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Mathematical Proceedings of the Cambridge Philosophical Society (2002), 133 : 345-355 Cambridge University Press
Copyright © 2002 Cambridge Philosophical Society
doi:10.1017/S030500410200590X
Published online by Cambridge University Press 12 Nov 2002
 
Mathematical Proceedings of the Cambridge Philosophical Society (2002), 133:2:345-355 Cambridge University Press
Copyright © 2002 Cambridge Philosophical Society
doi:10.1017/S030500410200590X

Non-differentiability of devil's staircases and dimensions of subsets of Moran sets

WENXIA LI a1 1 , DONGMEI XIAO a1 and F. M. DEKKING a3
a1 Department of Mathematics, Central China Normal University, Wuhan 430079, P.R. China. e-mail: wxli@mail.ccnu.edu.cn, dmxiao@mail.ccnu.edu.cn
a2 Department of Mathematics, Central China Normal University, Wuhan 430079, P.R. China. e-mail: dmxiao@mail.ccnu.edu.cn
a3 Thomas Stieltjes Institute of Mathematics Delft University of Technology ITS (CROSS), Mekelweg 4, 2628 CD Delft, The Netherlands. e-mail: f.m.dekking@its.tudelft.nl

Abstract

Let C be the homogeneous Cantor set invariant for x[rightward arrow]ax and x[rightward arrow]1−a+ax. It has been shown by Darst that the Hausdorff dimension of the set of non-differentiability points of the distribution function of uniform measure on C equals (dimH C)2 = (log 2/log a)2. In this paper we generalize the essential ingredient of the proof of this result. Let [Omega] = {0, 1, …, r}. Let F be a Moran set associated with {0 < ai < 1, i [set membership] [Omega]} and [Omega]w = [Omega]×[Omega]×[three dots above]. Let ø be the associated coding map from [Omega]e onto F. Fix a non-empty set [Gamma] [subset, equals] [Omega] with [Gamma][not equal]Ø and let z([sigma], n) denote the position of the nth occurrence of the elements of [Gamma] in [sigma] [set membership] [Omega]w.

(Received October 13 2000)


Footnotes

1 Supported by the National Science Foundation of China 10071027.


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