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

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.