nivat s conjecture and pattern complexity in algebraic subshifts

• DOI: 10.48550/arxiv.1806.07107
• Publication Date: 2019-07-01
• Authors: Etienne Moutot; Etienne Moutot; Jarkko Kari
• Keywords: FOS: Computer and information sciences; Discrete Mathematics (cs.DM); ta111; Dynamical Systems (math.DS); 0102 computer and information sciences; [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]; 01 natural sciences; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); Mathematics - Dynamical Systems; 0101 mathematics; Computer Science - Discrete Mathematics

11 pages<br/>We study Nivat's conjecture on algebraic subshifts and prove that in some of them every low complexity configuration is periodic. This is the case in the Ledrappier subshift (the 3-dot system) and, more generally, in all two-dimensional algebraic subshifts over $\mathbb{F}_p$ defined by a polynomial without line polynomial factors in more than one direction. We also find an algebraic subshift that is defined by a product of two line polynomials that has this property (the 4-dot system) and another one that does not.<br/>