  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.
