Solutions of the Multivariate Inverse Frobenius-Perron Problem

Area of Science:

• Mathematics
• Probability Theory
• Dynamical Systems

Background:

• The inverse Frobenius-Perron problem seeks a map M whose iterations converge to a target distribution ρ.
• Understanding such maps is crucial for analyzing the long-term behavior of dynamical systems.

Purpose of the Study:

• To characterize all solutions to the inverse Frobenius-Perron problem.
• To introduce a novel factorization for these solutions.

Main Methods:

• The study employs a factorization approach combining forward and inverse Rosenblatt transformations.
• Invariance of the uniform distribution under a map (uniform map) is a key component.

Main Results:

• All solutions to the inverse Frobenius-Perron problem can be expressed via a specific factorization.
• Every solution is demonstrated to be equivalent to the selection of a uniform map.
• The factorization is illustrated with one-dimensional examples and extended to one and two dimensions.

Conclusions:

• The proposed factorization provides a comprehensive framework for understanding solutions to the inverse Frobenius-Perron problem.
• The equivalence to uniform maps simplifies the construction and analysis of such deterministic maps.