Emergent Mind
Cobham's theorem for substitutions
(1010.4009)
Published Oct 19, 2010
in
math.CO
,
cs.DM
,
and
math.DS
Abstract
The seminal theorem of Cobham has given rise during the last 40 years to a lot of works around non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then, a sequence $x\in A\mathbb{N}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately periodic.
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