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

It is well known that for different classes of transformations, including the class of piecewise C-2 expanding maps T : [ 0, 1] O, 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 phi beta = - beta log vertical bar T-1 vertical bar, where beta is an element of R. In particular, we prove that our extended Ulam's method is a powerful tool for computing the topological pressure P( T, phi beta) and the density of the equilibrium state.