Brian R Hunt and Vadim Yu Kaloshin 1999 Nonlinearity 12 1263 doi:10.1088/0951-7715/12/5/303
Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces
Brian R Hunt
and Vadim Yu Kaloshin
Recommended by Professor P Cvitanovic
We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections 
into 
N. We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of to X. The same quantity also bounds the factor by which the Hausdorff dimension of X can decrease under these projections. Such a bound is motivated by our discovery that the Hausdorff dimension of X need not be preserved by typical projections, in contrast to classical results on the preservation of a Hausdorff dimension by projections between finite-dimensional spaces. We give an example for any positive number d of a set X with box-counting and Hausdorff dimension d in the real Hilbert space 
2 such that for all projections into N, no matter how large N is, the Hausdorff dimension of (X) is less than d (and in fact, is less than two, no matter how large d is).
- PACS
- MSC
-
37F35 Conformal densities and Hausdorff dimension
52A21 Finite-dimensional Banach spaces (including special norms, zonoids, etc.) (See also 46Bxx)
- Subjects
- Dates
-
Issue 5 (September 1999)
Received 9 September 1998, in final form 4 June 1999
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Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces
Brian R Hunt and Vadim Yu Kaloshin 1999 Nonlinearity 12 1263
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