Self-similar tiling systems, topological factors and stretching factors

TL;DR

This paper investigates the relationship between self-similar tiling systems and their stretching factors, proving that if they share a non-periodic common factor, their stretching factors are multiplicatively dependent.

Contribution

It establishes a new connection between the existence of a common non-periodic factor and the multiplicative dependence of stretching factors in self-similar tiling systems.

Findings

Shared non-periodic factor implies multiplicative dependence of stretching factors

Stretching factors are multiplicatively dependent if systems share a non-periodic factor

Provides a criterion linking factors and stretching properties in tiling systems

Abstract

In this paper we prove that if two self-similar tiling systems, with respective stretching factors $\lambda_1$ and $\lambda_2$, have a common factor which is a non periodic tiling system, then $\lambda_1$ and $\lambda_2$ are multiplicatively dependent.