Fixed Point (Map)

A point $x^*$ which is mapped to itself under a Map $G$, so that $x^* = G(x^*)$. Such points are sometimes also called Invariant Points, or Fixed Elements (Woods 1961). Stable fixed points are called elliptical. Unstable fixed points, corresponding to an intersection of a stable and unstable invariant Manifold, are called Hyperbolic (or Saddle). Points may also be called asymptotically stable (a.k.a. superstable).

See also Critical Point, Involuntary

References

Shashkin, Yu. A. Fixed Points. Providence, RI: Amer. Math. Soc., 1991.

Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, p. 14, 1961.