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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An analogue of Cobham's theorem for fractals


Authors: Boris Adamczewski and Jason Bell
Journal: Trans. Amer. Math. Soc. 363 (2011), 4421-4442
MSC (2010): Primary 28A80, 11B85
Published electronically: March 4, 2011
MathSciNet review: 2792994
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Abstract: We introduce the notion of $ k$-self-similarity for compact subsets of $ \mathbb{R}^n$ and show that it is a natural analogue of the notion of $ k$-automatic subsets of integers. We show that various well-known fractals such as the triadic Cantor set, the Sierpiński carpet or the Menger sponge turn out to be $ k$-self-similar for some integers $ k$. We then prove an analogue of Cobham's theorem for compact sets of $ \mathbb{R}$ that are self-similar with respect to two multiplicatively independent bases $ k$ and $ \ell$. Namely, we show that $ X$ is both a $ k$- and an $ \ell$-self-similar compact subset of $ \mathbb{R}$ if and only if it is a finite union of closed intervals with rational endpoints.


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Additional Information

Boris Adamczewski
Affiliation: CNRS, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
Email: Boris.Adamczewski@math.univ-lyon1.fr

Jason Bell
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia, Canada V5A 1S6
Email: jpb@math.sfu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05357-2
PII: S 0002-9947(2011)05357-2
Received by editor(s): July 5, 2009
Received by editor(s) in revised form: January 28, 2010, and March 23, 2010
Published electronically: March 4, 2011
Additional Notes: The first author was supported by the ANR through the project “DyCoNum”–JCJC06 134288. He also thanks Jean-Paul Allouche for pointing out relevant references.
The second author thanks NSERC for its generous support.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.