In mathematics, a fixed point of a function f is a value x such that f(x) = x. For example, Wolfram notes that cos(0.7390851332) = 0.7390851332. There need not be such a fixed point, of course, but it is interesting when there is.

## Unique fixed points

The Banach fixed-point theorem applies to complete metric spaces – sets where there is a concept of distance and where convergent sequences have a limit.

If f is a function that shrinks things – if, say, the distance between f(x) and f(y) is at most 50%, or 90%, or 99.99999% of the distance between x and y – then the function f will not only have a fixed point, but that fixed point will be unique.

Intuitively, this is clear. You apply the function f to the whole set an infinite number of times, and everything shrinks down to a single point satisfying f(x) = x.

Comic 688 by xkcd, with the fixed point highlighted in pink

It follows that if you stand inside a country, city, or home holding (horizontally) a map of that country, city, or home, then exactly one point in the country, city, or home is exactly underneath the corresponding point on the map. More usefully, this theorem has been used to assign meaning to recursive type expressions in computer science.

## Iterated function systems

Iterated function systems are one way of defining fractals. An iterated function system is a family of functions, each of which shrinks things (i.e. is contractive). This family of functions defines a Hutchinson operator on sets of points, and that operator is in turn contractive for a certain definition of distance between sets (see Hutchinson, 1981).

Consequently, that operator has a unique “fixed point,” which is in fact a set of points, like the famous Barnsley fern. The set of points can be (approximately) generated by an iterative process (see code example here):

## Least fixed points

As another example of fixed points in action, the Kleene fixed-point theorem assigns meaning to recursively defined computer programs, like these:

• x = x, a circular definition which is satisfied by infinitely many things, defines x = ⊥, the least defined of all those things
• z = 0 ⊙ (1 + z), where “” means putting a number in front of a list, defines the infinite list z = 0, 1, 2, 3, 4, 5, …
• g = function (x) { if (x = 0) then 1 else x * g(x − 1) }, defines g to be the factorial function

Computer programs are modelled by Scott-continuous functions on a partial order where ⊥ is the least element (⊥ means “no information” and is the meaning of any computer program stuck in an infinite loop). Scott-continuous functions may have more than one fixed point, but they have a unique least fixed point, which is the limit of the sequence ⊥ ⊑ f(⊥) ⊑ f(f(⊥)) ⊑ …

This sequence corresponds to what actually happens when the computer attempts to execute recursive definitions like the three in red above.

## Brouwer’s fixed point theorem

Even more interesting is Brouwer’s fixed-point theorem. Any continuous function mapping a compact convex set to itself has at least one fixed point.

This is hard to prove in general, though easy to see in the case that the set is simply the interval [0, 1]. We have f(0) − 0 ≥ 0 and f(1) − 1 ≤ 0, so that f(x) − x = 0 for at least one x in the range 0 to 1:

An implication is that, when I stir my cup of coffee, at least one speck of liquid winds up exactly where it started.

## Fixed points and dynamics

If the function f represents a state transition function, then a fixed point corresponds to a state that remains the same over time. For example, the logistic mapx (1 − x) has fixed points 0 and 0.75. However, these are both unstable – if you start out close to them, repeated application of the function takes you further away:

• 0.000001, 0.000004, 0.000016, 0.000064, 0.000256, 0.001024, 0.00409, 0.016295, 0.064117, 0.240023, 0.729649, 0.789046, 0.66581, 0.890028, 0.391511, 0.952921, 0.179452, 0.588995, 0.96832, 0.122706, 0.430598, …
• 0.749999, 0.750002, 0.749996, 0.750008, 0.749984, 0.750032, 0.749936, 0.750128, 0.749744, 0.750512, 0.748975, 0.752045, 0.745893, 0.758147, 0.733441, 0.782021, 0.681856, 0.867713, 0.459147, 0.993324, 0.026525, …

On the other hand, the logistic map 2.5 x (1 − x) has a stable fixed point (an attractor) at 0.6. Starting at, say, 0.5, repeatedly applying the function gets you closer and closer to 0.6:

• 0.5, 0.625, 0.585938, 0.606537, 0.596625, 0.601659, 0.599164, 0.600416, 0.599791, 0.600104, 0.599948, 0.600026, 0.599987, 0.600007, 0.599997, 0.600002, 0.599999, 0.6, 0.6, 0.6, 0.6, …

Fixed points are everywhere! Not only that, but the concept is a fruitful one in a diverse collection of fields. For more on fixed points, see this book (available electronically here).