Complexity of Short Rectangles and Periodicity
European Journal of Combinatorics
The Morse-Hedlund Theorem states that a bi-infinite sequence eta in a finite alphabet is periodic if and only if there exists n is an element of N such that the block complexity function P-eta(n) satisfies P-eta(n) <= n. In dimension two, Nivat conjectured that if there exist n, k is an element of N such that the n x k rectangular complexity P-eta(n, k) satisfies P-eta(n, k) <= nk, then eta is periodic. Sander and Tijdeman showed that this holds for k <= 2. We generalize their result, showing that Nivat's Conjecture holds for k <= 3. The method involves translating the combinatorial problem to a question about the nonexpansive subspaces of a certain Z(2) dynamical system, and then analyzing the resulting system. (C) 2015 Elsevier Ltd. All rights reserved.
Cyr, Van and Kra, Bryna. "Complexity of Short Rectangles and Periodicity." European Journal of Combinatorics 52, no. Pt. A (2016) : 146-173.