Diagonals in 2D-tilings and coincidence densities of substitutions

TL;DR

This paper investigates the properties of diagonals in 2D tilings generated by substitutions, revealing their recurrence, frequency, and connection to coincidence densities, influenced by eigenvalues and letter order.

Contribution

It introduces new insights into the behavior of diagonals in 2D tilings, linking their properties to eigenvalues and substitution order, and explores their recurrence and frequency issues.

Findings

Diagonals can be non-uniformly recurrent.

Letter frequencies on diagonals may not exist.

Diagonal properties are connected to coincidence densities.

Abstract

We study the diagonals of two-dimensional tilings generated by direct product substitutions. The properties of these diagonals are primarily determined by the eigenvalues of the substitution matrix, but also the order of the letters in the substitution plays a role. We show that the diagonals may fail to be uniformly recurrent, and that the frequencies of letters on the diagonal may not exist. We also highlight the connection with the density of coincidences and overlap distributions.