| Title:
|
Some new examples of recurrence and non-recurrence sets for products of rotations on the unit circle (English) |
| Author:
|
Grivaux, Sophie |
| Author:
|
Roginskaya, Maria |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
3 |
| Year:
|
2013 |
| Pages:
|
603-627 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
recurrence for dynamical systems |
| Keyword:
|
non-recurrence for dynamical systems |
| Keyword:
|
rotations of the unit circle |
| Keyword:
|
syndetic set |
| Keyword:
|
Bohr topology on $\mathbb {Z}$ |
| Keyword:
|
Bohr set |
| Keyword:
|
$r$-Bohr set |
| MSC:
|
37A45 |
| MSC:
|
37B05 |
| MSC:
|
37B20 |
| idZBL:
|
Zbl 06282101 |
| idMR:
|
MR3125645 |
| DOI:
|
10.1007/s10587-013-0043-z |
| . |
| Date available:
|
2013-10-07T11:59:18Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143480 |
| . |
| Reference:
|
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