# Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

@article{CeccheriniSilberstein2020InvariantSA, title={Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts}, author={Tullio Ceccherini-Silberstein and Michel Coornaert and Xuan Kien Phung}, journal={arXiv: Dynamical Systems}, year={2020} }

Let $G$ be a group and let $V$ be an algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $V$. We introduce algebraic sofic subshifts $\Sigma \subset A^G$ and study endomorphisms $\tau \colon \Sigma \to \Sigma$. We generalize several results for dynamical invariant sets and nilpotency of $\tau$ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau$ is nilpotent if and only if its limit set, i.e., the…

## 10 Citations

### On linear shifts of finite type and their endomorphisms

- MathematicsJournal of Pure and Applied Algebra
- 2021

### On Dynamical Finiteness Properties of Algebraic Group Shifts

- MathematicsIsrael Journal of Mathematics
- 2022

Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $\Sigma \subset V^G$ which generalize both the class of algebraic sofic…

### On symbolic group varieties and dual surjunctivity

- Mathematics
- 2021

Let G be a group. Let X be an algebraic group over an algebraically closed field K. Denote by A = X(K) the set of rational points of X. We study algebraic group cellular automata τ : A → A whose…

### ON SYMBOLIC ALGEBRAIC GROUP VARIETIES AND DUAL SURJUNCTIVITY

- Mathematics
- 2021

Let G be a group. Let X be an algebraic group over an algebraically closed field K. Denote by A = X(K) the set of rational points of X. We study algebraic group cellular automata τ : A → A whose…

### LEF-groups and endomorphisms of symbolic varieties

- Mathematics
- 2021

Let G be a group and let X be an algebraic variety over an algebraically closed field k of characteristic zero. Denote A = X(k) the set of rational points of X. We investigate invertible algebraic…

### On images of subshifts under injective morphisms of symbolic varieties

- Mathematics
- 2021

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, resp. a…

### Stable finiteness of twisted group rings and noisy linear cellular automata

- MathematicsArXiv
- 2022

For linear non-uniform 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…

### A geometric generalization of Kaplansky's direct finiteness conjecture

- MathematicsArXiv
- 2021

A geometric direct finiteness theorem is established for endomorphisms of symbolic algebraic varieties Whenever G is a sofic group or more generally a surjunctive group, this result implies a generalization of Kaplansky’s direct Finiteness conjecture for the near ring R(k,G).

### Shadowing for families of endomorphisms of generalized group shifts

- MathematicsDiscrete & Continuous Dynamical Systems
- 2021

It is shown that the valuation action of $\Gamma$ on $\Sigma$ satisfies a natural intrinsic shadowing property and generalizations are also established for families of endomorphisms of admissible group subshifts.

### Weakly surjunctive groups and symbolic group varieties

- Mathematics
- 2021

In this paper, we introduce the classes of weakly surjunctive and linearly surjunctive groups which include all sofic groups and more generally all surjunctive groups. We investigate various…

## References

SHOWING 1-10 OF 57 REFERENCES

### On linear shifts of finite type and their endomorphisms

- MathematicsJournal of Pure and Applied Algebra
- 2021

### On sofic groups, Kaplansky's conjectures, and endomorphisms of pro-algebraic groups

- MathematicsJournal of Algebra
- 2020

### On the Garden of Eden theorem for endomorphisms of symbolic algebraic varieties

- MathematicsPacific Journal of Mathematics
- 2020

Let $G$ be an amenable group and let $X$ be an irreducible complete algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $X$ and let $\tau \colon A^G \to…

### On Dynamical Finiteness Properties of Algebraic Group Shifts

- MathematicsIsrael Journal of Mathematics
- 2022

Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $\Sigma \subset V^G$ which generalize both the class of algebraic sofic…

### On symbolic group varieties and dual surjunctivity

- Mathematics
- 2021

Let G be a group. Let X be an algebraic group over an algebraically closed field K. Denote by A = X(K) the set of rational points of X. We study algebraic group cellular automata τ : A → A whose…

### Thermodynamic formalism for countable Markov shifts

- MathematicsErgodic Theory and Dynamical Systems
- 1999

We establish a generalized thermodynamic formalism for topological Markov shifts with a countable number of states. We offer a definition of topological pressure and show that it satisfies a…

### The Nilpotency Problem of One-Dimensional Cellular Automata

- MathematicsSIAM J. Comput.
- 1992

The present work proves that it is algorithmically undecidable whether a given one-dimensional cellular automaton is nilpotent, the basis of the proof of Rice's theorem for CA limit sets.

### A notion of effectiveness for subshifts on finitely generated groups

- MathematicsTheor. Comput. Sci.
- 2017

### On injective endomorphisms of symbolic schemes

- MathematicsCommunications in Algebra
- 2019

Abstract Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties, we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular…

### Simulations and the Lamplighter group

- MathematicsArXiv
- 2020

It is proved that the tiling problem for the simulating graph is at least as difficult as that for the simulated graph, and the undecidability criterion by simulation covers cases not covered by Jeandel's criterion based on translation-like action of a product of finitely generated infinite groups.