#### DMCA

## Constant-length substitutions and countable scrambled sets

Venue: | Nonlinearity |

Citations: | 5 - 1 self |

### Citations

319 | Period three implies chaos,”
- Li, Yorke
- 1975
(Show Context)
Citation Context ...systems. Starting from a circle rotation we also construct a dynamical system having Li–Yorke pairs, none of which is recurrent. 1. Introduction One definition of topological chaos, based on ideas in =-=[LY]-=-, emerged twenty–five years ago. It is not the only definition of chaos by far. It relies on the existence of uncountable ‘scrambled’ subsets. Here we investigate what might be called ‘the edge of Li–... |

115 |
Substitution Dynamical Systems-Spectral Analysis,
- Queffelec
- 1987
(Show Context)
Citation Context ...t subshift of A Z admitting all words {τ n (a) : n ∈ N, a ∈ A}; when τ is primitive the subshift it generates is minimal. For more details and complements about this subsection we refer the reader to =-=[Q]-=-. To end this subsection we recall the following result due to B. Mossé [Mo] (see also [MS]). Theorem 2.1. Let τ be a primitive substitution. Suppose Xτ is infinite. Then, τ : Xτ → τ(Xτ) is a one-to-o... |

103 |
Minimal flows and their extensions,
- Auslander
- 1988
(Show Context)
Citation Context ...urrent under T × T. They call semi–distal a system without strong Li–Yorke pairs. Distal, almost distal and semi–distal systems are minimal when transitive. A system described some time ago by Floyd (=-=[Au]-=-, p. 26) was recently remarked to be semi–distal but not almost distal ([Y], [AA]); it is an extension of an adding machine in which fibers are intervals or singletons. Here we give several examples w... |

48 | On Li-Yorke pairs,
- Blanchard, Glasner, et al.
- 2002
(Show Context)
Citation Context ...s an uncountable scrambled set. Li–Yorke chaos has been recently proved to result from various dynamical properties: positive entropy; 2–scattering; transitivity together with one periodic orbit (see =-=[BGKM]-=- and [HY]). The opposite situation exists too: equicontinuous and distal systems have no Li–Yorke pairs. The aim of this article is to describe various systems that are not Li–Yorke chaotic while havi... |

45 |
The spectrum of dynamical systems arising from substitutions of constant length,
- DEKKING
- 1978
(Show Context)
Citation Context ...nstant length p such that Xτ is not finite. The p–odometer is a factor of the subshift Xτ. The proof of this fact is based on the following central result of the theory of substitutions. Theorem 2.4 (=-=[De]-=-, [Mo]). Let τ be a primitive substitution such that Xτ is not finite. Then for any x ∈ Xτ there are a unique sequence of points (x (i) )i∈N ⊆ Xτ and a unique sequence of positive integers (δi)i∈N ⊆ [... |

28 |
Devaney’s chaos or 2-scattering implies Li-Yorke’s chaos,
- Huang, Ye
- 2002
(Show Context)
Citation Context ...table scrambled set. Li–Yorke chaos has been recently proved to result from various dynamical properties: positive entropy; 2–scattering; transitivity together with one periodic orbit (see [BGKM] and =-=[HY]-=-). The opposite situation exists too: equicontinuous and distal systems have no Li–Yorke pairs. The aim of this article is to describe various systems that are not Li–Yorke chaotic while having Li–Yor... |

18 |
Reconnaissabilite des substitutions et complexite des suites automatiques,
- Mosse
- 1996
(Show Context)
Citation Context ...rimitive the subshift it generates is minimal. For more details and complements about this subsection we refer the reader to [Q]. To end this subsection we recall the following result due to B. Mossé =-=[Mo]-=- (see also [MS]). Theorem 2.1. Let τ be a primitive substitution. Suppose Xτ is infinite. Then, τ : Xτ → τ(Xτ) is a one-to-one continuous map. If τ is of constant length p, then T p ◦ τ = τ ◦ T and τ(... |

