The smoothness of the stationary measure

TL;DR

This paper investigates how small smooth changes in the defining functions of an iterated function scheme affect the smoothness of its stationary measure, including implications for Hausdorff dimensions.

Contribution

It establishes theoretical relationships between perturbations of the scheme and the resulting smoothness of the stationary measure and its Hausdorff dimension.

Findings

Stationary measure smoothness depends on scheme and weight function smoothness.

Results connect perturbation smoothness to measure and dimension regularity.

Provides conditions for smoothness of Hausdorff dimension of limit set.

Abstract

We study the smoothness of the stationary measure with respect to smooth perturbations of the iterated function scheme and the weight functions that define it. Our main theorems relate the smoothness of the perturbation of: the iterated function scheme and the weight functions; to the smoothness of the perturbation of the stationary measure. The results depend on the smoothness of: the iterated function scheme and the weights functions; and the space on which the stationary measure acts as a linear operator. As a consequence we also obtain the smoothness of the Hausdorff dimension of the limit set and of the Hausdorff dimension of the stationary measure.