On Dual surjunctivity and applications

TL;DR

This paper investigates dual surjunctivity in groups, demonstrating its implications for Kaplansky's conjecture, cellular automata, and stability properties, thus advancing understanding of algebraic and dynamical systems.

Contribution

It introduces the notion of dual surjunctivity, explores its properties, and establishes its stability under various algebraic constructions and dynamical systems.

Findings

Dual surjunctive groups satisfy Kaplansky's conjecture in positive characteristic.

Injective cellular automata images are subshifts of finite type.

Dual surjunctivity is preserved under ultraproducts, elementary equivalence, and certain semidirect products.

Abstract

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.