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

Published online by Cambridge University Press:  12 November 2002

WENXIA LI ,
DONGMEI XIAO  and
F. M. DEKKING
WENXIA LI
Affiliation:
Department of Mathematics, Central China Normal University, Wuhan 430079, P.R. China. e-mail: [email protected], [email protected]
DONGMEI XIAO
Affiliation:
Department of Mathematics, Central China Normal University, Wuhan 430079, P.R. China. e-mail: [email protected], [email protected]
F. M. DEKKING
Affiliation:
Thomas Stieltjes Institute of Mathematics Delft University of Technology ITS (CROSS), Mekelweg 4, 2628 CD Delft, The Netherlands. e-mail: [email protected]
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Abstract

Let C be the homogeneous Cantor set invariant for x→ax and x→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 (dimHC)2 = (log 2/log a)2. In this paper we generalize the essential ingredient of the proof of this result. Let Ω = {0, 1, …, r}. Let F be a Moran set associated with {0 < ai < 1, i ∈ Ω} and Ωw = Ω×Ω×⃛. Let ø be the associated coding map from Ωe onto F. Fix a non-empty set Γ ⊆ Ω with Γ≠Ø and let z(σ, n) denote the position of the nth occurrence of the elements of Γ in σ ∈ Ωw.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 133 , Issue 2 , September 2002 , pp. 345 - 355
Copyright
2002 Cambridge Philosophical Society

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