The $S$-adic Pisot conjecture on two letters

TL;DR

This paper extends the Pisot substitution conjecture to the $S$-adic setting on two letters, utilizing Rauzy fractals and strong coincidences to establish the result.

Contribution

It provides the first proof of the $S$-adic Pisot conjecture for two-letter systems, expanding the understanding of symbolic dynamics and fractal structures.

Findings

Proves the $S$-adic Pisot conjecture for two-letter systems

Demonstrates the role of Rauzy fractals in the proof

Shows that strong coincidences hold in this framework

Abstract

We prove an extension of the well-known Pisot substitution conjecture to the $S$-adic symbolic setting on two letters. The proof relies on the use of Rauzy fractals and on the fact that strong coincidences hold in this framework.