On the box dimensions of graphs of typical continuous functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:391

Article

Hyde, J.¹  Laschos, V.²  Olsen, L.¹  Petrykiewicz, I.³  Shaw, A.¹ 

[1] Univ St Andrews, Dept Math, St Andrews KY16 9SS, Fife, Scotland

[2] Univ Bath, Dept Math Sci, Bath BA2 7AY, Avon, England

[3] Univ Grenoble 1, Inst Fourier, F-38402 St Martin Dheres, France

关键词: Box dimension;    Continuous function;    Baire category;   

DOI  :  10.1016/j.jmaa.2012.02.044

来源: Elsevier

【 摘 要 】

Let X subset of R be a bounded set; we emphasize that we are not assuming that X is compact or Borel. We prove that for a typical (in the sense of Baire) uniformly continuous function f on X, the lower box dimension of the graph of f is as small as possible and the upper box dimension of the graph of f is as big as possible. We also prove a local version of this result. Namely, we prove that for a typical uniformly continuous function f on X, the lower local box dimension of the graph of f at all points x is an element of X is as small as possible and the upper local box dimension of the graph of f at all points x is an element of X is as big as possible. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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