How smooth is a devil’s staircase?

F.M. Dekking ^{∗} & Wenxia Li ^{†}

Abstract

Let the Cantor set C in R be defined by C = ⋃_{j=0}^{r} h_{j}(C) with a disjoint union, where the h_{j}’s are similitude mappings with ratios 0 < a_{j} < 1. Let μ on C be the selfsimilar probability measure corresponding to the probability vector (a_{0}^{ξ}, a_{1}^{ξ}, ⋯, a_{r}^{ξ}), where ξ = dim_{H} C is the Hausdorff dimension of C. Let S be the set of points at which the probability distribution function F (x) of μ has no derivative, finite or infinite. We prove that dim_{H} S = (dim_{H} C)^{2} and dim_{P} S = dim_{B} S = dim_{H} C. Key Words: Hausdorff dimension; Packing dimension; Non-differentiability; Cantor function; Cantor measure. AMS Subject Classifications: 28A80, 28A78.

∗Thomas Stieltjes Institute of Mathematics, Delft University of Technology, ITS (CROSS), Mekelweg 4, 2628 CD Delft, The Netherlands. Email: f.m.dekking@its.tudelft.nl

†Department of Mathematics, Nonlinear Laboratory, Central China Normal University, Wuhan 430079, P.R. China. Supported by the National Science Foundation of China 10071027. Email:

wxli@mail.ccnu.edu.cn

1 Introduction

Let h_{i}(x) = ax + i(1 − a), i = 0, 1 with x ∈ [0, 1] and 0 < a < \frac{1}{2}. Then there exists a unique non-empty compact set C ⊂ [0, 1] such that

C = h_{0}(C) ⋃ h_{1}(C).

It is well-known that the Hausdorff dimension of C equals dim_{H} C = −\frac{log 2}{log a}. Let μ be the uniform probability measure on C. Consider the distribution function which is often referred to as the Devil’s staircase (for a = \frac{1}{3}):

F (x) = μ([0, x]), x ∈ [0, 1].

It is easy to check that the derivative of F (x) is zero for all x ∈ [0, 1] \ C and the upper derivative of F (x) is infinite on C. Let S be the set of points at which F (x) is not differentiable, i.e., the set of points in C at which the lower derivative of F (x) is finite. S can be decomposed into

S = N^{+} ⋃ N^{−} ⋃{t : t is an endpoint of C},  (1)

where N^{+}(N^{−}) is the set of non-end points of C at which the lower right (left) derivative of F (x) is finite. Each t ∈ C can be encoded by a unique 0 − 1 sequence, denoted by t̃= (t(1), t(2), ⋯), which satisfies {t} = ⋂_{n=1}^{∞} h_{t(1)}∘⋯∘h_{t(n)}([0, 1]). Now let z(t, n) denote the position of the n-th zero in t̃. The set N^{+} (symmetrically for N^{−}) is characterized by R.Darst [1] as follows:

[a] if t ∈ N^{+}, then lim sup_{n→∞} \frac{z(t,n+1)}{z(t,n)} ≥ −\frac{log a}{log 2};

[b] if lim sup_{n→∞} \frac{z(t,n+1)}{z(t,n)} > −\frac{log a}{log 2}, then t ∈ N^{+}.

By means of the above [a] and [b] R. Darst [1] proves that

dim_{H} S = dim_{H} N^{+} = [\frac{log 2}{log a}]^{2} = (dim_{H} C)^{2}.

It is not difficult to show (see [1]) that dim_{H} S = dim_{H} N^{+} = [\frac{log(r+1)}{log a}]^{2} = (dim_{H} C)^{2} still holds for a little bit more general Cantor set C with C = ⋃_{j=0}^{r} h_{j}(C), where h_{j}(x) = ax + (1 − a)\frac{j}{r}, j = 0, 1, ⋯, r and 0 < a < \frac{1}{r+1} .

