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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 be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f − z is holomorphic, and has an isolated zero at z0. We define the fixed-point index of f at z0, denoted i, to be the multiplicity of the zero of the function f − 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
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 to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x0.
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