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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.