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Nivat's Conjecture and expansive dynamics
If you have a question about this talk, please contact Ben Green.
The Morse-Hedlund Theorem states that an infinite word in a finite alphabet is periodic if and only if there is exists a positive integer n such that the complexity (the number of words of length n) is bounded by n, and a natural approach to this theorem is via analyzing the dynamics of the Z-action associated to the word. In two dimensions, a conjecture of Nivat states that if there exist positive integers n and k such that the complexity (the number of n by k rectangles) is bounded by nk. Associating a Z^2 dynamical system to the infinite word, we show that periodicity is equivalent to a statement about the expansive subspaces of the action. As a corollary, we prove a weaker form of Nivat’s conjecture, under a stronger bound on the complexity function. This is joint work with Van Cyr.
This talk is part of the Discrete Analysis Seminar series.
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