The measure transfer for subshifts induced by a morphism of free monoids

Abstract

Every non-erasing monoid morphism $\sigma :{\mathcal{A}}^{\ast }\to {\mathcal{B}}^{\ast }$ induces a {\em measure transfer map} ${\sigma }_X^{\mathcal{M}}:\mathcal{M}(X)\to \mathcal{M}(\sigma (X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(\sigma (X))$, associated to any subshift $X\subset {\mathcal{A}}^{\mathbb{Z}}$ and its image subshift $\sigma (X)\subset {\mathcal{B}}^{\mathbb{Z}}$ 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 \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure ${\sigma }_X^{\mathcal{M}(\mu )}$ on any cylinder $[w]$ (for $w\in {\mathcal{B}}^{\ast }$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $\sigma :{\mathcal{A}}^{\ast }\to {\mathcal{B}}^{\ast }$ is injective on the shift-orbits of some subshift $X\subset {\mathcal{A}}^{\mathbb{Z}}$, then ${\sigma }^{{\mathcal{M}}_X}$ is injective. \smallskip The assumption on $\sigma$ 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 ${\sigma }_X^{\mathcal{M}}$.

References in corpus (1)

• A Pansiot-type subword complexity theorem for automorphisms of free groups