16 |
If a D0L language is k-power free then it is circular
- Mignosi, Séébold
- 1993
(Show Context)
Citation Context ...bshift it generates is minimal. For more details and complements about this subsection we refer the reader to [Q]. To end this subsection we recall the following result due to B. Mossé [Mo] (see also =-=[MS]-=-). Theorem 2.1. Let τ be a primitive substitution. Suppose Xτ is infinite. Then, τ : Xτ → τ(Xτ) is a one-to-one continuous map. If τ is of constant length p, then T p ◦ τ = τ ◦ T and τ(Xτ) is a proper... |

12 |
Decidability of periodicity for infinite words.
- Pansiot
- 1986
(Show Context)
Citation Context ...ition {T i (τ(Xτ)) : i ∈ {0, ..., p − 1}} will be called the fundamental partition of Xτ. Let τ be a primitive substitution. Let us now recall a way to prove that Xτ is infinite or not. J.-J. Pansiot =-=[Pa]-=-, and, T. Harju and M. Linna [HL] proved (in our settings) that it is decidable whether Xτ is infinite or not. We need some definitions. Let L(τ) be the set of finite words having an occurrence in som... |

11 |
On the periodicity of morphisms on free monoids.
- Harju, Linna
- 1986
(Show Context)
Citation Context ... p − 1}} will be called the fundamental partition of Xτ. Let τ be a primitive substitution. Let us now recall a way to prove that Xτ is infinite or not. J.-J. Pansiot [Pa], and, T. Harju and M. Linna =-=[HL]-=- proved (in our settings) that it is decidable whether Xτ is infinite or not. We need some definitions. Let L(τ) be the set of finite words having an occurrence in some x ∈ Xτ. We say a word u ∈ L(Xτ)... |

6 |
Directed graphs and substitutions. Theory Comput
- Holton, Zamboni
(Show Context)
Citation Context ...n that any infinite subshift has non–trivial asymptotic pairs. Moreover, it has been proved in [Q] that the number of different orbits of asymptotic pairs in the Cartesian product is finite (see also =-=[HZ]-=-). This property is also true for any infinite subshift with sub–affine symbolic complexity. In [BDH] the authors give an upper bound for the number of different orbits of asymptotic pairs for substit... |

4 |
Asymptotic orbits for primitive substitutions,” Theoret
- Barge, Diamond, et al.
(Show Context)
Citation Context ...hat the number of different orbits of asymptotic pairs in the Cartesian product is finite (see also [HZ]). This property is also true for any infinite subshift with sub–affine symbolic complexity. In =-=[BDH]-=- the authors give an upper bound for the number of different orbits of asymptotic pairs for substitution subshifts. In the next propositions we show how asymptotic and Li–Yorke pairs arise in our subs... |

2 |
Distality concepts for Ellis actions
- Akin, Auslander
(Show Context)
Citation Context ...system is one in which every non–diagonal pair is distal. A dynamical system having no Li–Yorke pairs is called almost distal [BGKM]; Sturmian systems and the Morse system are elementary examples. In =-=[AA]-=- Akin and Auslander introduce strong Li–Yorke pairs, i.e., those Li–Yorke pairs that are recurrent under T × T. They call semi–distal a system without strong Li–Yorke pairs. Distal, almost distal and ... |

1 |
personal communication. Institut de Mathématiques de Luminy (UPR 9016 du CNRS, FRUMAM) ; case 907, 163 avenue de Luminy, 13288 Marseille Cedex 09, France E-mail address: blanchar@iml.univ-mrs.fr Laboratoire Amiénois de Mathématiques Fondamentales et Ap
- Ye
(Show Context)
Citation Context ...airs. Distal, almost distal and semi–distal systems are minimal when transitive. A system described some time ago by Floyd ([Au], p. 26) was recently remarked to be semi–distal but not almost distal (=-=[Y]-=-, [AA]); it is an extension of an adding machine in which fibers are intervals or singletons. Here we give several examples with the same property, one comes from a substitution of constant length and... |