Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / B / 7 / iii / Solution

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.