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.

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

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]

The Lefschetz–Hopf theorem

The importance of the fixed-point index is largely due to its role in the Lefschetz–Hopf theorem, which states:

\sum_{x \in \mathrm{Fix}(f)} i(f,x) = \Lambda_f,

where Fix(f) is the set of fixed points of f, and Λ_f is the Lefschetz number of f.

Since the quantity on the left-hand side of the above is clearly zero when f has no fixed points, the Lefschetz–Hopf theorem trivially implies the Lefschetz fixed point theorem.

Notes

↑ A. Katok and B. Hasselblatt(1995), Introduction to the modern theory of dynamical systems, Cambridge University Press, Chapter 8.

References

Robert F. Brown: Fixed Point Theory, in: I. M. James, History of Topology, Amsterdam 1999, ISBN 0-444-82375-1, 271–299.

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