A piecewise linear spline maximum entropy method for Frobenius-Perron operators of multi-dimensional transformations

In 1976, Mathematician Tien-Yien Li published a solution to a conjecture by Ulam concerning a finite approximation to the Frobenius-Perron operator. This accomplishment was a breakthrough for the numerical approximation of the invariant densities that describe the statistical behaviour of dynamical systems. Twenty years later, Jiu Ding and Aihui Zhou extended this method to multi-dimensional transformations. Since then, several different methods have been developed to approximate these invariant densities. Here, we present a piecewise linear spline maximum entropy method for the approximation of invariant densities corresponding to multi-dimensional transformations. Applications are considered and numerical results are explored.