Van Cyr

Complexity of short rectangles and periodicity

The Morse-Hedlund Theorem states that a bi-infinite sequence ? in a finite alphabet is periodic if and only if there exists n ? N such that the block complexity function P ? ( n ) satisfies P ? ( n ) ? n . In dimension two, Nivat conjectured that if there exist n , k ? N such that the n × k rectangular complexity P ? ( n , k ) satisfies P ? ( n , k ) ? n k , then ? is periodic. Sander and Tijdeman showed that this holds for k ? 2 . We generalize their result, showing that Nivats 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.