Nivat’s conjecture is a long-standing open combinatorial problem. It concerns two-dimensional configurations, that is, maps Z 2 →A Z2→A where A A is a finite set of symbols. Such configurations are often understood as colorings of a two-dimensional square grid. Let P c (m,n) Pc(m,n) denote the number of distinct m×n m×n block patterns occurring in a configuration c. Configurations satisfying P c (m,n)≤mn Pc(m,n)≤mn for some m,n∈N m,n∈N are said to have low rectangular complexity. Nivat conjectured that such configurations are necessarily periodic.
Recently, Kari and the author showed that low complexity configurations can be decomposed into a sum of periodic configurations. In this paper we show that if there are at most two components, Nivat’s conjecture holds. As a corollary we obtain an alternative proof of a result of Cyr and Kra: If there exist m,n∈N m,n∈N such that P c (m,n)≤mn/2 Pc(m,n)≤mn/2 , then c is periodic. The technique used in this paper combines the algebraic approach of Kari and the author with balanced sets of Cyr and Kra.