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 $m$.

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.

(over) Expectation: There exist $A,B\subset S$ such that $A\cup B=S$, $m(A)\neq 0,$ (also $m(B)\neq 0$ if possible) and $f_1=f_{|A}, f_2=f_{|B}$ such that $f_1$ preserves $m$.