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

Bayart, Frédéric; Heurteaux, Yanick

doi:10.1007/978-0-8176-8400-6_2

Cited by 20 publications

(22 citation statements)

References 17 publications

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“…Clearly C α [0, 1] is a Banach space. Clausel and Nikolay [8,Theorem 2] proved that the graph of the prevalent f ∈ C α [0, 1] is of Hausdorff dimension 2 − α, see also [6] for a generalization. Studying the level sets seems to be a more delicate matter.…”

Section: Finer Results With Generalized Hausdorff Measuresmentioning

confidence: 99%

“…The next result was proved by Bayart and Heurteaux, see [6,Theorem 3]. Theorem 1.10 (Bayart-Heurteaux).…”

Section: Fibers Of Maximal Dimensionmentioning

confidence: 86%

“…Note that if X : K → R d is a fractional Brownian motion restricted to some K ⊂ [0, 1] and f ∈ C(K, R d ), then Peres and Sousi [36] determined the almost sure Hausdorff dimension of graph(X + f ) in terms of f and the Hurst index of X. It is not difficult to extend the proof of [6,Theorem 3] to vector valued functions, and Theorem 1.7 handles the case dim H K = 0. These yield the following theorem.…”

Section: Fibers Of Maximal Dimensionmentioning

confidence: 99%

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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: Finer Results With Generalized Hausdorff Measuresmentioning

confidence: 99%

“…The next result was proved by Bayart and Heurteaux, see [6,Theorem 3]. Theorem 1.10 (Bayart-Heurteaux).…”

Section: Fibers Of Maximal Dimensionmentioning

confidence: 86%

Section: Fibers Of Maximal Dimensionmentioning

confidence: 99%

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

Balka

Darji

Elekes

2016

Advances in Mathematics

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“…This improves the analogous results concerning box and packing dimension, see Fact 2.4. The following generalization is due to Bayart and Heurteaux [4].…”

Section: Introductionmentioning

confidence: 99%

“…The proof of Theorem 1.16 is based on the energy method, see [4,Theorem 3]. A lower estimate for the Hausdorff dimension of graph(X + f ) is given there, where X : K → R is a fractional Brownian motion restricted to K ⊂ R m and f ∈ C(K, R) is a continuous drift.…”

Section: Introductionmentioning

confidence: 99%

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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