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: The
earliest one was to the existence of periodic orbits for 3 bodies.
@ General references: Granas & Dugundji
03;
Farmakis & Moskowitz 13 [and applications].
@ Various types: Prykarpatsky a0902 [Leray-Schauder, Borsuk-Ulam type generalization].
> Online resources:
see Wikipedia page.
Simple Case: Real Functions of One Variable
$ Def: Given any interval
[a, b] ⊂ \(\mathbb R\), 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*: Hi → Hi, for i = 1, ..., n,
and from these we can define some numbers ni, which are something like the trace of f*, and
L = ∑i (−1)i ni , 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 = \(\mathbb R\),
H1 = 0,
H2 = \(\mathbb R\),
and
L = ∑i (−1)i ni = 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.
@ References: van Lon MS-a1509
[quantum mechanical path integral methods, and other index theorems].
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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