群上元胞自动机的后满射性与平衡性
Post-surjectivity and balancedness of cellular automata over groups
AI总结：
本文研究有限生成群上元胞自动机的后满射性，证明预单射与后满射蕴含可逆性，sofic群上后满射性单独蕴含可逆性，且可逆元胞自动机保持均匀测度，即平衡性。
摘要：
我们讨论任意有限生成群上的元胞自动机。若对于任意一对渐近配置，一个配置的每个原像都与另一个配置的某个原像渐近，则称该元胞自动机为后满射的。众所周知的对偶概念是预单射性：若不同的渐近配置具有不同的像，则元胞自动机是预单射的。我们证明，预单射且后满射的元胞自动机是可逆的。此外，在sofic群上，仅后满射性即可蕴含可逆性。我们还证明，任意群上的可逆元胞自动机是平衡的，即它们保持配置空间上的均匀测度。
英文摘要：
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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