Periodicity and complexity in higher dimensions

Jeudi 24 janvier 2013 14:00-15:00 - Bryna Kra - Northwestern

Résumé : The Morse-Hedlund Theorem states that an infinite word in a finite alphabet is periodic if and only if there is is a positive integer n such that the complexity (the number of words of length n) is bounded by n. 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, then the two dimensional word is periodic in some direction. Associating a Z^2 dynamical system to the 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.

Lieu : bât. 425 - 121-123

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