## Abstract

We show that in the category of effective ℤ-dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for d ≥ 3 there exist d-dimensional shifts of finite type which are universal for 1-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective Zd-system for d ≥ 2, and in particular SFTs cannot be universal for subactions of rank ≥ 2. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs. We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.

Original language | American English |
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Pages (from-to) | 301-314 |

Number of pages | 14 |

Journal | Discrete and Continuous Dynamical Systems - Series S |

Volume | 2 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2009 |

Externally published | Yes |

## Keywords

- Cellular automaton
- Decidability
- Medvedev degree
- Shift of finite type
- Topological dynamics