A class of self-affine tiles in Rd that are d-dimensional tame balls

摘要

We study a family of self-affine tiles in Rd (d >= 2) with noncollinear digit sets, which naturally generalizes a two-dimensional class studied originally by Deng and Lau and its extension to R3 by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be d-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a d-dimensional ball if and only if its interior is connected.

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