Sink fixed point of a map ($|{\lambda }_j|<1\text{ for every multiplier}$)

A hyperbolic sink fixed point of a smooth finite-dimensional map has every Jacobian eigenvalue inside the unit circle. Nearby iterates converge to it, and zero multipliers are allowed for a noninvertible map. Unit-modulus multipliers require nonlinear or exact analysis rather than this strict linear criterion. A complex pair gives spiral contraction, while negative real multipliers alternate directions.