrically. For a smooth local diffeomorphism, this splitting continues to local stable and unstable manifolds. It differs from the determinant-negative test for a planar continuous flow: an area-preserving map can have a saddle with positive determinant one, because its multipliers can be real reciprocal numbers. The criterion uses the unit circle, not the sign of the real parts.