Nivat's Conjecture and Pattern Complexity in Algebraic Subshifts

2018/06/19 by Kari, Jarkko, Moutot, Etienne
#Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.07107

Abstract

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 \mathbbF_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.

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