In mathematics, a fixed-point theorem is a
result saying that a function F will have
at least one fixed point (a point x for which
F(x) = x), under some conditions on F that
can be stated in general terms.
Results of this kind are amongst the most
generally useful in mathematics.
== In mathematical analysis ==
The Banach fixed-point theorem gives a general
criterion guaranteeing that, if it is satisfied,
the procedure of iterating a function yields
a fixed point.By contrast, the Brouwer fixed-point
theorem is a non-constructive result: it says
that any continuous function from the closed
unit ball in n-dimensional Euclidean space
to itself must have a fixed point, but it
doesn't describe how to find the fixed point
(See also Sperner's lemma).
For example, the cosine function is continuous
in [−1,1] and maps it into [−1, 1], and
thus must have a fixed point.
This is clear when examining a sketched graph
of the cosine function; the fixed point occurs
where the cosine curve y=cos(x) intersects
the line y=x.
Numerically, the fixed point is approximately
x=0.73908513321516 (thus x=cos(x) for this
value of x).
The Lefschetz fixed-point theorem (and the
Nielsen fixed-point theorem) from algebraic
topology is notable because it gives, in some
sense, a way to count fixed points.
There are a number of generalisations to Banach
fixed-point theorem and further; these are
applied in PDE theory.
See fixed-point theorems in infinite-dimensional
spaces.
The collage theorem in fractal compression
proves that, for many images, there exists
a relatively small description of a function
that, when iteratively applied to any starting
image, rapidly converges on the desired image.
== In algebra and discrete mathematics ==
The Knaster–Tarski theorem states that any
order-preserving function on a complete lattice
has a fixed point, and indeed a smallest fixed
point.
See also Bourbaki–Witt theorem.
The theorem has applications in abstract interpretation,
a form of static program analysis.
A common theme in lambda calculus is to find
fixed points of given lambda expressions.
Every lambda expression has a fixed point,
and a fixed-point combinator is a "function"
which takes as input a lambda expression and
produces as output a fixed point of that expression.
An important fixed-point combinator is the
Y combinator used to give recursive definitions.
In denotational semantics of programming languages,
a special case of the Knaster–Tarski theorem
is used to establish the semantics of recursive
definitions.
While the fixed-point theorem is applied to
the "same" function (from a logical point
of view), the development of the theory is
quite different.
The same definition of recursive function
can be given, in computability theory, by
applying Kleene's recursion theorem.
These results are not equivalent theorems;
the Knaster–Tarski theorem is a much stronger
result than what is used in denotational semantics.
However, in light of the Church–Turing thesis
their intuitive meaning is the same: a recursive
function can be described as the least fixed
point of a certain functional, mapping functions
to functions.
The above technique of iterating a function
to find a fixed point can also be used in
set theory; the fixed-point lemma for normal
functions states that any continuous strictly
increasing function from ordinals to ordinals
has one (and indeed many) fixed points.
Every closure operator on a poset has many
fixed points; these are the "closed elements"
with respect to the closure operator, and
they are the main reason the closure operator
was defined in the first place.
Every involution on a finite set with an odd
number of elements has a fixed point; more
generally, for every involution on a finite
set of elements, the number of elements and
the number of fixed points have the same parity.
Don Zagier used these observations to give
a one-sentence proof of Fermat's theorem on
sums of two squares, by describing two involutions
on the same set of triples of integers, one
of which can easily be shown to have only
one fixed point and the other of which has
a fixed point for each representation of a
given prime (congruent to 1 mod 4) as a sum
of two squares.
Since the first involution has an odd number
of fixed points, so does the second, and therefore
there always exists a representation of the
desired form.
== List of fixed-point theorems ==
== Footnotes
