Topological properties of a class of self-affine tiles in ℝ³

Deng, Guotai; Liu, Chuntai; Ngai, Sze-Man

doi:10.1090/tran/7055

Topological properties of a class of self-affine tiles in $\mathbb {R}^3$
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by Guotai Deng, Chuntai Liu and Sze-Man Ngai PDF

Trans. Amer. Math. Soc. 370 (2018), 1321-1350 Request permission

Abstract:

We construct a class of connected self-affine tiles in $\mathbb {R}^3$ and prove that it contains a subclass of tiles that are homeomorphic to a unit ball in $\mathbb {R}^3$. Our construction is obtained by generalizing a two-dimensional one by Deng and Lau. The proof of ball-likeness is inspired by the construction of a homeomorphism from Alexander’s horned ball to a 3-ball.

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