Solutions of the Multivariate Inverse Frobenius--Perron Problem

We address the inverse Frobenius--Perron problem: given a prescribed target distribution \(\rho\), find a deterministic map \(M\) such that iterations of \(M\) tend to \(\rho\) in distribution. We show that all solutions may be written in terms of a factorization that combines the forward and inverse Rosenblatt transformations with a uniform map, that is, a map under which the uniform distribution on the \(d\)-dimensional hypercube as invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via \(1\)-dimensional examples, and then use the factorization to present solutions in \(1\) and \(2\) dimensions induced by a range of uniform maps.