Geodesic
Post-surjectivity and balancedness of cellular automata over groups
Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 3.

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We discuss cellular automata over arbitrary finitely generated groups. We call a cellular automaton post-surjective if for any pair of asymptotic configurations, every pre-image of one is asymptotic to a pre-image of the other. The well known dual concept is pre-injectivity: a cellular automaton is pre-injective if distinct asymptotic configurations have distinct images. We prove that pre-injective, post-surjective cellular automata are reversible. Moreover, on sofic groups, post-surjectivity alone implies reversibility. We also prove that reversible cellular automata over arbitrary groups are balanced, that is, they preserve the uniform measure on the configuration space. 
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     title = {Post-surjectivity and balancedness of cellular automata over groups},
     journal = {Discrete mathematics & theoretical computer science},
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Capobianco, Silvio; Kari, Jarkko; Taati, Siamak. Post-surjectivity and balancedness of cellular automata over groups. Discrete mathematics & theoretical computer science, Tome 19 (2017-2018) no. 3. doi : 10.23638/DMTCS-19-3-4. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-3-4/

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