The horizon problem for prevalent surfaces

Falconer, K. J.; Fraser, Jonathan M.

doi:10.1017/s030500411100048x

Cited by 31 publications

(13 citation statements)

References 14 publications

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“…First McClure [29] proved that the packing dimension, and thus the upper box dimension of the graph of the prevalent f ∈ C[0, 1] is 2. The analogous result for the lower box dimension was proved in [15], [18], and [39], independently.…”

Section: Introductionsupporting

confidence: 73%

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

Balka

Darji

Elekes

2016

Advances in Mathematics

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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, R d ) has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent f ∈ C(K, R d ) 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 fcontains a dense open set having full measure with respect to the occupation measure λ m • f −1 , where dim H and λ m 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 dim H f −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.

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Section: Introductionsupporting

confidence: 73%

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

Balka

Darji

Elekes

2016

Advances in Mathematics

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“…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: 59%

Dimensions of graphs of prevalent continuous maps

Balka¹

2016

J. Fractal Geom.

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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.

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“…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

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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),

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The horizon problem for prevalent surfaces

Cited by 31 publications

References 14 publications

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

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

Dimensions of graphs of prevalent continuous maps

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

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