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# The effect of projections on fractal sets and measures in Banach spaces

Ott, William and Hunt, Brian and Kaloshin, Vadim (2006) The effect of projections on fractal sets and measures in Banach spaces. Ergodic Theory and Dynamical Systems, 26 (3). pp. 869-891. ISSN 0143-3857 http://resolver.caltech.edu/CaltechAUTHORS:OTTetds06

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## 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 (Nonlinearity 12 (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.

Item Type: Article Copyright © 2006 Cambridge University Press. Reprinted with permission. (Received 19 December 2003 and accepted in revised form 18 June 2005) Published online 18 April 2006. CaltechAUTHORS:OTTetds06 http://resolver.caltech.edu/CaltechAUTHORS:OTTetds06 http://dx.doi.org/10.1017/S0143385705000714 No commercial reproduction, distribution, display or performance rights in this work are provided. 8828 CaltechAUTHORS Archive Administrator 20 Sep 2007 26 Dec 2012 09:42

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