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A matrix solution to the inverse Perron-Frobenius problem


Authors: P. Góra and A. Boyarsky
Journal: Proc. Amer. Math. Soc. 118 (1993), 409-414
MSC: Primary 58F11; Secondary 28D05
DOI: https://doi.org/10.1090/S0002-9939-1993-1129877-8
MathSciNet review: 1129877
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Abstract: Let $ f$ be a probability density function on the unit interval $ I$. The inverse Perron-Frobenius problem involves determining a transformation $ \tau :I \to I$ such that the one-dimensional dynamical system $ {x_{i + 1}} = \tau ({x_i})$ has $ f$ as its unique invariant density function. A matrix method is developed that provides a simple relationship between $ \tau $ and $ f$, where $ f$ is any piecewise constant density function. The result is useful for modelling and predicting chaotic data.


References [Enhancements On Off] (What's this?)

  • [1] N. Friedman and A. Boyarsky, Construction of ergodic transformations, Adv. in Math. 45 (1982), 213-254. MR 673802 (84f:28013)
  • [2] S. H. Ershov and G. G. Malinetskiĭ, The solution of the inverse problem for the Perron-Frobenius equation, U.S.S.R. Comput. Math. and Math. Phys. 28 (1988), 136-141. MR 973203 (90d:58074)
  • [3] A. Boyarsky and G. Haddad, All absolutely continuous invariant measures of piecewise linear Markov maps are piecewise constant, Adv. in Appl. Math. 2 (1981), 284-289. MR 626863 (82i:58041)
  • [4] M. Casdagli, Nonlinear prediction of chaotic time series, Phys. D 35 (1989), 335-356. MR 1004201 (90e:58100)
  • [5] S. Pelikan, Invariant densities for random maps of the interval, Trans. Amer. Math. Soc. 281 (1984), 813-825. MR 722776 (85i:58070)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1129877-8
Article copyright: © Copyright 1993 American Mathematical Society

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