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Hausdorff and Packing Dimension of Fibers and Graphs of Prevalent Continuous Maps
| Content Provider | Semantic Scholar |
|---|---|
| Author | Elekes, Márton |
| Copyright Year | 2014 |
| Abstract | The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let K be an uncountable compact metric space. We prove that the prevalent f ∈ C(K,Rd) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent f ∈ C(K,Rd) has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for the prevalent f ∈ C([0, 1]m,Rd) the set of y ∈ f([0, 1]m) for which dimH f −1(y) = m contains a dense open set having full measure with respect to the occupation measure λm ◦ f−1, where dimH and λm denote the Hausdorff dimension and the m-dimensional Lebesgue measure, respectively. We also prove an analogous result when [0, 1]m is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions f ∈ C[0, 1] for which positively many level sets are singletons form a non-shy set in C[0, 1]. In order to do so, we generalize a theorem of Antunović, Burdzy, Peres and Ruscher. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.renyi.hu/~emarci/Haar_null_level_sets140808.pdf |
| Alternate Webpage(s) | https://users.renyi.hu/~emarci/Haar_null_level_sets140808.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Arabic numeral 0 Graph - visual representation Haar wavelet Hausdorff dimension Map Maximal set Packing dimension Peres–Horodecki criterion Self-similarity Set packing Tissue fiber |
| Content Type | Text |
| Resource Type | Article |