The Cantor sets C described above are all homogeneous in the sense that all similitude mappings h_{j}(x) have the same scaling factor a and the gaps between the images h_{j}([0, 1]), j = 0, 1, ⋯, r, have the same length. In the following, let r be a positive integer and let the Cantor set C in R be defined by

C = ⋃_{j=0}^{r} h_{j}(C),

where h_{j}(x) = a_{j}x + b_{j}, j = 0, 1, ⋯, r, with 0 < a_{j} < 1. Without loss of generality we shall assume that b_{0} = 0 and a_{r} + b_{r} = 1. We furthermore assume that the images h_{j}([0, 1]), j = 0, 1, ⋯, r are pairwise disjoint and are lined up from left to right. In this paper, we will determine the Hausdorff, box and packing dimensions of the set of non-differentiability points of the distribution function F (x) = μ([0, x]) of the self-similar probability measure μ associated to the mappings (h_{j})_{j=0}^{r}. In order to encode the elements of C, we introduce some notations. Let Ω = {0, 1, ⋯, r}. We will write

(i) Ω^{ω} = {σ = (σ(1), σ(2), ⋯) : σ(j) ∈ Ω};

(ii) Ω^{k} = {σ = (σ(1), σ(2), ⋯, σ(k)) : σ(j) ∈ Ω} for k ∈ N and Ω^{∗} = ⋃_{k=1}^{∞}Ω^{k};

(iii) | · | is used to denote the length of word. For any σ, τ ∈ Ω^{∗} write σ ∗ τ = (σ(1), ⋯, σ(|σ|), τ(1), ⋯, τ(|τ|)), and write τ ∗σ = (τ(1), ⋯, τ(|τ|), σ(1), σ(2), ⋯) for any τ ∈ Ω^{∗}, σ ∈ Ω^{ω};

(iv) σ|k = (σ(1), σ(2), ⋯, σ(k)) for σ ∈ Ω^{ω} and k ∈ N.

Denote h_{σ}(x) = h_{σ(1)} ∘ ⋯ ∘ h_{σ(k)}(x) for σ ∈ Ω^{k} and x ∈ R. Then for σ ∈ Ω^{k}, the intervals h_{σ∗0}([0, 1]), h_{σ∗1}([0, 1]), ⋯, h_{σ∗r}([0, 1]) are contained in h_{σ}([0, 1]) in this order where the left endpoint of h_{σ∗0}([0, 1]) coincides with the left endpoint of h_{σ}([0, 1]), and the right endpoint of h_{σ∗r}([0, 1]) coincides with the right endpoint of h_{σ}([0, 1]). Moreover the length of interval h_{σ}([0, 1]) equals λ(h_{σ}([0, 1])) = ∏_{j=1}^{k} a_{σ(j)} =: a_{σ} for σ ∈ Ω^{k}. For j = 1, 2, ⋯, we define

C_{j} = ⋃_{σ∈Ω^{j}} h_{σ}([0, 1]).

Then C_{j} ↓ C as j → ∞ and x ∈ C can be encoded by a unique σ ∈ Ω^{ω} satisfying {x} = ⋂_{k=1}^{∞} h_{σ|k}([0, 1]). We usually denote this unique code of x by x̃ and use x(k) to denote the k-th component of x̃. It is well-known that dim_{H} C = dim_{B} C = dim_{P} C = ξ where ξ is given by

∑_{j=0}^{r} a_{j}^{ξ} = 1.  (2)

Let μ be the self-similar probability measure on C corresponding to the probability vector (a_{0}^{ξ}, a_{1}^{ξ}, ⋯, a_{r}^{ξ}) , i.e., the measure satisfying

μ(h_{σ}([0, 1])) = ∏_{j=1}^{k} a_{σ(j)}^{ξ} = a_{σ}^{ξ}, for any σ ∈ Ω^{k}, k ∈ N.

Consider the distribution function of the probability measure μ, also called Cantor function:

F (x) = μ([0, x]), x ∈ [0, 1].

Figure 1: The graph of F (x) for the case a_{0} = 0.5, b_{0} = 0, a_{1} = 0.2 and b_{1} = 0.8.

It is easy to check that the derivative of F (x) is zero for all x ∈ [0, 1] \ C. We will show that the upper derivative of F (x) is infinite on C. Let S be the set of points at which F (x) is not differentiable, i.e., the set of points in C at which the lower derivative of F (x) is finite. The set S can be decomposed in the same way as in (1). The endpoints of h_{σ}([0, 1]) for a σ ∈ Ω^{∗} will be called the endpoints of C. Obviously any endpoint e of C lies in C and except for a finite number of terms, its coding ẽ consists of either only the symbol 0 if e is the left endpoint of some h_{σ}([0, 1]), or only the symbol r if e is the right endpoint of some h_{σ}([0, 1]). In this paper, we will prove that dim_{H} S = dim_{H} N^{+} = dim_{H} N^{−} = (dim_{H} C)^{2} = ξ^{2} and dim_{B} S = dim_{P} S = dim_{H} C = ξ.

2 Codes of non-differentiability points

In this section, we characterize the set S of non-differentiability points by means of the behavior of their codings. We focus on N^{+}. Results on N^{−} can be obtained symmetrically.

