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 ''z''₀ be a fixed point of ''f''. Then the function ''f''(''z'') − ''z'' is holomorphic, and has an isolated zero at ''z''₀. We define the fixed-point index of ''f'' at ''z''₀, denoted ''i''(''f'', ''z''₀), to be the multiplicity of the zero of the function ''f''(''z'') − ''z'' at the point ''z''₀.

In real Euclidean space, the fixed-point index is defined as follows: If ''x''₀ is an isolated fixed point of ''f'', then let ''g'' be the function defined by

:$g(x) = \frac.$

Then ''g'' has an isolated singularity at ''x''₀, and maps the boundary of some deleted neighborhood of ''x''₀ to the unit sphere. We define ''i''(''f'', ''x''₀) to be the Brouwer degree of the mapping induced by ''g'' on some suitably chosen small sphere around ''x''₀.A. Katok and B. Hasselblatt(1995), Introduction to the modern theory of dynamical systems, Cambridge University Press, Chapter 8.

The Lefschetz–Hopf theorem

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

:$\sum_ i(f,x) = \Lambda_f,$

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

References

* Robert F. Brown: ''Fixed Point Theory'', in: I. M. James, ''History of Topology'', Amsterdam 1999, , 271–299.

{{DEFAULTSORT:Fixed-Point Index
Fixed points (mathematics)
Topology