Recognizability for sequences of morphisms

Abstract

We investigate different notions of recognizability for a free monoid morphism σ : A * → B * . Full recognizability occurs when each (aperiodic) point in B Z admits at most one tiling with words σ(a), a ∈ A. This is stronger than the classical notion of recognizability of a substitution σ : A * → A * , where the tiling must be compatible with the language of the substitution. We show that if |A| = 2, or if σ's incidence matrix has rank |A|, or if σ is permutative, then σ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé (1992) and Bezuglyi, Kwiatkowski and Medynets (2009), by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an S-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the S-adic shift it generates, and the measurable Bratteli-Vershik dynamical system that it defines.

On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

“…The next result shows that minimal dendric subshifts are primitive unimodular proper S-adic subshifts. Similar results are proved with the same method in [10,12,14] but not highlighting all the properties stated below, so we provide a proof for the sake of self-containedness. It relies on S-adic representations built from return words [27,28] together with the remarkable property of return words of dendric subshifts stated in Theorem 3.7.…”

“…Observe that in the previous result, we can relax the assumption of minimality. Indeed, one checks that the same proof works if we assume that (X, S) is aperiodic (recognizability then holds by [14]) and that min a∈A |τ [1,n) (a)| goes to infinity.…”

“…We recall the definition of an S-adic subshift as stated in [14], see also [11] for more on S-adic subshifts. Let τ = (τ n :…”

“…One of our main results states that any continuous integer-valued function defined on X is cohomologous to some integer linear combination of characteristic functions of letter cylinders (Theorem 4.1). This relies on the fact that such subshifts being aperiodic (see Proposition 3.5) and recognizable by [14], they have a sequence of Kakutani-Rohlin tower partitions with suitable topological properties. We then deduce an explicit computation of their dimension group (Theorem 4.5).…”

The Hölder property for the spectrum of translation flows in genus two

“…) t s. This definition depends on the notion of recognizability for the sequence of substitutions, see [4], which generalizes bilateral recognizability of B. Mossé [21] for a single substitution, see also Sections 5.5 and 5.6 in [24]. By definition of the space X a + , for every n ≥ 1, every x ∈ X a + has a representation of the form…”