DOUBLING PROPERTIES OF SELF-AFFINE MEASURES SUPPORTED ON GATZOURAS–LALLEY FRACTALS

Abstract

In this paper, we study the doubling properties of self-affine measures supported on a class of self-affine sets named Gatzouras–Lalley sets. First, we show that every self-affine measure is doubling if the Gatzouras–Lalley sets satisfy “strong separation” in both horizontal and vertical directions. Then we provide a sufficient condition for self-affine measures supported on Gatzouras–Lalley sets to be doubling if the sets do not satisfy the “separation condition” in the horizontal direction. Finally, we investigate measures without “vertical separation”, and give sufficient conditions for them to be doubling.