---
res:
bibo_abstract:
- 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 $\tau > 0$.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: WILLIAM
foaf_name: OTT, WILLIAM
foaf_surname: OTT
- foaf_Person:
foaf_givenName: BRIAN
foaf_name: HUNT, BRIAN
foaf_surname: HUNT
- foaf_Person:
foaf_givenName: Vadim
foaf_name: Kaloshin, Vadim
foaf_surname: Kaloshin
foaf_workInfoHomepage: http://www.librecat.org/personId=FE553552-CDE8-11E9-B324-C0EBE5697425
orcid: 0000-0002-6051-2628
bibo_doi: 10.1017/s0143385705000714
bibo_issue: '3'
bibo_volume: 26
dct_date: 2006^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/0143-3857
- http://id.crossref.org/issn/1469-4417
dct_language: eng
dct_publisher: Cambridge University Press@
dct_title: The effect of projections on fractal sets and measures in Banach spaces@
...