Abstract:
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.