---
_id: '8514'
abstract:
- lang: eng
text: We study the extent to which the Hausdorff dimension of a compact subset of
an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional
space. It is possible that the dimension drops under all such mappings, but the
amount by which it typically drops is controlled by the ‘thickness exponent’ of
the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275).
More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness
exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally)
Lipschitz functions from $B$ to $\mathbb{R}^{m}$ that contains the space of bounded
linear functions. We prove that for almost every (in the sense of prevalence)
function $f \in M$, the Hausdorff dimension of $f(X)$ is at least $\min\{ m, d
/ (1 + \tau) \}$. We also prove an analogous result for a certain part of the
dimension spectra of Borel probability measures supported on $X$. The factor $1
/ (1 + \tau)$ can be improved to $1 / (1 + \tau / 2)$ if $B$ is a Hilbert space.
Since dimension cannot increase under a (locally) Lipschitz function, these theorems
become dimension preservation results when $\tau = 0$. We conjecture that many
of the attractors associated with the evolution equations of mathematical physics
have thickness exponent zero. We also discuss the sharpness of our results in
the case $\tau > 0$.
article_processing_charge: No
article_type: original
author:
- first_name: WILLIAM
full_name: OTT, WILLIAM
last_name: OTT
- first_name: BRIAN
full_name: HUNT, BRIAN
last_name: HUNT
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
citation:
ama: OTT W, HUNT B, Kaloshin V. The effect of projections on fractal sets and measures
in Banach spaces. *Ergodic Theory and Dynamical Systems*. 2006;26(3):869-891.
doi:10.1017/s0143385705000714
apa: OTT, W., HUNT, B., & Kaloshin, V. (2006). The effect of projections on
fractal sets and measures in Banach spaces. *Ergodic Theory and Dynamical Systems*.
Cambridge University Press. https://doi.org/10.1017/s0143385705000714
chicago: OTT, WILLIAM, BRIAN HUNT, and Vadim Kaloshin. “The Effect of Projections
on Fractal Sets and Measures in Banach Spaces.” *Ergodic Theory and Dynamical
Systems*. Cambridge University Press, 2006. https://doi.org/10.1017/s0143385705000714.
ieee: W. OTT, B. HUNT, and V. Kaloshin, “The effect of projections on fractal sets
and measures in Banach spaces,” *Ergodic Theory and Dynamical Systems*, vol.
26, no. 3. Cambridge University Press, pp. 869–891, 2006.
ista: OTT W, HUNT B, Kaloshin V. 2006. The effect of projections on fractal sets
and measures in Banach spaces. Ergodic Theory and Dynamical Systems. 26(3), 869–891.
mla: OTT, WILLIAM, et al. “The Effect of Projections on Fractal Sets and Measures
in Banach Spaces.” *Ergodic Theory and Dynamical Systems*, vol. 26, no. 3,
Cambridge University Press, 2006, pp. 869–91, doi:10.1017/s0143385705000714.
short: W. OTT, B. HUNT, V. Kaloshin, Ergodic Theory and Dynamical Systems 26 (2006)
869–891.
date_created: 2020-09-18T10:48:52Z
date_published: 2006-06-01T00:00:00Z
date_updated: 2021-01-12T08:19:48Z
day: '01'
doi: 10.1017/s0143385705000714
extern: '1'
intvolume: ' 26'
issue: '3'
language:
- iso: eng
month: '06'
oa_version: None
page: 869-891
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
issn:
- 0143-3857
- 1469-4417
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: The effect of projections on fractal sets and measures in Banach spaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 26
year: '2006'
...