I learned this concept in feedback linearization problems, where it is used for state transformation. For example, $x_1,...,x_n$ are states of a system. Now we have a new set of variables: $z_1,...,z_n$. The problem is: whether $z_1,...,z_n$ can be a set of states of the system? The answer is: as long as the function $z_1(x),...,z_n(x)$ is diffeomorphism, it can be used to represent the system.
It should be noted that: from $z_1(x),...,z_n(x)$ to $x_1,...,x_n$, DOFs are preserved. But the value region changes. If $x_i \in \Omega _1$ then $z_i \in \Omega_2$. So my question on math stackexchange is not proper. Because $x_1,...,x_n$ may not be an arbitrary vector in $\mathbb{R}^n$. And of course, $z_1(x),...,z_n(x)$ may not be an arbitrary vector in $\mathbb{R}^n$. But keep in mind that the DOFs are preserved.