Decidability and universality of quasiminimal subshifts

Abstract

We introduce the quasiminimal subshifts, subshifts having only finitely many subsystems. With N-actions, their theory essentially reduces to the theory of minimal systems, but with Z-actions, the class is much larger. We show many examples of such subshifts, and in particular construct a universal system with only a single proper subsystem, refuting a conjecture of [Delvenne, Kůrka, Blondel, '05].5 The one-point subshift is the simplest minimal subshift, but any minimal subshift can be used here, though naturally at the expense of countability.6 Here, we use the term extension in the sense of containment (monomorphisms), and not in the sense of factoring (epimorphisms). This usage is somewhat nonstandard in the theory of dynamical systems, but it is fitting for quasiminimal systems, since they are inductively built, in finitely many steps, from smaller quasiminimal systems by adding new points.7 Strictly speaking, we could also consider S Z with an N-action (obtaining a rather unnatural definition of a subshift), but the convention is that S M uses the natural shift action of M .

Sturmian and infinitely desubstitutable words accepted by an ω-automaton

“…The following result already appeared in [15], but an erratum clarified that some cases were not covered [16]. It is a parallel to a result in [5].…”

“…Carton and Thomas provided a method to decide this question in the case of substitutive or morphic words on Büchi ω-automata, using verification theory and semigroups of congruence [5]. This result was partially reproved by Salo [15], using a more combinatorial point of view. For the last 20 years, the substitutive approach (iterating a single homomorphism) has been generalized to the S-adic approach [6] that lets one alternate betweeen multiple substitutions.…”

Decidable problems in substitution shifts

“…Quasiminimal shifts We now investigate the question of whether a substitution shift has a finite number of subshifts. Following the terminology introduced in [26] these shifts are called quasiminimal .…”

Entropy pair realization

“…We assume familiarity with the process of constructing all words in the orbit closure of the ruler sequence, and how it manifests for subshifts based on the sequence, see e.g. [12].…”