Keleti, T., Mathe, A. E. and Zindulka, O. (2014) Hausdorff dimension of metric spaces and Lipschitz maps onto cubes. International Mathematics Research Notices, Volume 2014 (Number 2). pp. 289-302. doi:10.1093/imrn/rns223 ISSN 1073-7928.
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Abstract
We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than k can always be mapped onto a k-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces.
As an application, we essentially answer a question of Urbański by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset A of a complete separable metric space is ⌊dimHA⌋ if dimHA is finite but not an integer, dimHA or dimHA−1 if dimHA is an integer and at least ω0 if Graphic.
| Item Type: | Journal Article |
|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics |
| Journal or Publication Title: | International Mathematics Research Notices |
| Publisher: | Oxford University Press |
| ISSN: | 1073-7928 |
| Official Date: | March 2014 |
| Dates: | Date Event March 2014 Published 13 October 2012 Available 14 September 2012 Accepted 22 March 2012 Submitted |
| Volume: | Volume 2014 |
| Number: | Number 2 |
| Page Range: | pp. 289-302 |
| DOI: | 10.1093/imrn/rns223 |
| Status: | Peer Reviewed |
| Publication Status: | Published |
| Access rights to Published version: | Restricted or Subscription Access |
| URI: | https://wrap.warwick.ac.uk/60118/ |
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