NDA@SZTAKI

Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

Balka, RichĂĄrd

Darji, U. B.

Elekes, MĂĄrton

ELSEVIER

2016

QA Mathematics / matematika

QA166-QA166.245 Graphs theory / grĂĄfelmĂŠlet

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 a prevalent f∈C(K,Rd) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that a 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 a prevalent f∈C([0,1]m,Rd) the set of y∈f([0, 1]m) for which dimHf-1(y)=m contains a dense open set having full measure with respect to the occupation measure λmo 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. As a complementary result we prove that the functions f∈C[0, 1] for which dimHf-1(y)=1 for all y∈(min f, max f) form a non-shy set in C[0, 1].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. © 2016 Elsevier Inc.

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http://real.mtak.hu/44190/1/1_s2.0_S0001870816000499_main_u...

http://real.mtak.hu/44190/2/1408.2176.pdf

Balka, RichĂĄrd and Darji, U. B. and Elekes, MĂĄrton (2016) Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps. ADVANCES IN MATHEMATICS, 293. pp. 221-274. ISSN 0001-8708

http://real.mtak.hu/44190/

http://dx.doi.org/10.1016/j.aim.2016.02.005

MTMT:3043062; doi:10.1016/j.aim.2016.02.005