Abelian maximal pattern complexity of two-dimensional words

In this paper, we study the maximal pattern complexity of two dimensional words up to Abelian equivalence. We establish a lower bound for the Abelian maximal pattern complexity of two-dimensional words that are nondoubly periodic by projection, under either the existence of a transverse recurrence direction or strong recurrence. We further show that this bound is sharp. As a consequence, we obtain the exact complexity for strongly recurrent binary non-doubly periodic words and a boundedness criterion for double periodicity.