Topological Properties of a Class of Self-Affine Tiles in R3
Transactions of the American Mathematical Society
We construct a class of connected self-affine tiles in R3 and prove that it contains a subclass of tiles that are homeomorphic to a unit ball in R3. 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.
Deng, Guotai, Chuntai Liu, Sze-Man Ngai.
"Topological Properties of a Class of Self-Affine Tiles in R3."
Transactions of the American Mathematical Society, 370 (2): 1321-1350.
doi: 10.1090/tran/7055 source: https://www.ams.org/journals/tran/2018-370-02/S0002-9947-2017-07055-0/