General theory and examples of the inverse Frobenius-Perron problem

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The general solution of the inverse Frobenius-Perron problem considering the construction of a fully chaotic dynamical system with given invariant density is obtained within the class of one-dimensional unimodal maps. Some interesting connections between this solution and the approach via conjugation transformations are eluminated. The developed method is applied to obtain a wide 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.