### Abstract

Let Y⊆{-1,1} ^{Z∞×n} be the mosaic solution space of an n-layer cellular neural network. We decouple Y into n subspaces, say Y ^{(1)}, Y ^{(2)},..., Y ^{(n)}, and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, Y ^{(i)} is a sofic shift for 1≤i≤n. This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether Y ^{(i)} and Y ^{(j)} are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a multi-layer cellular neural network, each layer's structure. As an extension, we can decouple Y into arbitrary k-subspaces, where 2≤k≤n, and demonstrates each subspace's structure.

Original language | English |
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Pages (from-to) | 4563-4597 |

Number of pages | 35 |

Journal | Journal of Differential Equations |

Volume | 252 |

Issue number | 8 |

DOIs | |

State | Published - 15 Apr 2012 |

### Keywords

- Dimension group
- Finite equivalence
- Shift equivalence
- Sofic shift
- Strong shift equivalence

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## Cite this

*Journal of Differential Equations*,

*252*(8), 4563-4597. https://doi.org/10.1016/j.jde.2012.01.006