Proposition 2.1 The upper derivative of F (x) is infinite for all x ∈ C.

Proof For any t in C, t not a right endpoint, let the code be t̃= (t(1), t(2), ⋯). Then t̃ has infinitely many entries lying in Ω\{r}. Suppose t̃ has a entry from Ω\{r} in position j. Then t lies in the interval h_{t̃|(j−1)}([0, 1]) but is not equal to the right endpoint u of h_{t̃|(j−1)}([0, 1]), where

ũ = (t(1), ⋯, t(j − 1), r, r, ⋯).

Note that u is also the right endpoint of h_{ũ|j}([0, 1]) and that t ∉ h_{ũ|j}([0, 1]). Thus we have that t, u ∈ h_{t̃|(j−1)}([0, 1]) and (t, u] ⊇ h_{ũ|j}([0, 1]). Consider the slope of the line segment from the point P = (t, F (t)) on the graph of F to the point Q = (u, F (u)). We have

\frac{F (u) − F (t)}{u − t} = \frac{μ((t, u])}{u − t} ≥ \frac{μ(h_{ũ|j}([0, 1]))}{|h_{t̃|(j−1)}([0, 1])|} = \frac{a_{t̃|(j−1)}^{ξ}a_{r}^{ξ}}{a_{t̃|(j−1)}}
= a_{r}^{ξ}(a_{t̃|(j−1)})^{ξ−1} → ∞ as j → ∞.

Symmetrically the upper left derivative of t at a non-left-end point of C is infinite. QED

Proposition 2.2 Let Γ = \{0, 1, ⋯, r − 1\}. Let \underline{a} = min_{j∈Γ} a_{j} and ā = max_{j∈Γ} a_{j}. Let t ∈ C be not an endpoint of C and let z(t, n) denote the position of the n-th occurrence of elements of Γ in t̃, then (I) if t ∈ N^{+}, then lim sup_{n→∞} [\frac{n}{z(t,n)}(1 − ξ^{−1}) (\frac{log ā}{log a_{r}} − 1) + \frac{z(t,n+1)}{z(t,n)} − ξ^{−1}] ≥ 0;

(II) if lim sup_{n→∞} [\frac{n}{z(t,n)}(1 − ξ^{−1}) (\frac{log \underline{a}}{log a_{r}} − 1) + \frac{z(t,n+1)}{z(t,n)} − ξ^{−1}] > 0, then t ∈ N^{+}.

Proof We first prove the statement (I), i.e., the lower-right derivative of F (x) is infinite at a non-endpoint t of C when lim sup_{n→∞} [\frac{n}{z(t,n)}(1 − ξ^{−1}) (\frac{log ā}{log a_{r}} − 1) + \frac{z(t,n+1)}{z(t,n)} − ξ^{−1}] < 0. Consider such a point t with t̃= (t(1), t(2), ⋯). Let k be a positive integer such that

\frac{n}{z(t, n)}(1 − ξ^{−1}) (\frac{log ā}{log a_{r}} − 1) + \frac{z(t, n + 1)}{z(t, n)} − ξ^{−1} < q,  (3)

for some negative real number q whenever n ≥ k. Let u be a positive number such that u is smaller than the distance between t and [0, 1] \ C_{t̃|l} with l = z(t, k). Let x be a point in the segment (t, t + u). Then t, x ∈ h_{t̃|l}([0, 1]). We will see that (F (x) − F (t))/(x − t) is large relative to k, so t is not in N^{+}. Let i denote the level at which x ∉ h_{t̃|i}([0, 1]) but x ∈ h_{t̃|(i−1)}([0, 1]). Note that also t ∈ h_{t̃|(i−1)}([0, 1]). Thus x−t ≤ |h_{t̃|(i−1)}([0, 1])| = a_{t̃|(i−1)}; also i = z(t, n) for some n > k. Put j = z(t, n + 1) − 1, and by v we denote the right endpoint of h_{t̃|j}([0, 1]), which implies that ṽ = (t(1), ⋯, t(j), r, r, ⋯) and (t, v] ⊇ h_{ṽ|(j+1)}([0, 1]). Then we have t < v < x and F (v) − F (t) = μ((t, v]) ≥ μ(h_{ṽ|(j+1)}([0, 1])) = (a_{t̃|j})^{ξ}a_{r}^{ξ}. Therefore, defining

