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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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An analogue of Cobham’s theorem for fractals
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by Boris Adamczewski and Jason Bell PDF
Trans. Amer. Math. Soc. 363 (2011), 4421-4442 Request permission

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
  • MR Author ID: 704234
  • 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
  • MR Author ID: 632303
  • Email: jpb@math.sfu.ca
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 4421-4442
  • MSC (2010): Primary 28A80, 11B85
  • DOI: https://doi.org/10.1090/S0002-9947-2011-05357-2
  • MathSciNet review: 2792994