Solutions of the Multivariate Inverse Frobenius–Perron Problem
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We address the inverse Frobenius–Perron problem: given a prescribed target distribution ρ, find a deterministic map M such that iterations of M tend to ρ 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.
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