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Simultaneously non-dense orbits under different expanding maps

Färm, David LU (2010) In Dynamical Systems 25(4). p.531-545
Abstract
Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. It is well-known that in many cases such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, i.e. countable intersections of such sets also have full Hausdorff dimension. This result applies to a class of maps including multiplication by integers modulo 1 and x -> 1/x modulo 1. We prove that the same properties hold for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
non-integer expansions, numbers, badly approximable, Schmidt games, interval maps, Hausdorff dimension
in
Dynamical Systems
volume
25
issue
4
pages
531 - 545
publisher
Taylor & Francis
external identifiers
  • wos:000284411900005
  • scopus:78649496062
ISSN
1468-9367
DOI
10.1080/14689367.2010.482519
language
English
LU publication?
yes
id
a6a3c3b6-2eda-4791-b03f-c1865be3bad9 (old id 1752528)
date added to LUP
2010-12-30 08:39:04
date last changed
2018-05-29 11:48:53
@article{a6a3c3b6-2eda-4791-b03f-c1865be3bad9,
  abstract     = {Given a point and an expanding map on the unit interval, we consider the set of points for which the forward orbit under this map is bounded away from the given point. It is well-known that in many cases such sets have full Hausdorff dimension. We prove that such sets have a large intersection property, i.e. countable intersections of such sets also have full Hausdorff dimension. This result applies to a class of maps including multiplication by integers modulo 1 and x -> 1/x modulo 1. We prove that the same properties hold for multiplication modulo 1 by a dense set of non-integer numbers between 1 and 2.},
  author       = {Färm, David},
  issn         = {1468-9367},
  keyword      = {non-integer expansions,numbers,badly approximable,Schmidt games,interval maps,Hausdorff dimension},
  language     = {eng},
  number       = {4},
  pages        = {531--545},
  publisher    = {Taylor & Francis},
  series       = {Dynamical Systems},
  title        = {Simultaneously non-dense orbits under different expanding maps},
  url          = {http://dx.doi.org/10.1080/14689367.2010.482519},
  volume       = {25},
  year         = {2010},
}