On Periodic Decompositions, One-sided Nonexpansive Directions and Nivat's Conjecture

TL;DR

This paper investigates the structure of nonexpansive lines in symbolic dynamics, providing conditions under which Szabados's conjecture holds and relating it to Nivat's conjecture for low complexity configurations.

Contribution

It establishes conditions where Szabados's conjecture is valid and links nonexpansive lines to one-sided nonexpansive directions, advancing understanding of Nivat's conjecture.

Findings

Conditions where Szabados's conjecture holds

Equivalence of nonexpansive lines and one-sided nonexpansive directions

Nivat's conjecture holds for low convex complexity configurations under certain conditions

Abstract

Nivat's conjecture is a famous open problem in symbolic dynamics. The existence of nonexpansive lines that when endowed with a given orientation are one-sided nonexpansive directions is at the heart of some advances. In his Ph.D. thesis, Michal Szabados conjectured that for a minimal periodic decomposition the nonexpansive lines are exactly the lines that contain a period of some periodic configuration in such decomposition. In this paper, we provide conditions where (i) Szabados's conjecture holds and (ii) a given line is nonexpansive if and only if the same line endowed with a given orientation is a one-sided nonexpansive direction. As a corollary of our main result, we get that Nivat's conjecture holds for low convex complexity configurations if and only if it holds for low convex complexity configurations satisfying (i) and (ii).