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A hyperbolic saddle fixed point of a map has one eigenvalue of modulus less than one and one greater than one. The corresponding stable and unstable directions persist locally as invariant sets or manifolds under the usual smoothness and invertibility assumptions. For a noninvertible map a zero stable fixed-point multiplier is possible, and the word saddle fixed point of a map describes forward contraction in one direction and expansion in the other; backward dynamics may not be unique.

A hyperbolic sink fixed point of a map has both Jacobian eigenvalues strictly inside the unit circle. Its iterates locally attract nearby initial states; complex fixed-point multipliers can make the approach spiral. Zero fixed-point multipliers are strongly contracting and can occur for noninvertible maps. A fixed-point multiplier of modulus one requires nonlinear or exact analysis.

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.