Dimensions of graphs of prevalent continuous maps

2015/03/03 by Balka, Richárd
#28C10 #46E15 #54E52 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 28A78 #Probability (math.PR) #Secondary: 60B05

paper · doi:10.48550/arxiv.1503.00865

Abstract

Let K be an uncountable compact metric space and let C(K,ℝ^d) denote the set of continuous maps f\colon K → ℝ^d endowed with the maximum norm. The goal of this paper is to determine various fractal dimensions of the graph of the prevalent f∈ C(K,ℝ^d). As the main result of the paper we show that if K has finitely many isolated points then the lower and upper box dimension of the graph of the prevalent f∈ C(K,ℝ^d) are \underlinedim_B K+d and dim_B K+d, respectively. This generalizes a theorem of Gruslys, Jonušas, Mijovic, Ng, Olsen, and Petrykiewicz. We prove that the graph of the prevalent f∈ C(K,ℝ^d) has packing dimension dim_P K+d, generalizing a result of Balka, Darji, and Elekes. Balka, Darji, and Elekes proved that the Hausdorff dimension of the graph of the prevalent f∈ C(K,ℝ^d) equals dim_H K+d. We give a simpler proof for this statement based on a method of Fraser and Hyde.

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