On the Interior of “Fat” Sierpiński Triangles

Abstract

For λ ∈ [0.6439, 0.6441]∪[0.6458, 0.6466]∪[0.6470, 0.6472], the “fat” Sierpiński triangle—the unique compact invariant set (“attractor”) Λλ of the iterated function system on , where p0 = (0, 0), p1 = (1, 0), p2 = (1/2, 1)—has interior points and hence positive Lebesgue measure and Hausdorff dimension 2. For these λ, standard techniques for determining the Hausdorff dimension (or Lebesgue measure) of Λλ do not apply, and its Hausdorff dimension has not been known for any specific such λ. The novelty of our approach is that instead of extending techniques developed for small λ, we significantly extend geometric methods developed by Broomhead, Montaldi, and Sidorov for larger values of λ.