Lattès map

Example

Let $\Lambda \subset \C$ be a lattice, let $T = \C / \Lambda$ be the complex torus, and let $\wp$ be the Weierstrass elliptic function of $\Lambda$, with invariants $g_2, g_3$. The duplication formula expresses $\wp(2u)$ as a rational function 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 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.