Important or interesting mathematical objects that are fixed points

Question

I've observed that many interesting or important objects that arise in mathematics later turn out to be fixed points for some function (possibly in a way that is not obvious from their original definition). What are some examples of this phenomenon?

Here are a few I could think of off the top of my head:

• The golden ratio $\varphi \approx 1.618033...$ is a fixed point for the function $x \mapsto 1/(x-1)$, the other fixed point being $-1/\varphi$.

• The Thue-Morse sequence is obtained by starting with $0$, and at each stage appending the Boolean complement of what you currently have (so $0$, $01$, $0110$, $01101001$, ...). This turns out to be a fixed point of the function $\{ 0,1 \}^\mathbb{N} \to \{ 0,1 \}^\mathbb{N}$ which replaces each $0$ with $01$ and each $1$ with $10$.

• An initial algebra for an endofunctor $F\colon \mathbf{C} \to \mathbf{C}$ turns out to be a fixed point of $F$ (Lambek's theorem).

Since everything is a fixed point for the identity function, maybe we'll restrict to objects which turn out to be the unique fixed point of something, or at least one of a small number.

Answer 1

I'm not sure if this counts, but one proof of the Picard-Lindelöf theorem shows that the unique solution to the ODE

$$y'=f(x,y), y(x_0)=y_0$$

can be thought of as the unique fixed point of the function operator $L$ defined by

$$L(\phi(x))=y_0+\int_{x_0}^x f(t,\phi(t))\text{ }dt$$

As shown in the Wikipedia article, if $f$ satisfies the assumptions of the theorem, the fixed point will exist as a consequence of the Banach fixed point theorem.

Answer 2

The multiples of the exponential function $\exp\colon\Bbb R\longrightarrow\Bbb R$ (that is, the functions of the form $x\mapsto\lambda e^x$) are the fixed points of $D\colon C^\infty(\Bbb R)\longrightarrow C^\infty(\Bbb R)$, with $D(f)=f'$.

Answer 3

The Dottie number is a fixed point of the cosine function. It is in fact the only real fixed point (other fixed points are complex numbers).