On the Uniform Doubling of Hutchinson Orbits of Contractive Mappings

Abstract

We are interested in the preservation of doubling properties along the Hutchinson orbit generated by successive applications of contraction mappings on a metric measure space. We construct some elementary examples, built on Muckenhoupt weights, showing that even when the initial and the limit points of the orbit are doubling, no iteration of the IFS remains doubling. We also obtain positive results under some quantitative assumptions on the separation of the images through the IFS. We also explore the completeness, in the Hausdorff-Kantorovich metric, of a version of the doubling property which is suitable for the application of Hutchinson type contractions.