Noncommutative point spaces of symbolic dynamical systems

Abstract

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 $P^1$’—is isomorphic to a union of a classical projective line with a Cantor set. While there is a continuum of monomial $P^1$'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 parametrizing spaces of truncated point modules of finitely presented monomial algebras are rational functions, and classify isomorphisms and automorphisms of projectively simple monomial algebras.