### Abstract

Let Tr be the r-dimensional torus, and let f: Tr → Tr be a map. If Per(f) denotes the set of periods of f, the minimal set of periods of f, denoted by MPer(f), is defined as ∩g≃f Per(g) where g Tr ← Tr is homotopic to f. First, we characterize the set MPer(f) in terms of the Nielsen numbers of the iterates of f. Second, we distinguish three types of the set MPer(f) and show that for each type and any given dimension r, the variation of MPer(f) is uniformly bounded in a suitable sense. Finally, we classify all the sets MPer(f) for self-maps of the 3-dimensional torus.

Original language | English |
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Pages (from-to) | 301-320 |

Journal | Discrete and Continuous Dynamical Systems |

Volume | 4 |

Issue number | 2 |

Publication status | Published - 1 Dec 1998 |

### Keywords

- Minimum set of periods
- Nielsen fixed point theory
- Periodic points
- Three dimensional torus
- Torus homeomorphisms
- Torus maps

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## Cite this

Jiang, B., & Llibre, J. (1998). Minimal sets of periods for torus maps.

*Discrete and Continuous Dynamical Systems*,*4*(2), 301-320.