Dynamical systems and ergodic theory Definition

Herman ring

Builds on Fatou set Rational function Annulus Conformal map Irrational number

Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a rational function of degree at least $2$ . A Herman ring of $f$ is a component $U$ of the Fatou set of $f$ with the following properties.

There is $p \ge 1$ with $f^p(U) = U$ .
There are $r > 1$ and a conformal map $\varphi$ from $U$ onto the round annulus $A_r = \set{w \in \C : 1 < \abs{w} < r}$ .
There is an irrational number $\theta$ with $\varphi(f^p(z)) = e^{2\pi i \theta} \varphi(z)$ for all $z \in U$ .

Remarks

The circles $\abs{w} = s$ with $1 < s < r$ pull back to invariant closed curves of $f^p$ in $U$ . Non-example: a polynomial $P$ has no Herman ring. If $U$ were one, an invariant curve $\gamma$ in $U$ would bound a disc $D$ , and by the maximum principle the iterates of $P$ would be bounded on $D$ . Then $D$ would lie in the Fatou set, so $U$ could not be a component of it with a hole. Herman constructed the first examples of rational maps with Herman rings. Shishikura later built them by quasiconformal surgery from maps with Siegel discs.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Annulus concept
Conformal map concept
Fatou set concept
Irrational number concept
Rational function concept
Herman ring definition

Tag	`QYHKS4B7M0`
URI	`m2://QYHKS4B7M0`
Kind	definition
Writing	W0 stub
MSC	37F50, 37F10
Defines	Herman ring
Uses	Fatou set, Rational function, Annulus, Conformal map, Irrational number

Herman ring

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here