TY - JOUR

T1 - Aperiodic two-dimensional words of small abelian complexity

AU - Puzynina, Svetlana

PY - 2019/10/11

Y1 - 2019/10/11

N2 - In this paper we prove an abelian analog of the famous Nivat's conjecture linking complexity and periodicity for two-dimensional words: We show that if a two-dimensional recurrent word contains at most two abelian factors for each pair (n;m) of integers, then it has a periodicity vector. Moreover, we show that a two-dimensional aperiodic recurrent word must have more than two abelian factors infinitely often. On the other hand, there exist aperiodic recurrent words with abelian complexity bounded by 3, as well as aperiodic words having abelian complexity 1 for some pairs (m;n).

AB - In this paper we prove an abelian analog of the famous Nivat's conjecture linking complexity and periodicity for two-dimensional words: We show that if a two-dimensional recurrent word contains at most two abelian factors for each pair (n;m) of integers, then it has a periodicity vector. Moreover, we show that a two-dimensional aperiodic recurrent word must have more than two abelian factors infinitely often. On the other hand, there exist aperiodic recurrent words with abelian complexity bounded by 3, as well as aperiodic words having abelian complexity 1 for some pairs (m;n).

UR - http://www.scopus.com/inward/record.url?scp=85074024712&partnerID=8YFLogxK

UR - https://www.combinatorics.org/ojs/index.php/eljc/article/view/v26i4p15/7940

M3 - Article

AN - SCOPUS:85074024712

VL - 26

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 4

M1 - P4.15

ER -