The Hausdorff dimension of graphs of density continuous functions

Buczolich, Zoltán; Ostaszewski, Krzysztof

doi:10.1090/S0002-9939-1991-1045588-X

The Hausdorff dimension of graphs of density continuous functions
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by Zoltán Buczolich and Krzysztof Ostaszewski PDF

Proc. Amer. Math. Soc. 112 (1991), 1037-1043 Request permission

Abstract:

The density topology on the real line consists of all measurable sets whose all points are their points of Lebesgue density one. A real-valued function of a real variable that is continuous with respect to the density topology on both the domain and the range is termed density continuous. By constructing graphs of density continuous functions as invariant sets of systems of affine maps on the unit square we show that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two.

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