On a conjecture about the absence of an initial balanced pair for Pisot substitutions

TL;DR

This paper proves a conjecture that the balanced pair algorithm fails for a specific pair of Pisot substitutions due to the absence of an initial balanced pair, using a novel coding method.

Contribution

It introduces a new method based on simultaneous coding to prove the failure of the balanced pair algorithm for the conjectured Pisot substitutions.

Findings

Confirmed the conjecture that no initial balanced pair exists for the given substitutions.

Demonstrated the effectiveness of simultaneous coding in analyzing substitution systems.

Abstract

Sellami and Sirvent conjectured that the balanced pair algorithm fails for the following pair of Pisot substitutions: \[ \varphi_0: \begin{array}{l} a \mapsto abc b \mapsto a c \mapsto ac \end{array} \quad \text{ and } \quad \varphi_1: \begin{array}{l} a \mapsto cba b \mapsto a c \mapsto ca \end{array}. \] The conjecture stated the balanced pair algorithm fails because there is no initial balanced pair. In the present note we prove this conjecture using a method based on simultaneous coding of the pair of the fixed points of the morphisms.