Pointwise estimate for the Hardy-Littlewood maximal operator on the orbits of contractive mappings

Let Mn denote the Hardy-Littlewood maximal operator on the n-th iteration of a given iterated function system (IFS). We give sufficient conditions on the IFS in order to obtain a pointwise estimate for Mn in terms of the composition of M0 and a discrete Hardy-Littlewood type maximal operator. As a corollary we prove the uniform preservation of Muckenhoupt condition along the Hutchinson orbits induced by such an IFS.