Balka, Richárd and Darji, U. B. and Elekes, Márton (2016) Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps. ADVANCES IN MATHEMATICS, 293. pp. 221274. ISSN 00018708
Text
1_s2.0_S0001870816000499_main_u.pdf Restricted to Registered users only Download (810kB)  Request a copy 


Text
1408.2176.pdf Download (507kB)  Preview 
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 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 dimHf1(y)=m contains a dense open set having full measure with respect to the occupation measure λmo f1, where dimH and λm denote the Hausdorff dimension and the mdimensional Lebesgue measure, respectively. We also prove an analogous result when [0, 1]m is replaced by any selfsimilar 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 nonshy 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 dimHf1(y)=1 for all y∈(min f, max f) form a nonshy 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.
Item Type:  Article 

Uncontrolled Keywords:  Ultrametric space; Shy; SECONDARY; Primary; PREVALENT; Packing dimension; Occupation measure; Lipschitz map; Level Set; Hölder map; Hausdorff dimension; Haar null; GRAPH; generic; FIBER; Continuous map; Brownian motion; Baire category; SPACES; EXISTENCE; LEVEL SETS; VARIABLE DRIFT; BROWNIANMOTION; HAAR NULL SETS; Brownian motion; Holder map; Level set 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika Q Science / természettudomány > QA Mathematics / matematika > QA166QA166.245 Graphs theory / gráfelmélet 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  03 Jan 2017 07:43 
Last Modified:  03 Jan 2017 07:43 
URI:  http://real.mtak.hu/id/eprint/44190 
Actions (login required)
Edit Item 