Dynamical systems and ergodic theory Definition

Fatou set

Builds on Rational function Riemann sphere Normal family Iterated function Open set

Definition

Let $f \colon \hat{\C} \to \hat{\C}$ be a rational function of degree at least $2$ on the Riemann sphere $\hat{\C}$ . Let $f^n$ be its $n$ -th iterate. A point $z \in \hat{\C}$ belongs to the Fatou set $F(f)$ if the following holds. There is an open neighbourhood $U$ of $z$ such that $\set{f^n|_U : n \ge 0}$ is a normal family. Normality refers to maps into $\hat{\C}$ with the spherical metric. So every sequence in the family has a subsequence that converges locally uniformly on $U$ , possibly to the constant $\infty$ . A connected component of $F(f)$ is a Fatou component.

Remarks

By definition $F(f)$ is open. Example: for $f(z) = z^2$ , the open unit disc lies in $F(f)$ , since there $f^n(z) = z^{2^n} \to 0$ locally uniformly. Non-example: $z = 1$ is not in $F(z^2)$ ; every neighbourhood of $1$ contains points tending to $0$ and points tending to $\infty$ , so no subsequence of iterates has a continuous limit there. The Fatou set is completely invariant: $z \in F(f)$ if and only if $f(z) \in F(f)$ . The same definition with $\C$ in place of $\hat{\C}$ is used for transcendental entire functions.

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Iterated function concept
Normal family concept
Open set concept
Rational function concept
Riemann sphere concept
Fatou set definition

Tag	`REER3CG7DS`
URI	`m2://REER3CG7DS`
Kind	definition
Writing	W0 stub
MSC	37F10
Defines	Fatou set
Uses	Rational function, Riemann sphere, Normal family, Iterated function, Open set

Fatou set

Definition

Remarks

Reading path

Further reading

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