Orthonormal expansions of invariant densities for expanding maps

AbstractWe give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding Cω Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [Problems in Modern Mathematics, Interscience, New York, 1960] of phase space discretisation.Our method hinges instead on the expansion of the density function with respect to an L2 orthonormal basis, and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map. The efficiency of the method, and its extension to Ck expanding maps, are also discussed