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”) Λ_λ 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 Λ_λ 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.

2000 AMS Subject Classification::

• Primary 28A80
• Secondary 37B99

Keywords::

• Sierpiński gasket
• interior
• iterated-function system
• fractal
• attractor

Notes

¹We use here the fact that the conclusions are unaffected by linear coordinate changes: the symmetry argument is easiest for an equilateral triangle, but otherwise, we work with an isosceles triangle.

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