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
DOI:
https://doi.org/10.1090/S0002-9939-2013-11843-5
Published electronically:
November 14, 2013
MathSciNet review:
3133987
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be the set of hyper-cubes of size
in
with vertices having coordinates of the form
. For
we define
as the linear expansion map from
to
in the obvious way. We extend the map
to a map on all of
by defining
for
.
Let be a compact subset of
and
be an integer. We define the
-kernel of
as





Recently Adamczewski and Bell showed an analogue of Cobham's theorem for one-dimensional fractals. Let and
be multiplicatively independent positive integers. They proved that the compact set
is both
- and
-self-similar if and only if
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 and
be multiplicatively independent positive integers. The compact set
is both
- and
-self-similar if and only if
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.