Fixed-point index

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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.

In real Euclidean space, the fixed-point index is defined as follows: If x₀ is an isolated fixed point of f, then let g be the function defined by

${\displaystyle g(x)={\frac {x-f(x)}{||x-f(x)||}}.}$

Then g has an isolated singularity at x₀, and maps the boundary of some deleted neighborhood of x₀ to the unit sphere. We define i(f, x₀) to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x₀.^[1]