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On dynamical finiteness properties of algebraic group shifts

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Abstract

Let G be a group and let V be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts Σ ⊂ VG which generalize both the class of algebraic sofic subshifts of VG and the class of closed group subshifts over finite group alphabets. When G is a polycyclic-by-finite group, we show that VG satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known results of Kitchens and Schmidt to cover the case of many non-compact group alphabets.

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Acknowledgments

We would like to express our deep gratitude to the anonymous reviewer for the careful reading of our manuscript and for numerous insightful comments and suggestions.

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  1. Département d’informatique et de recherche opérationnelle, Université de Montréal, Montréal, Québec, H3T 1J4, Canada

    Xuan Kien Phung

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  1. Xuan Kien Phung
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Correspondence to Xuan Kien Phung.

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Phung, X.K. On dynamical finiteness properties of algebraic group shifts. Isr. J. Math. 252, 355–398 (2022). https://doi.org/10.1007/s11856-022-2351-1

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  • DOI: https://doi.org/10.1007/s11856-022-2351-1

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