Complexity of short rectangles and periodicity

The Morse-Hedlund Theorem states that a bi-infinite sequence η in a finite alphabet is periodic if and only if there exists n ∈ N such that the block complexity function Pη(n) satisfies Pη(n) ≤ n. In dimension two, Nivat conjectured that if there exist n, k ∈ N such that the n × k rectangular complexity Pη(n, k) satisfies Pη(n, k) ≤ nk, then η 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 Z2 dynamical system, and then analyzing the resulting system.

• Type: Journal
• Year: 2016
• Where: EJC
• Authors: Van Cyr, Bryna Kra
• Keywords: EJC 2016, Information Technology