What is the general form of an area-preserving map of $\mathbb{R}_2$?

Question

A Hamiltonian flow generates such a map. But what is the most general form of such a map? Any theorem? Any procedure to generate such a map?

Of course, I want the map to be continuous.

Answer 1

If you consider continuously differentiable maps, then by the integration by substitution formula we have that $f:\mathbb{R}^n\to\mathbb{R}^n$ is (locally) volume preserving if and only if the equality$$\det(df_x)=\pm1$$holds for every $x\in\mathbb{R}^n$.