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On images of subshifts under embeddings of symbolic varieties
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 43 / Issue 9 / September 2023
- Published online by Cambridge University Press:
- 01 August 2022, pp. 3131-3149
- Print publication:
- September 2023
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- Article
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We show that the image of a subshift X under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, respectively a sofic subshift, if and only if so is X. Similarly, let G be a countable monoid and let A, B be Artinian modules over a ring. We prove that for every closed subshift submodule
$\Sigma \subset A^G$ and every injective G-equivariant uniformly continuous module homomorphism 
$\tau \colon \! \Sigma \to B^G$, a subshift 
$\Delta \subset \Sigma $ is of finite type, respectively sofic, if and only if so is the image 
$\tau (\Delta )$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.
