On periodicity of two-dimensional words

A two-dimensional word is a function on Z2 with finite number of values. The main problem we are interested in is the periodicity of two-dimensional words satisfying some local conditions. In this paper we prove that every bounded centered function on the infinite rectangular grid is periodic. A function is called centered if the sum of its values in every ball is equal to 0. Similar results are obtained for the infinite triangular and hexagonal grids.