Jeroen S.W. Lamb:

On the canonical projection method for one-dimensional quasicrystals
and invertible substitution rules

abstract:

We show that if a non-periodic two-symbol sequence obtained by the
canonical projection method has an infinite number of predecessors
with respect to a substitution rule $\sigma$, then $\sigma$ is
an invertible substitution rule. Vice-versa, we show that every
non-periodic two-symbol sequence that has an infinite number of
predecessors with respect to a nontrivial invertible substitution rule, can 
be obtained by the canonical projection method.