A multi-dimensional analogue of Cobham’s theorem for fractals
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- by Davy Ho-Yuen Chan and Kevin G. Hare PDF
- Proc. Amer. Math. Soc. 142 (2014), 449-456 Request permission
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 \[ \{T_c(X) \ \mid \ c \in \mathcal {C}_1 \cup \mathcal {C}_k \cup \mathcal {C}_{k^2} \cup \mathcal {C}_{k^3} \cup \cdots \}. \] 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.
References
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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
- MR Author ID: 690847
- Email: kghare@uwaterloo.ca
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 449-456
- MSC (2010): Primary 11B85, 28A80
- DOI: https://doi.org/10.1090/S0002-9939-2013-11843-5
- MathSciNet review: 3133987