Simultaneously non-dense orbits under different expanding maps
(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:
https://lup.lub.lu.se/record/1752528
- author
- Färm, David LU
- organization
- publishing date
- 2010
- 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
- 2016-04-01 09:53:04
- date last changed
- 2022-01-25 17:35:58
@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}}, keywords = {{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}}, doi = {{10.1080/14689367.2010.482519}}, volume = {{25}}, year = {{2010}}, }