On periodic decompositions, one-sided nonexpansive directions and Nivat's conjecture
Cleber Fernando Colle · Discrete and Continuous Dynamical Systems · 2023
Nivat's conjecture is a famous open problem in symbolic dynamics. Recent advances toward a proof for Nivat's conjecture involve a precise control of nonexpansive lines and one-sided nonexpansive directions. In his Ph.D. thesis, Michal Szabados showed that every low pattern complexity configuration can be decomposed into a finite sum of periodic configurations. If we consider a not fully periodic configuration with a minimal periodic decomposition, that is, a periodic decomposition with the smallest possible number of periodic configurations, a nonexpansive line contains a period of some periodic configuration in such minimal periodic decomposition. Michal Szabados conjectured that the converse also holds. In this paper, we provide conditions where (i) Szabados's conjecture holds and (ii) a given line is a nonexpansive line if and only if the same line endowed with a given orientation is a one-sided nonexpansive direction. As an application of our main result, we get that Nivat's conjecture holds if and only if it holds for configurations satisfying (i) and (ii).