On periodicity of generalized two-dimensional infinite words

A generalized two-dimensional word is a function on Z with a finite number of values. The main problem we are interested in is periodicity of twodimensional words satisfying some local conditions. Let A be a matrix of order n. The function φ : Z → R is a generalized centered function of radius r with the matrix A if ∑ y ∈ Z2 : 0 < |y − x| ≤ r φ(y) = φ(x)A for every x ∈ Z, where for x = (x1, x2), y = (y1, y2) we have |y − x| = |y1 − x1| + |y2 − x2|. We prove that every generalized centered function of radius r > 1 with a finite number of values is periodic. For r = 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.