Abstract
We consider the complexity of bi-infinite words in one and two dimensions. A result of Morse and Hedlund in one dimension states that if the complexity, p ξ ( n ), of a word satisfies p ξ ( n )⩽ n for some n , then the word ξ is periodic. The corresponding question in two dimensions (whether p ξ ( m , n )⩽ mn implies that ξ is periodic) is known as the Nivat conjecture. In this paper, we strengthen the one-dimensional result of Morse and Hedlund and prove a weak form of the Nivat conjecture, namely that if for a bi-infinite two-dimensional word ξ , p ξ ( m , n )⩽ mn /16 then ξ is periodic.
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