Dimensions of prevalent continuous functions

Gruslys, Vytautas; Jonušas, Julius; Vuksan, Mijovic,; Ng, O.; Olsen, L.; Petrykiewicz, Izabela

doi:10.1007/s00605-011-0365-6

Cited by 16 publications

(15 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First McClure proved in [23] that the packing dimension (and hence the upper box dimension) of the graph of a prevalent f ∈ C[0, 1] is 2. For the lower box dimension the analogous result was proved independently in [8], [12], and [26]. Moreover, Gruslys et al [12] proved the following theorem.…”

Section: Introductionsupporting

confidence: 57%

Dimensions of graphs of prevalent continuous maps

Balka¹

2016

J. Fractal Geom.

View full text Add to dashboard Cite

Let K be an uncountable compact metric space and let C(K, R d ) denote the set of continuous maps f : 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 dim 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.2010 Mathematics Subject Classification. Primary: 28A78, 28C10, 46E15, Secondary: 60B05, 54E52.

show abstract

Section: Introductionsupporting

confidence: 57%

Dimensions of graphs of prevalent continuous maps

Balka¹

2016

J. Fractal Geom.

View full text Add to dashboard Cite

show abstract

“…Indeed, it was shown in [10] that the graph of a prevalent function in (C[0, 1], d ∞ ) has upper box dimension 2. Also, Theorem 1.5 (1) was very recently obtained in [5], and a slight weakening of Theorem 1.5 (1) (with '1-prevalent' replaced just by 'prevalent') was given in [13] using a completely different method without a probe.…”

Section: Resultsmentioning

confidence: 99%

The horizon problem for prevalent surfaces

Falconer

Fraser

2011

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

We investigate the box dimensions of the horizon of a fractal surface defined by a function f ∈ C[0, 1] 2 . In particular we show that a prevalent surface satisfies the 'horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most α, for α ∈ [2, 3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions.

show abstract

“…In this paper, we have decided to focus to the notion of Hausdorff dimension of graphs. Nevertheless, we can mention that there are also many papers that deal with the generic value of the dimension of graphs when the notion of dimension is for example the lower box dimension (see [5,9,12,16]) or the packing dimension (see [10,14]).…”

Section: Introductionmentioning

confidence: 99%

On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets

Bayart

Heurteaux

2013

Further Developments in Fractals and Related Fields

View full text Add to dashboard Cite

Let K be a compact set in R d with positive Hausdorff dimension. Using a Fractional Brownian Motion, we prove that in a prevalent set of continuous functions on K, the Hausdorff dimension of the graph is equal to dim H (K) + 1. This is the largest possible value. This result generalizes a previous work due to J.M. Fraser and J.T. Hyde ([6]) which was exposed in the conference Fractal and Related Fields 2. The case of α-Hölderian functions is also discussed.2 −kα cos(2 k x),

show abstract

Dimensions of prevalent continuous functions

Cited by 16 publications

References 14 publications

Dimensions of graphs of prevalent continuous maps

Dimensions of graphs of prevalent continuous maps

The horizon problem for prevalent surfaces

On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets

Contact Info

Product

Resources

About