Abstract
We address the inverse Frobenius–Perron problem: given a prescribed target distribution , find a deterministic map M such that iterations of M tend to 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 is invariant. Indeed, every solution is equivalent to the choice of a uniform map. We motivate this factorization via one-dimensional examples, and then use the factorization to present solutions in one and two dimensions induced by a range of uniform maps.
Highlights
A basic question in the theory of discrete dynamical systems, and in statistical mechanics, is whether a chaotic iterated function M : X → X that maps a space X ⊆ Rd back onto X has an equilibrium distribution with probability density function (PDF) ρ( x )
In the language of [7], Theorem 1 shows that the inverse Frobenius–Perron problem (IFPP) for any distribution ρ has a solution, every solution map is conjugate to a uniform map, and the conjugating function is precisely the inverse Rosenblatt transformation
We have shown that the solution of the IFPP—finding an iterative map with a given invariant distribution—can be constructed from uniform maps through the factorization established in Theorem 1, M = R −1 ◦ U ◦ R
Summary
A basic question in the theory of discrete dynamical systems, and in statistical mechanics, is whether a chaotic iterated function M : X → X that maps a space X ⊆ Rd back onto X has an equilibrium distribution with probability density function (PDF) ρ( x ). For a given Rosenblatt transformation, there is a one-to-one correspondence between the solution of the IFPP and the choice of a uniform map This reformulation of the IFPP in terms of two well-studied constructs leads to practical analytic and numerical solutions by exploiting existing, well-developed methods for Rosenblatt transformations and deterministic iterations that target the uniform distribution. The factorization allows us to establish the equivalence of solutions of the IFPP and other methods that employ a deterministic map within the generation of ergodic sequences. This standardizes and simplifies existing solution methods by showing that they are special cases of constructing the Rosenblatt transformation (or its inverse) and the selection of a uniform map. A summary and discussion of results is presented in Section 7, including a discussion of some existing computational methods that can be viewed as implicitly implementing the factorization solution of the IFPP presented here
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