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Nonexpansive $ \mathbb{Z}^2$-subdynamics and Nivat's Conjecture


Authors: Van Cyr and Bryna Kra
Journal: Trans. Amer. Math. Soc. 367 (2015), 6487-6537
MSC (2010): Primary 37B50; Secondary 68R15, 37B10
Published electronically: February 4, 2015
Previous version of record: Original version posted January 30, 2015
Corrected version of record: Current version corrects error introduced by publisher in the abstract.
MathSciNet review: 3356945
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Abstract: For a finite alphabet $ \mathcal {A}$ and $ \eta \colon \mathbb{Z}\to \mathcal {A}$, the Morse-Hedlund Theorem states that $ \eta $ is periodic if and only if there exists $ n\in \mathbb{N}$ such that the block complexity function $ P_\eta (n)$ satisfies $ P_\eta (n)\leq n$, and this statement is naturally studied by analyzing the dynamics of a $ \mathbb{Z}$-action associated with $ \eta $. In dimension two, we analyze the subdynamics of a $ \mathbb{Z}^2$-action associated with $ \eta \colon \mathbb{Z}^2\to \mathcal {A}$ and show that if there exist $ n,k\in \mathbb{N}$ such that the $ n\times k$ rectangular complexity $ P_{\eta }(n,k)$ satisfies $ P_{\eta }(n,k)\leq nk$, then the periodicity of $ \eta $ is equivalent to a statement about the expansive subspaces of this action. As a corollary, we show that if there exist $ n,k\in \mathbb{N}$ such that $ P_{\eta }(n,k)\leq \frac {nk}{2}$, then $ \eta $ is periodic. This proves a weak form of a conjecture of Nivat in the combinatorics of words.


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Additional Information

Van Cyr
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Address at time of publication: Department of Mathematics, 361 Olin, Bucknell University, Lewisburg, Pennsylvania 17837
Email: cyr@math.northwestern.edu, van.cyr@bucknell.edu

Bryna Kra
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: kra@math.northwestern.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06391-0
Keywords: Nivat's Conjecture, $\mathbb{Z}^2$-subshift, nonexpansive subdynamics, block complexity, periodicity
Received by editor(s): March 29, 2013
Received by editor(s) in revised form: April 10, 2013, and August 23, 2013
Published electronically: February 4, 2015
Additional Notes: The second author was partially supported by NSF grant $1200971$.
Article copyright: © Copyright 2015 American Mathematical Society