Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle

Sophie Grivaux; Maria Roginskaya

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 3, page 603-627
  • ISSN: 0011-4642

Abstract

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We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle 𝕋 . A set of integers is called r -Bohr if it is recurrent for all products of r rotations on 𝕋 , and Bohr if it is recurrent for all products of rotations on 𝕋 . It is a result due to Katznelson that for each r ≥ 1 there exist sets of integers which are r -Bohr but not ( r + 1 ) -Bohr. We present new examples of r -Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on ℤ , and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.

How to cite

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Grivaux, Sophie, and Roginskaya, Maria. "Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle." Czechoslovak Mathematical Journal 63.3 (2013): 603-627. <http://eudml.org/doc/260645>.

@article{Grivaux2013,
abstract = {We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb \{T\}$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb \{T\}$, and Bohr if it is recurrent for all products of rotations on $\mathbb \{T\}$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on $\mathbb \{Z\}$, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.},
author = {Grivaux, Sophie, Roginskaya, Maria},
journal = {Czechoslovak Mathematical Journal},
keywords = {recurrence for dynamical systems; non-recurrence for dynamical systems; rotations of the unit circle; syndetic set; Bohr topology on $\mathbb \{Z\}$; Bohr set; $r$-Bohr set; recurrence for dynamical systems; non-recurrence for dynamical systems; rotations of the unit circle; syndetic set; Bohr topology on ; Bohr set; -Bohr set},
language = {eng},
number = {3},
pages = {603-627},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle},
url = {http://eudml.org/doc/260645},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Grivaux, Sophie
AU - Roginskaya, Maria
TI - Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 3
SP - 603
EP - 627
AB - We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb {T}$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb {T}$, and Bohr if it is recurrent for all products of rotations on $\mathbb {T}$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on $\mathbb {Z}$, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.
LA - eng
KW - recurrence for dynamical systems; non-recurrence for dynamical systems; rotations of the unit circle; syndetic set; Bohr topology on $\mathbb {Z}$; Bohr set; $r$-Bohr set; recurrence for dynamical systems; non-recurrence for dynamical systems; rotations of the unit circle; syndetic set; Bohr topology on ; Bohr set; -Bohr set
UR - http://eudml.org/doc/260645
ER -

References

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