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arXiv · 2209.03008

A class of self-affine tiles in $\mathbb{R}^d$ that are $d$-dimensional tame balls

Guotai Deng·Chuntai Liu·Sze-man Ngai
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Abstract

We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ 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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Guotai Deng, Chuntai Liu, Sze-man Ngai. 2022-09-07. A class of self-affine tiles in $\mathbb{R}^d$ that are $d$-dimensional tame balls. https://doi.org/10.1016/j.aim.2022.108716

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