  We study point modules of monomial algebras associated with symbolic
dynamical systems, parametrized by proalgebraic varieties which 'linearize' the
underlying dynamical systems. Faithful point modules correspond to transitive
sub-systems, equivalently, to monomial algebras associated with infinite words.
In particular, we prove that the space of point modules of every prime monomial
algebra with Hilbert series $1/(1-t)^2$ -- which is thus thought of as a
'monomial $\mathbb{P}^1$' -- is isomorphic to a union of a classical projective
line with a Cantor set. While there is a continuum of monomial $\mathbb{P}^1$'s
with non-equivalent graded module categories, they all share isomorphic
parametrizing spaces of point modules. In contrast, free algebras are
geometrically rigid, and are characterized up to isomorphism from their spaces
of point modules. Furthermore, we derive enumerative and ring-theoretic
consequences from our analysis. In particular, we show that the formal power
series counting the irreducible components of the moduli schemes of truncated
point modules of finitely presented monomial algebras are rational functions,
and classify isomorphisms and automorphisms of projectively simple monomial
algebras.
