  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.
