  We show how geometric methods from the general theory of fractal dimensions
and iterated function systems can be deployed to study symbolic dynamics in the
zero entropy regime. More precisely, we establish a dimensional
characterization of the topological notion of amorphic complexity. For
subshifts with discrete spectrum associated to constant length substitutions,
this characterization allows us to derive bounds for the amorphic complexity by
interpreting the subshift as the attractor of an iterated function system in a
suitable quotient space. As a result, we obtain the general finiteness and
positivity of amorphic complexity in this setting and provide a closed formula
in case of a binary alphabet.
