Yet another characterization of the Pisot substitution conjecture

TL;DR

This paper establishes geometric conditions for subshifts to be isomorphic to domain exchanges and torus translations, introduces a new topology for Pisot substitutions, and provides algorithms and examples confirming the Pisot substitution conjecture.

Contribution

It offers a new geometric criterion and an automata-based algorithm for pure discrete spectrum in Pisot substitutions, advancing understanding of the conjecture.

Findings

Families of substitutions satisfying the Pisot conjecture are identified.

A new topology on the discrete line characterizes pure discrete spectrum.

An example of an S-adic system with pure discrete spectrum everywhere is provided.

Abstract

We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and we give a simple necessary and sufficient condition for the symbolic system to have pure discrete spectrum. This condition gives rise to an algorithm based on computation of automata. To see the power of this criterion, we provide families of substitutions that satisfies the Pisot substitution conjecture: 1) $a \mapsto a^kbc$, $b \mapsto c$, $c \mapsto a$, for $k \in \mathbb{N}$ and 2) $a \mapsto a^lba^{k-l}$, $b \mapsto c$, $c \mapsto a$, for $k \in \mathbb{N}_{\geq 1}$, for $0 \leq l \leq k$ using different methods. And we also provide an example of $\mathcal{S}$-adic system with pure discrete spectrum everywhere.