Martin Lustig’s research while affiliated with Aix-Marseille University and other places

Every non-erasing monoid morphism σ:A∗→B∗ induces a measure transfer map σXM:M(X)→M(σ(X)) between the measure cones M(X) and M(σ(X)), associated to any subshift X⊆AZ and its image subshift σ(X)⊆BZ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective (Bédaride et al 2024 Ergod. Theor. Dynam. Syst. 44 3120–54). Furthermore, an efficient technique to compute the value of the transferred measure σXM(μ) on any cylinder [w] (for w∈B∗) is presented. Theorem. If a non-erasing morphism σ:A∗→B∗ is injective on the shift-orbits of some subshift X⊆AZ, then σXM is injective. The assumption on σ that it is ‘injective on the shift-orbits of X’ is strictly weaker than ‘recognizable in X’, and strictly stronger than ‘recognizable for aperiodic points in X’. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map σXM.

Based on previous work of the authors, to any S -adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S -adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d \geq 2$ , an S -adic development of a minimal, aperiodic, uniquely ergodic subshift X , where all level alphabets $\mathcal A_n$ have cardinality d, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subseteq \mathcal A_n^{\mathbb {Z}}$ .

Every non-erasing monoid morphism $\sigma: {\cal A}^* \to {\cal B}^*$ induces a measure transfer map $\sigma_X^{\cal M}: {\cal M}(X) \to {\cal M}(\sigma(X))$ between the measure cones ${\cal M}(X)$ and ${\cal M}(\sigma(X))$, associated to any subshift $X \subseteq {\cal A}^\mathbb Z$ and its image subshift $\sigma(X) \subseteq {\cal B}^\mathbb Z$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective. Furthermore, an efficient technique to compute the value of the transferred measure $\sigma_X^{\cal M}(\mu)$ on any cylinder [w] (for $w \in {\cal B}^\mathbb Z$) is presented. Theorem: If a non-erasing morphism $\sigma: {\cal A}^* \to {\cal B}^*$ is recognizable in some subshift $X \subseteq {\cal A}^\mathbb Z$, then $\sigma^{\cal M}_X$ is bijective. The notion of a "recognizable" subshift is classical in symbolic dynamics, and due to its long history and various transformations and sharpening over time, it plays a central role in the theory. In order to prove the above theorem we show here: Proposition: A non-erasing morphism $\sigma: {\cal A}^* \to {\cal B}^*$ is recognizable in some subshift $X \subseteq {\cal A}^\mathbb Z$ if and only if (1) the induced map from shift-orbits of X to shift-orbits of $\sigma(X)$ is injective, and (2) $\sigma$ preserves the shift-period of any periodic word in X.

Based on previous work of the authors, to any S-adic development of a subshift X a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$.

For any non-erasing free monoid morphism $\sigma: \cal A^* \to \cal B^*$, and for any subshift $X \subset \cal A^\Z$ and its image subshift $Y = \sigma(X) \subset \cal B^\Z$, the associated complexity functions $p_X$ and $p_Y$ are shown to satisfy: there exist constants $c, d, C > 0$ such that $c \cdot p_X(d \cdot n) \,\, \leq \,\, p_Y(n) \,\, \leq \,\, C \cdot p_X(n)$ holds for all sufficiently large integers $n \in \N$, provided that $\sigma$ is recognizable in X. If $\sigma$ is in addition letter-to-letter, then $p_Y$ belongs to $\Theta(p_X)$ (and conversely). Otherwise, however, there are examples where $p_X$ is not in $\cal O(p_Y)$. It follows that in general the value $h_X$ of the topological entropy of X is not preserved when applying a morphism $\sigma$ to X, even if $\sigma$ is recognizable in X. As a consequence, there is no meaningful way to define the topological entropy of a current on a free group $F_N$; only the distinction of currents $\mu$ with topological entropy h_{\tiny\supp(\mu)} = 0 and h_{\tiny\supp(\mu)} > 0 is well defined.

We explain and restate the results from our recent paper [2] in standard language for substitutions and S-adic systems in symbolic dynamics. We then produce as rather direct application an S-adic system (with finite set of substitutions S on d letters) that is minimal and has d distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure $\mu ([w])$, for any word $w \in \mathcal A^*$ and any invariant measure $\mu$ on the subshift $X_\sigma$ defined by any everywhere growing but not necessarily primitive or irreducible substitution $\sigma : \mathcal A^* \rightarrow \mathcal A^*$. Several examples are considered in detail, and model computations are presented.

In this note we show that for any subshift X of finite S-rank every invariant measure $\mu$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.

We define train track maps for graphs-of-groups and show that a fundamental finiteness property (which allows one to control the growth of illegal turns) for classical train track maps extends to this generalization.