  In his Ph.D. thesis, Michal Szabados conjectured that for a not fully
periodic configuration with a minimal periodic decomposition the nonexpansive
lines are exactly the lines that contain a period for some periodic
configuration in such decomposition. In this paper, we study Szabados's
conjecture. First, we show that we may consider a minimal periodic
decomposition where each periodic configuration is defined on a finite
alphabet. Then we prove that Szabados's conjecture holds for a wide class of
configurations, which includes all not fully periodic low convex pattern
complexity configurations.
