Constraints on a Function Mapping a distribution to Itself

I have a function $f: \mathbb{R}^d \to \mathbb{R}^d$ smooth and invertible defined as $y = f(x)$. The input $x$ has independent components following a uniform distribution:
$$x_i \sim \mathcal{U}[-0.5, 0.5], \quad \text{for } i=1, \dots, d.$$The output $y$ is also distributed following the same distribution $y_i \sim \mathcal{U}[-0.5, 0.5]$.

My question is : What does that tell me on the function $f$ ? We can already see that $|det J_f|=1$ but are there more constraints since it looks like it needs to be more than simply volume preserving ?