Existence, mixing and approximation of invariant densities for expanding maps on Rr

Abstract

This paper generalises Gora and Boyarsky’s bounded variation(BV) approach to the ergodic properties of expanding transformations, and analysies the convergence of Ulam’s method for the numerical approximation of absolutely continuous invariant measures. We first prove an existence theorem for BV invariant densities for piecewise expanding maps on subsets of Rr; the maps must be C2, but may have infinitely many branches and need not be Markov. Under and additional “onto” assumption, explicit bounds on the spectral gap in the associated Perron-Frobenius operator are proved. The corresponding contraction rates are in the BV norm, rather than a projective metric. With this quantitative information, we are then able to prove convergence and explicit upper bounds on the approximation error in Ulam’s method for approximating invariant measures. Because the BV approach is rather concete, the methods of this paper can be applied in practice; this is illustrated by an application of the main results to the Jacobi-Perron transformation on R2.