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.
Best Answer
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 $D(f)=f'$.