Solutions of the Multivariate Inverse Frobenius--Perron Problem

TL;DR

This paper characterizes all solutions to the inverse Frobenius--Perron problem by expressing them through a factorization involving the Rosenblatt transformations and a uniform map, providing explicit solutions in one and two dimensions.

Contribution

It introduces a comprehensive factorization framework for solving the inverse Frobenius--Perron problem using forward and inverse Rosenblatt transformations combined with uniform maps.

Findings

All solutions can be expressed via a factorization involving Rosenblatt transformations and a uniform map.

Explicit solutions are provided for one and two-dimensional cases.

The approach unifies and generalizes previous methods for constructing solutions.

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.