Doubling properties of self-similar measures and Markov measures on Sierpiński carpets

Abstract

As is well known, when a self-similar set satisfies the strong separation condition, all self-similar measures are doubling. In this paper, we further prove that all Markov measures are doubling on a self-similar set with the strong separation condition. Subsequently, we focus on self-similar measures and Markov measures on Sierpinski carpets. Without the strong separation condition, Sierpinski carpets can be divided into different types. In each case, we fully characterize doubling self-similar measures and doubling Markov measures on a Sierpinski carpet.