A matrix solution to the inverse Perron-Frobenius problem

Abstract

Let f f be a probability density function on the unit interval I I . The inverse Perron-Frobenius problem involves determining a transformation τ : I → I \tau :I \to I such that the one-dimensional dynamical system x i + 1 = τ ( x i ) {x_{i + 1}} = \tau ({x_i}) has f f as its unique invariant density function. A matrix method is developed that provides a simple relationship between τ \tau and f f , where f f is any piecewise constant density function. The result is useful for modelling and predicting chaotic data.