---
tag: 9PK608W8MS
kind: example
name: Lattès map
msc: [37F10, 14H52]
illustrates: [5M4S7P04GR]
uses: [5M4S7P04GR, 0XDK9CGYKJ, FF9ZPCZ59K, Y7MA6KBE9M, WN9V7WSJKF]
sources:
  - url: https://en.wikipedia.org/wiki/Latt%C3%A8s_map
same_as:
  wikidata: Q6497180
  wikipedia: Lattès map
---

## Example

Let $\Lambda \subset \C$ be a [[lattice|WN9V7WSJKF]], let $T = \C / \Lambda$ be the [[complex torus|Y7MA6KBE9M]], and let $\wp$ be the Weierstrass [[elliptic function|FF9ZPCZ59K]] of $\Lambda$, with invariants $g_2, g_3$.
The duplication formula expresses $\wp(2u)$ as a [[rational function|0XDK9CGYKJ]] of $\wp(u)$:
$$\wp(2u) = L(\wp(u)), \qquad L(x) = \frac{x^4 + \frac{g_2}{2} x^2 + 2 g_3 x + \frac{g_2^2}{16}}{4x^3 - g_2 x - g_3}.$$
The map $L$ of degree $4$ is a *Lattès map*.
It is the shadow of the doubling map $u \mapsto 2u$ on $T$ under the projection $\wp \colon T \to \hat{\C}$.

For a lattice with $g_2 = 4$ and $g_3 = 0$, which exists because $g_2^3 - 27 g_3^2 = 64 \ne 0$, the formula gives
$$L(x) = \frac{(x^2 + 1)^2}{4x(x^2 - 1)}.$$

Its [[Julia set|5M4S7P04GR]] is the whole Riemann sphere.
The reason: the doubling map on $T$ multiplies small distances by $2$.
So the image under $u \mapsto 2^n u$ of any small disc in $T$ covers $T$ for $n$ large.
Applying $\wp$, for every nonempty open set $U \subseteq \hat{\C}$ some iterate $L^n$ maps $U$ onto $\hat{\C}$.
The Fatou set is invariant and the Julia set is nonempty, so an open subset of the Fatou set never maps onto $\hat{\C}$.
Hence the Fatou set of $L$ is empty.

## Remarks

Lattès maps are the standard examples of rational maps whose Julia set is the whole sphere.
In general, a Lattès map is a rational map induced by an affine self-map $u \mapsto a u + b$ of a complex torus, with $\abs{a}^2 \ge 2$ an integer.
The map passes to the sphere through the quotient of the torus by a finite group of automorphisms.

## Explanations

### L1 human

On a doughnut-shaped torus, doubling every point stretches each small patch until it wraps around the whole torus.
The Weierstrass function folds the torus onto the sphere, and the doubling becomes a rational map of the sphere.
Every small region of the sphere gets spread over the whole sphere after a few steps, so the map is chaotic everywhere.
There is no calm region at all: the Julia set is the entire sphere.