Fixed-point index

In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points.

The index can be easily defined in the setting of complex analysis: Let f(z) be a holomorphic mapping on the complex plane, and let z₀ be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z₀. We define the fixed point index of f at z₀, denoted i(f, z₀), to be the multiplicity of the zero of the function f(z) − z at the point z₀.