Symbolic recurrence plot for uniform binary substitutions

TL;DR

This paper investigates the structure of diagonal lines in symbolic recurrence plots derived from fixed points of uniform binary substitutions, revealing how primitivity and aperiodicity influence line length densities.

Contribution

It characterizes the distribution and density of diagonal line lengths in recurrence plots for primitive and non-primitive uniform binary substitutions.

Findings

Primitive aperiodic substitutions have zero density of all diagonal line lengths.

Existence of a line of specific length implies positive density of such lines.

Non-primitive substitutions contain lines of any length, but with zero density.

Abstract

Diagonal lines in symbolic recurrence plots are closely related to the identification and characterization of specific biprolongable words within a sequence. In this paper we focus on the recurrence plot of a fixed point of a uniform binary substitution. We show that, if the substitution is primitive and aperiodic, the set of all diagonal line lengths of the recurrence plot has zero density. However, if a line of a specific length exists in the recurrence plot, the density of (the set of starting points of) all diagonal lines with that length is strictly positive. On the other hand, we demonstrate that the recurrence plot of a non-primitive substitution contains lines of any given length. Nonetheless, for any given length, the density of lines with that length is zero.