Searching for a maximally nonlinear but smooth bijective mapping $[0,1]^N \mapsto [0,1]^N$ Question: I am searching for a maximally nonlinear but still smooth (or at least differentiable) bijective (!) mapping $[0,1]^N \mapsto [0,1]^N$ between multivariate uniform spaces.
So far I've been toying around with convex linear combinations and invertible elementwise nonlinearities (e.g. cosine) but usually the mapping is not bijective (on the uniform space).

Let $f:[0,1]\mapsto [0,1]$ be the cantor function. Then, 
$$ g(x) = \frac{f(x) + x}{2} $$ 
is a bijective and continuous mapping from $[0,1]$ into itself. Obviously $$ G(x_1,\dotsc,x_N) = (g(x_1), \dotsc, g(x_N)) $$
is a bijective and continuous mapping from $[0,1]^N$ into itself.
$G$ is quite non-linear in the sense that it isn't even absolutely continuous. However, as $g$ is monotonous, it has bounded variation.
