Random beta-transformations on fat Sierpinski gaskets

Abstract

We consider the iterated function system (IFS) fq(z)=z +q β , q ∈{(0,0),(1,0),(0,1)}. As is well known, for β = 2 the attractor, Sβ, is a fractal called the Sierpi´nski gasket (or sieve) and for β>2 it is also a fractal. Our goal is to study random β-transformations on the attractor for this IFS with 1 <β≤ 3/2. In this case, Sβ is a triangle. We show that all β-expansions of a point z in Sβ can be generated by a random map Kβ defined on {0,1}N ×{0,1,2}N ×Sβ and Kβ has a unique invariant measure of maximal entropy. Furthermore, we show the existence of a Kβ-invariant probability measure of the form m1 ⊗ m2 ⊗ μβ, wherem1,m2 are product measures on {0,1}N,{0,1,2}N, respectively, and μβ is absolutely continuous with respect to the two-dimensional Lebesgue measure λ2.