### Resumen

In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincarés theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

Idioma original | English |
---|---|

Páginas (desde-hasta) | 565-594 |

Número de páginas | 30 |

Publicación | Ergodic Theory and Dynamical Systems |

Volumen | 30 |

N.º | 2 |

DOI | |

Estado | Published - abr 2010 |

### Huella dactilar

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Citar esto

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**Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line.** / Aliste-Prieto, José.

Resultado de la investigación: Article

TY - JOUR

T1 - Translation numbers for a class of maps on the dynamical systems arising from quasicrystals in the real line

AU - Aliste-Prieto, José

PY - 2010/4

Y1 - 2010/4

N2 - In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincarés theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

AB - In this paper, we study translation sets for non-decreasing maps of the real line with a pattern-equivariant displacement with respect to a quasicrystal. First, we establish a correspondence between these maps and self maps of the continuous hull associated with the quasicrystal that are homotopic to the identity and preserve orientation. Then, by using first-return times and induced maps, we provide a partial description for the translation set of the latter maps in the case where they have fixed points and obtain the existence of a unique translation number in the case where they do not have fixed points. Finally, we investigate the existence of a semiconjugacy from a fixed-point-free map homotopic to the identity on the hull to the translation given by its translation number. We concentrate on semiconjugacies that are also homotopic to the identity and, under a boundedness condition, we prove a generalization of Poincarés theorem, finding a sufficient condition for such a semiconjugacy to exist depending on the translation number of the given map.

UR - http://www.scopus.com/inward/record.url?scp=77951219226&partnerID=8YFLogxK

U2 - 10.1017/S0143385709000145

DO - 10.1017/S0143385709000145

M3 - Article

AN - SCOPUS:77951219226

VL - 30

SP - 565

EP - 594

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 2

ER -