On Periodic Decompositions and Nonexpansive Lines

TL;DR

This paper investigates Szabados's conjecture relating nonexpansive lines to periodic configurations within minimal decompositions, proving it for a broad class of configurations including low convex pattern complexity cases.

Contribution

It demonstrates that Szabados's conjecture holds for a wide class of configurations, including those with low convex pattern complexity, and refines the understanding of periodic decompositions.

Findings

Szabados's conjecture is valid for all not fully periodic low convex pattern complexity configurations.

Minimal periodic decompositions can be considered with configurations defined on finite alphabets.

The conjecture's validity extends to a broad class of configurations.

Abstract

In his Ph.D. thesis, Michal Szabados conjectured that for a not fully periodic configuration with a minimal periodic decomposition the nonexpansive lines are exactly the lines that contain a period for some periodic configuration in such decomposition. In this paper, we study Szabados's conjecture. First, we show that we may consider a minimal periodic decomposition where each periodic configuration is defined on a finite alphabet. Then we prove that Szabados's conjecture holds for a wide class of configurations, which includes all not fully periodic low convex pattern complexity configurations.