Fatou-Bieberbach domains were first constructed by Fatou in 1927; the construction was improved by Bieberbach in 1932. The idea is linearizing coordinates at attracting cycles.

Let $f:\bf C^2 \to C^2$ be an analytic automorphism, such that $(0,0)$ is an attracting fixed point, for instance the Hénon mapping
$$
f: (x,y) \mapsto (x^2-y,x/2).
$$
Denote by $L$ the derivative of $f$ at the origin.

There is then (usually, in particular in the case above) a unique analytic mapping $\phi:\bf C^2\to C^2$ with $\phi(0,0)=(0,0)$, $D\phi(0,0)=id$ and $L\circ \phi= \phi\circ f$.

In this case (and in many others), $\phi^{-1}$ is defined on the basin of attraction of the origin by the formula
$$
\phi(x,y)= \lim_{m \to \infty} L^{-\circ m} \circ f^{\circ m} (x,y).
$$

The map $\phi$ is injective, but its image is certainly not dense. The image is the basin of attraction of the origin. There are lots of points not attracted to the origin, for instance the set
$$
U=\{(x,y) |\ |x|\ge 2, |y|\le x\}.
$$
it is easy to see that $f(U)\subset U$, and that if you iterate $f$ in $U$ all orbits tend to infinity.

Showing that the limit above exists and defines an analytic map is not hard; I will write a proof if you are interested.

There is an enormous literature about linearization at periodic points.