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Solutions of the Multivariate Inverse Frobenius-Perron Problem.

Colin Fox1, Li-Jen Hsiao2, Jeong-Eun Kate Lee3

  • 1Department of Physics, University of Otago, Dunedin 9016, New Zealand.

Entropy (Basel, Switzerland)
|July 2, 2021
PubMed
Summary
This summary is machine-generated.

Researchers solved the inverse Frobenius-Perron problem by finding deterministic maps that lead to a target distribution. All solutions involve a factorization combining Rosenblatt transformations and uniform maps, with every solution equivalent to a uniform map choice.

Keywords:
Rosenblatt transformationergodic mapinverse Frobenius–Perron problemmultivariate probability distributiontransfer operatoruniform map

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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.