On linear shifts of finite type and their endomorphisms

Ceccherini‐Silberstein, Tullio; Coornaert, Michel; Phung, Xuan Kien

doi:10.1016/j.jpaa.2021.106962

Cited by 8 publications

(3 citation statements)

References 17 publications

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“…The Garden of Eden theorem for linear CA over the full shifts was first proved in [3]. It was known to characterize amenable groups [1] and the limit set of a CA [9].…”

Section: Garden Of Eden Theorem For Linear Nucamentioning

confidence: 99%

On the Garden of Eden theorem for non-uniform cellular automata

Phung

2024

Nonlinearity

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We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata (CA) over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform CA. A strengthened surjunctivity result is also obtained for multi-dimensional CA over strongly irreducible subshifts of finite type.

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“…The Garden of Eden theorem for linear CA over the full shifts was first proved in [3]. It was known to characterize amenable groups [1] and the limit set of a CA [9].…”

Section: Garden Of Eden Theorem For Linear Nucamentioning

confidence: 99%

On the Garden of Eden theorem for non-uniform cellular automata

Phung

2024

Nonlinearity

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“…In the line of some recent results which establish the multifold interaction between symbolic dynamics, group theory, and ring theory such as [2,10,31,35,36], etc. our main result is the following: Theorem B For every infinite group G, the following are equivalent:…”

Section: Definition 13mentioning

confidence: 99%

Stable finiteness of twisted group rings and noisy linear cellular automata

Phung

2023

Can. J. Math.-J. Can. Math.

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For linear nonuniform cellular automata (NUCA) which are local perturbations of linear CA over a group universe G and a finite-dimensional vector space alphabet V over an arbitrary field k, we investigate their Dedekind finiteness property, also known as the direct finiteness property, i.e., left or right invertibility implies invertibility. We say that the group G is $L^1$ -surjunctive, resp. finitely $L^1$ -surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in addition k is finite. In parallel, we introduce the ring $D^1(k[G])$ which is the Cartesian product $k[G] \times (k[G])[G]$ as an additive group but the multiplication is twisted in the second component. The ring $D^1(k[G])$ contains naturally the group ring $k[G]$ and we obtain a dynamical characterization of its stable finiteness for every field k in terms of the finite $L^1$ -surjunctivity of the group G, which holds, for example, when G is residually finite or initially subamenable. Our results extend known results in the case of CA.

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“…Over group universes, various injective endomorphisms of symbolic varieties are surjective and thus bijective as motivated by Gottschalk's surjunctivity conjecture [14] and the seminal paper of Gromov [15] (see also [3,7,8,22,24,27]). Notable applications of the surjunctivity property, that is, injectivity ⇒ surjectivity, include the well-known Kaplansky's stable finiteness conjecture [17] on group rings (see [1,3,10,13,22,25,26]).…”

Section: Introductionmentioning

confidence: 99%

On images of subshifts under embeddings of symbolic varieties

Phung

2022

Ergod. Th. Dynam. Sys.

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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.

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On linear shifts of finite type and their endomorphisms

Cited by 8 publications

References 17 publications

On the Garden of Eden theorem for non-uniform cellular automata

On the Garden of Eden theorem for non-uniform cellular automata

Stable finiteness of twisted group rings and noisy linear cellular automata

On images of subshifts under embeddings of symbolic varieties

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