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Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 10 / October 2024
- Published online by Cambridge University Press:
- 15 February 2024, pp. 2859-2900
- Print publication:
- October 2024
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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, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and 
$\Sigma $ is topologically mixing, we show that 
$\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.
