On dual surjunctivity and applications

0565043 - MÚ 2023 RIV CH eng J - Journal Article
Doucha, Michal - Gismatullin, J.
On dual surjunctivity and applications.
Groups Geometry and Dynamics. Roč. 16, č. 3 (2022), s. 943-961. ISSN 1661-7207. E-ISSN 1661-7215
R&D Projects: GA ČR(CZ) GJ19-05271Y
Institutional support: RVO:67985840
Keywords : (dual) surjunctive groups * cellular automata * expansive algebraic actions * Gottschalk’s conjecture * Kaplansky’s direct finiteness * sofic groups
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Open access
https://doi.org/10.4171/ggd/681

We explore the dual version of Gottschalk’s conjecture recently introduced by Capobianco, Kari, and Taati, and the notion of dual surjunctivity in general. We show that dual surjunctive groups satisfy Kaplansky’s direct finiteness conjecture for all fields of positive characteristic. By quantifying the notions of injectivity and post-surjectivity for cellular automata, we show that the image of the full topological shift under an injective cellular automaton is a subshift of finite type in a quantitative way. Moreover, we show that dual surjunctive groups are closed under ultraproducts, under elementary equivalence, and under certain semidirect products (using the ideas of Arzhantseva and Gal for the latter), they form a closed subset in the space of marked groups, fully residually dual surjunctive groups are dual surjunctive, etc. We also consider dual surjunctive systems for more general dynamical systems, namely for certain expansive algebraic actions, employing results of Chung and Li.
Permanent Link: https://hdl.handle.net/11104/0336598