Fixed Point Theorems

In General > s.a. posets.
* Simple example: If I hold a map of an area including the point where I stand, there must be a point on the map which is exactly above the corresponding point of the real world (even if I deform or crumple the map).
* Motivation: If A is any differential operator, the existence of solutions of the equation A f = 0 is equivalent to the existence of fixed points for A + I; We are interested in equations like df = 0 for the study of critical points (> see morse theory, etc).
* Applications: Earliest one was to the existence of periodic orbits for 3 bodies.
@ References: Granas & Dugundji 03.

Simple Case: Real Functions of One Variable
$ Def: Given any interval [a, b] R, any f : [a, b] → [a, b] must have at least one fixed point (the graph must cross at least once the line x = y).

Brouwer Fixed Point Theorem
$ Def: Any continuous f : Dⁿ → Dⁿ has (at least) one fixed point (Dⁿ is the n-dimensional ball).
* Special cases: For n = 2, it can be proved using the fundamental group, through the fact that S¹ is not a retract of D² (in general, it is done using higher order homotopy groups, or considering the index of the vector field v(x):= x – f(x), at different points with respect to different loops).

Lefschetz Fixed Point Theorem
* Idea: Let S be an n-dimensional manifold (or "almost"), and f a map f : S → S; Then f induces a map f_* of the homology groups,

f_*: H_i → H_i,    for    i = 1, ..., n,

and from these we can define some numbers n_i, which are something like the trace of f_*, and

L = _i (–1)ⁱ n_i ,    the Lefschetz number ;

The theorem states that this number can also be obtained as a sum of contributions of all the fixed points of f.
* Relationships: It is like a finite (as opposed to infinitesimal) generalization of the concept of Euler characteristic.
* Example: Consider the inversion map on S², A: S² → S²; We know that this does not have any fixed points; H₀ = R, H₁ = 0, H₂ = R, and

L = _i (–1)ⁱ n_i = 1 · 1 + (–1) · 0 + 1 · (–1) = 0 ;

If the map had been orientation-preserving, there would have been fixed points and L 0.
* Example: Every Lorentz transformation fixes at least one null direction.

Borel Fixed Point Theorem
$ Def: A connected solvable linear algebraic group over an algebraic closed field, when acting on a complete variety, has a fixed point.
@ References: Borel AM(56).

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 13 jun 2008