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Theory and examples of the inverse Frobenius-Perron problem for complete chaotic maps. (English) Zbl 0982.37024

Summary: The general solution of the inverse Frobenius-Perron problem considering the construction of a fully chaotic dynamical system with given invariant density is obtained for the class of one-dimensional unimodal complete chaotic maps. Some interesting connections between this general solution and the special approach via conjugation transformations are illuminated. The developed method is applied to obtain a class of maps having as invariant density the two-parametric beta-probability density function. Varying the parameters of the density a rich variety of dynamics is observed. Observables like autocorrelation functions, power spectra, and Lyapunov exponents are calculated for representatives of this family of maps and some theoretical predictions concerning the decay of correlations are tested.

MSC:

37E05 Dynamical systems involving maps of the interval
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
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