Ott, William; Hunt, Brian; Kaloshin, Vadim The effect of projections on fractal sets and measures in Banach spaces. (English) Zbl 1123.28012 Ergodic Theory Dyn. Syst. 26, No. 3, 869-891 (2006). The authors 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 B. R. Hunt and V. Yu. Kaloshin [Nonlinearity 12, No. 5, 1263–1275 (1999; Zbl 0932.28006)]. 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. The authors prove that for almost every function \(f \in M\), the Hausdorff dimension of \(f(X)\) is at least \(\min\{ m, d / (1 + \tau) \}\). In all of the results in this paper, ‘almost every’ is in the sense of prevalence, a generalization of ‘Lebesgue almost every’ to infinite-dimensional space [B. R. Hunt, T. Sauer and J. A. Yorke, Bull. Am. Math. Soc., New Ser. 28, No. 2, 306–307 (1993; Zbl 0782.28007)]. They 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 \(\tau = 0\). They conjecture that many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero. They also discuss the sharpness of the results in the case \(\tau > 0\). Reviewer: Kun Soo Chang (Seoul) Cited in 1 ReviewCited in 6 Documents MSC: 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 28A80 Fractals 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures Keywords:Hausdorff dimension; thickness exponent; Lipschitz functions; prevalence; dimension spectra; evolution equations Citations:Zbl 0932.28006; Zbl 0782.28007 PDFBibTeX XMLCite \textit{W. Ott} et al., Ergodic Theory Dyn. Syst. 26, No. 3, 869--891 (2006; Zbl 1123.28012) Full Text: DOI