Rates of convergence for iterates of positive linear operators

We are concerned with positive linear operators defined on C(X) , where X is a simplex or a hypercube.We assume that the operators preserve the affine functions.After identifying an eigenvalue a [0,1) of such an operator L , we show that the sequence| is dominated by a k multiplied by a factor depending on L , f and x .These general results are applied to several classical or recently introduced operators acting on simplices and hypercubes.