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

Abstract

Every non-erasing monoid morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ 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 $w \in \mathcal{B}^*$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $\sigma: \mathcal{A}^* \to \mathcal{B}^*$ 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 $\sigma_X^{\mathcal{M}}$.