Optimal partition choice for invariant measure approximation for one-dimensional maps

Abstract

The problem of finding absolutely continuous invariant measures (ACIMs) for a dynamical system can be formulated as a fixed point problem for a Markov operator (the Perron–Frobenius operator). This is an infinite-dimensional problem. Ulam's method replaces the Perron–Frobenius operator by a sequence of finite rank approximations whose fixed points are relatively easy to compute numerically. This paper concerns the optimal choice of Ulam approximations for one-dimensional maps; an adaptive partition selection is used to tailor the approximations to the structure of the invariant measure. The main idea is to select a partition which equally distributes the square root of the derivative of the invariant density amongst the bins of the partition. The results are illustrated for the logistic map where the ACIMs may have inverse square root singularities in their density functions. O(log n/n) convergence rates can be expected, whereas a non-adaptive algorithm yields O(n−1/2) at best. Studying the convergence of the adaptive algorithm allows an estimate to be made of the measure of the Jakobson parameter set (those logistic maps which admit an ACIM).