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A multi-dimensional analogue of Cobham's theorem for fractals


Authors: Davy Ho-Yuen Chan and Kevin G. Hare
Journal: Proc. Amer. Math. Soc. 142 (2014), 449-456
MSC (2010): Primary 11B85, 28A80
Published electronically: November 14, 2013
MathSciNet review: 3133987
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Abstract: Let $ \mathcal {C}_k$ be the set of hyper-cubes of size $ \frac {1}{k} \times \cdots \times \frac {1}{k}$ in $ [0,1]^d$ with vertices having coordinates of the form $ (a_1/k, a_2/k, \ldots , a_d/k)$. For $ c \in \mathcal {C}_k$ we define $ T_c$ as the linear expansion map from $ c$ to $ [0,1]^d$ in the obvious way. We extend the map $ T_c$ to a map on all of $ [0,1]^d$ by defining $ T_c(\overline {\mathbf {x}}) = \emptyset $ for $ \overline {\mathbf {x}} \not \in c$.

Let $ X$ be a compact subset of $ [0,1]^d$ and $ k > 1$ be an integer. We define the $ k$-kernel of $ X$ as

$\displaystyle \{T_c(X) \ \mid \ c \in \mathcal {C}_1 \cup \mathcal {C}_k \cup \mathcal {C}_{k^2} \cup \mathcal {C}_{k^3} \cup \cdots \}. $

If this set is finite, then we say that $ X$ has finite $ k$-kernel or, equivalently, that $ X$ is $ k$-self-similar. Some examples of this are the standard Cantor set, the Sierpiński carpet and the Sierpiński triangle.

Recently Adamczewski and Bell showed an analogue of Cobham's theorem for one-dimensional fractals. Let $ k$ and $ \ell $ be multiplicatively independent positive integers. They proved that the compact set $ X \subset [0,1]$ is both $ k$- and $ \ell $-self-similar if and only if $ X$ is a union of a finite number of intervals with rational endpoints.

In their paper, Adamczewski and Bell conjectured that a similar result should be true in higher dimensions. We prove their conjecture; in particular we prove:


Theorem. Let $ k$ and $ \ell $ be multiplicatively independent positive integers. The compact set $ X \subset [0,1]^d$ is both $ k$- and $ \ell $-self-similar if and only if $ X$ is a union of a finite number of rational polyhedra.


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Additional Information

Davy Ho-Yuen Chan
Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
Email: chydavy@gmail.com

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: kghare@uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9939-2013-11843-5
Received by editor(s): November 16, 2011
Received by editor(s) in revised form: March 28, 2012
Published electronically: November 14, 2013
Additional Notes: The research of the first author was supported by the Department of Mathematics, The Chinese University of Hong Kong
The research of the second author was partially supported by NSERC
Communicated by: Ken Ono
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.