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A characterization of the reversibility of linear cellular automata

  • Alonso Castillo-Ramirez
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Abstract

Let G be a group, let 𝕂 be a field, and let V be a 𝕂-vector space. We prove that there exists a bijective linear cellular automaton VGβ†’ VG whose inverse is not a cellular automaton if and only if G is not locally finite and dim𝕂V β‰₯ |𝕂|β„΅0. In particular, |𝕂|β„΅0 is the exact dimension threshold for non-reversibility. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.

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Cite

BibTeXRIS
BibTeX
@misc{castilloramirez2026characterization,
  title = {{A characterization of the reversibility of linear cellular automata}},
  author = {Alonso Castillo-Ramirez},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.20600},
  eprint = {2608.20600},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.20600},
}
RIS
TY  - UNPB
TI  - A characterization of the reversibility of linear cellular automata
AU  - Alonso Castillo-Ramirez
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2608.20600
UR  - https://arxiv.org/abs/2608.20600
AB  - Let G be a group, let 𝕂 be a field, and let V be a 𝕂-vector space. We prove that there exists a bijective linear cellular automaton V^Gβ†’ V^G whose inverse is not a cellular automaton if and only if G is not locally finite and dim_𝕂V β‰₯ |𝕂|^β„΅_0. In particular, |𝕂|^β„΅_0 is the exact dimension threshold for non-reversibility. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.
ER  -