Transfer operators for piecewise affine approximations of interval maps

We consider a natural approximation scheme for piecewise expanding, piecewise C1+Lipschitz, mixing Markov interval maps f by piecewise affine maps. We prove that the densities of the absolutely continuous invariant probability measures of the approximations converge exponentially fast to the density of the absolutely continuous invariant probability measure of f in the uniform norm. To do this we compare the relevant transfer operators of the approximations with that of f , and u s e recently developed perturbation techniques.