The measure transfer for subshifts induced by a morphism of free monoids
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| Format: | Preprint |
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2022
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| _version_ | 1866913684004012032 |
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| author | Bédaride, Nicolas Hilion, Arnaud Lustig, Martin |
| author_facet | Bédaride, Nicolas Hilion, Arnaud Lustig, Martin |
| contents | Every non-erasing monoid morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $σ_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(σ(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(σ(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $σ(X) \subset \mathcal{B}^{\mathbb{Z}}$ 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 $σ_X^{\mathcal{M}(μ)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented.
\smallskip \noindent {\bf Theorem:} If a non-erasing morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $σ^{\mathcal{M}_X}$ is injective.
\smallskip
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 $σ_X^{\mathcal{M}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_11234 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The measure transfer for subshifts induced by a morphism of free monoids Bédaride, Nicolas Hilion, Arnaud Lustig, Martin Dynamical Systems 37B10 (Primary) 37A25, 37E25 (Secondary) Every non-erasing monoid morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $σ_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(σ(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(σ(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $σ(X) \subset \mathcal{B}^{\mathbb{Z}}$ 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 $σ_X^{\mathcal{M}(μ)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $σ^{\mathcal{M}_X}$ is injective. \smallskip 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 $σ_X^{\mathcal{M}}$. |
| title | The measure transfer for subshifts induced by a morphism of free monoids |
| topic | Dynamical Systems 37B10 (Primary) 37A25, 37E25 (Secondary) |
| url | https://arxiv.org/abs/2211.11234 |