I am trying to understand the maps which are area-preserving and maps which are not. I know that the standard map is an area-preserving map since the Jacobian determinant is 1.
I also know that for the maps which the area is not preserved, they belong to the dissipative system (in contrast of Hamiltonian system).
Can anybody gives example of 2-d mapping that is not area-preserving map?
If $U$ is a non-empty open subset of the plane, a continuously-differentiable mapping $f:U \to U$ is area-preserving if and only if $|\det Df| = 1$.
A "randomly-chosen" plane mapping is not area-preserving. Among holomorphic mappings, for example, the only area-preserving examples are affine mappings $f(z) = \alpha z + \beta$ with $|\alpha| = 1$.
More interesting is the fact the space of area-preserving mappings is infinite-dimensional. If $\phi$ is a continuously-differentiable real-valued function of one variable, for example, the mapping $$ f(x, y) = (x, y + \phi(x)) $$ is area-preserving, as is the mapping given in polar coordinates by $$ f(r, \theta) = (r, \theta + \phi(r)). $$
