An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$

Question

Let $T(x,y):=(t_1(x,y),t_2(x,y))$ be a continuous bijection, namely a homeomorphism on $[0,1]^2$.
I am trying to find a $T$ such that $\det(J_T)=1$. (*)

The trivial cases are

$T(x,y)=(x,y)$, $T(x,y)=(1-x,y)$, $T(x,y)=(1-x,1-y)$, $T(x,y)=(x,1-y)$ and the maps obtained from the previous ones with $(t_1, t_2)$ changed to $(t_2, t_1)$.

Is there any examples of $T$ satisfying (*) besides those trivial ones given above? Thanks in advance.

Answer 1

This is a question which has been addressed by many mathematicians, including Gauß who solved the corresponding question for the plane, i.e., he gave a complete description of all homeomorphisms of the plane which are area´preserving (though he did miss one "degenerate" case). The interest in this question was motivated by mathematical cartography---describing all possible area-preserving map projections. The case of the unit square, where the boundary conditions intervene, is a bit more delicate. It is impossible to give a precise description of the general result in this context but roughly speaking you can take any horizontal foliation of curves joining the left hand border to the right hand one (i.e. topologically equivalent to the parallels to the $x$-axis) and find a unique area-preserving mapping of the square which maps the latter foliation into yours. This is shown in the arXiv paper 1108.4758. This is couched in terms of thermodynamics but the underlying mathematical problem is in each case identical---given two suitable one-parameter families of curves (parallels and meridians in cartography, isotherms and adiabats in thermodynamics), to determine whether they can be mapped onto the standard parallels to the axes via an area-preserving mapping. This shows not only that there is indeed a large number of such mappings but that one can also specify just how much freedom there is. By the way, the article of Gauß referred to above is in vol. 8 of his collected papers, p. 373.

Answer 2

Take any function $\phi$ that is constant on the boundary. Then define a vector field $v$ by $v=(\phi_y,-\phi_x)$. Let u denote the vector (x,y), and let $X(t,u)$ be the solution of the differential equation $X_t(t,u)=v(X(t,u))$ with initial condition $X(0,u)=u$. For any time t, you have $det(\partial X/\partial u)=1$.