A matrix solution to the inverse Perron-Frobenius problem

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.