Nivat's Conjecture and Pattern Complexity in Algebraic Subshifts

TL;DR

This paper investigates Nivat's conjecture within algebraic subshifts, demonstrating that certain low complexity configurations are necessarily periodic, with specific results for the Ledrappier subshift and related systems.

Contribution

It proves that in some algebraic subshifts, low complexity configurations are periodic, extending understanding of Nivat's conjecture in algebraic contexts.

Findings

Low complexity configurations are periodic in the Ledrappier subshift.

Certain algebraic subshifts defined by polynomials without line polynomial factors exhibit this property.

Existence of algebraic subshifts with different polynomial structures that do or do not have this property.

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 $\mathbb{F}_p$ 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.