The question is very vague therefore any kind of suggestions, reference, ideas are welcome.

Suppose $S$ is an oriented surface with or without boundary. Let $m$ be an area form. Let $f$ be a diffeomorphism of $S$ isotopic to the identity. Is there a way to decompose $f$ *naturally* such that one component preserves $m$. By naturally I mean the decomposition is continuous with respect to some topology on the group of diffeomorphisms isotopic to the identity. Ideally I would like to have $f=f_1+f_2$ such that $f_1$ preserves $m$.