Rigorous numerical approximation of Ruelle–Perron–Frobenius operators and topological pressure of expanding maps

Abstract

It is well known that for different classes of transformations, including the class of piecewise C2 expanding maps T : [0, 1] ↺, Ulam's method is an efficient way to numerically approximate the absolutely continuous invariant measure of T. We develop a new extension of Ulam's method and prove that this extension can be used for the numerical approximation of the Ruelle–Perron–Frobenius operator associated with T and the potential ϕβ = −β log |T∣|, where . In particular, we prove that our extended Ulam's method is a powerful tool for computing the topological pressure P(T, ϕβ) and the density of the equilibrium state.