On periodicity of generalized two-dimensional infinite words

A generalized two-dimensional word is a function on Z2Z2 with a finite number of values. The main problem we are interested in is periodicity of two-dimensional words satisfying some local conditions. Let A be a matrix of order n  . The function φ:Z2→Rnφ:Z2→Rn is a generalized centered function of radius r with the matrix A if∑y∈Z2:0<|y-x|⩽r-35ptφ(y)=φ(x)Afor every x∈Z2x∈Z2, where for x=(x1,x2)x=(x1,x2), y=(y1,y2)y=(y1,y2) we have |y-x|=|y1-x1|+|y2-x2||y-x|=|y1-x1|+|y2-x2|. We prove that every generalized centered function of radius r>1r>1 with a finite number of values is periodic. For r=1r=1 the existence of non-periodic generalized centered functions depends on the spectrum of the matrix A. Similar results are obtained for the infinite triangular and hexagonal grids.