Conjugacy Invariants of Subshifts: an Approach from Profinite Semigroup Theory

A new algebraic invariant for weak equivalence of sofic subshifts

It is studied how taking the inverse image by a sliding block code affects the syntactic semigroup of a sofic subshift. Two independent approaches are used: ζ-semigroups as recognition structures for sofic subshifts, and relatively free profinite semigroups. A new algebraic invariant is obtained for weak equivalence of sofic subshifts, by determining which classes of sofic subshifts naturally d...

The Complexity of the Topological Conjugacy Problem for Toeplitz Subshifts

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This resul...

A Matrix Formalism for Conjugacies of Higher-dimensional Shifts of Finite Type

We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of Z+-matrices. Using the decomposition theorem every topological conjugacy between two Zd-shifts of finite type can thus be factorized into a finite chain of matrix transformations acting on the transition matric...

Decidability and Tameness in the Theory of Finite Semigroups

Presentations of Schützenberger Groups of Minimal Subshifts

In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J -classes of free profinite semigroups. In this paper, the Schützenberger groups of such J -classes are investigated, in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for all non-periodic min...