Fixed point

Summary

A fixed point of a map is a point that the map sends to itself. In a dynamical system, a fixed point is a state that never changes, the discrete-time analogue of an equilibrium point. Many problems, such as solving an equation $g(x) = 0$, can be rewritten as finding a fixed point. Theorems such as the Banach fixed point theorem guarantee that one exists. Near a fixed point, the behaviour of the iterates tells whether nearby states are attracted or repelled.