Π̃_{n} = ∏_{i=1,t(i)≠r}^{z(t,n)−1} a_{t(i)},

we have

\frac{F (x) − F (t)}{x − t} ≥ \frac{(a_{t̃|j})^{ξ}a_{r}^{ξ}}{a_{t̃|(i−1)}} = \frac{a_{r}^{ξ} (∏_{i=1}^{z(t,n+1)−1} a_{t(i)})^{ξ}}{∏_{i=1}^{z(t,n)−1} a_{t(i)}} = \frac{a_{r}^{ξ} [a_{r}^{z(t,n+1)−1−n}a_{t(z(t,n))}Π̃_{n}]^{ξ}}{a_{r}^{z(t,n)−1−(n−1)}Π̃_{n}}
= a_{t(z(t,n))}^{ξ}Π̃_{n}^{ξ−1}a_{r}^{(z(t,n+1)−n)ξ−z(t,n)+n}
= a_{t(z(t,n))}^{ξ} [Π̃_{n}^{\frac{ξ−1}{z(t,n)}} a_{r}^{(\frac{z(t,n+1)}{z(t,n)} − \frac{n}{z(t,n)})ξ−1+ \frac{n}{z(t,n)}} ]^{z(t,n)}
≥ \underline{a}^{ξ} [ā^{\frac{(n−1)(ξ−1)}{z(t,n)}} a_{r}^{(\frac{z(t,n+1)}{z(t,n)} − \frac{n}{z(t,n)})ξ−1+ \frac{n}{z(t,n)}} ]^{z(t,n)}
= \underline{a}^{ξ}ā^{1−ξ} [(\frac{ā}{a_{r}} )^{\frac{n(ξ−1)}{z(t,n)}} a_{r}^{\frac{z(t,n+1)}{z(t,n)} ξ−1}]^{z(t,n)} .  (4)

Let

Q = (\frac{ā}{a_{r}} )^{\frac{n(ξ−1)}{z(t,n)}} a_{r}^{\frac{z(t,n+1)}{z(t,n)} ξ−1} .

Taking logs and by (3) we have

log Q = ξ log a_{r} [\frac{n}{z(t, n)}(1 − ξ^{−1}) (\frac{log ā}{log a_{r}} − 1) + \frac{z(t, n + 1)}{z(t, n)} − ξ^{−1}]
≥ ξq log a_{r} > 0.  (5)

Since t is a non-end point, z(t, n) → ∞ and the lower-right derivative of F (x) is infinite at t by (4) and (5). Now we turn to the proof of statement (II). Let t be such that

lim sup_{n→∞} [\frac{n}{z(t, n)}(1 − ξ^{−1}) (\frac{log \underline{a}}{log a_{r}} − 1) + \frac{z(t, n + 1)}{z(t, n)} − ξ^{−1}] > 0.

Then there exists a sequence {n_{k}} of positive integers such that

\frac{n_{k}}{z(t, n_{k})}(1 − ξ^{−1}) (\frac{log \underline{a}}{log a_{r}} − 1) + \frac{z(t, n_{k} + 1)}{z(t, n_{k})} − ξ^{−1} > c,  (6)

for some positive constant c. Let x_{k} be the left endpoint of h_{(t̃|j_{k})∗(t(j_{k}+1)+1)}([0, 1]), where j_{k} = z(t, n_{k}) − 1. Thus we have x̃_{k} = (t(1), ⋯, t(j_{k}), t(j_{k} + 1) + 1, 0, ⋯, 0, ⋯). Let u_{k} be the right endpoint of h_{t̃|(j_{k}+1)}([0, 1]). Then ũ_{k} = (t(1), ⋯, t(j_{k}), t(j_{k} + 1), r, r, r, ⋯). Thus [u_{k}, x_{k}] is the gap on the right side of h_{t̃|(j_{k}+1)}([0, 1]) and λ([u_{k}, x_{k}]) = x_{k}−u_{k} = a_{t̃|j_{k}}β_{t(j_{k}+1)} where by β_{j}, j = 0, 1, ⋯, r − 1, we denote length of the gap between images h_{j}([0, 1]) and h_{j+1}([0, 1]). Note that [t, x_{k}] ⊇ [u_{k}, x_{k}] and μ((t, x_{k}]) = μ((t, u_{k}]) + μ((u_{k}, x_{k}]) = μ((t, u_{k}]) ≤ μ(h_{t̃|(z(t,n_{k}+1)−1)}([0, 1])) since t̃|(z(t, n_{k} + 1) − 1) = ũ_{k}|(z(t, n_{k} + 1) − 1). Therefore we have

F (x_{k}) − F (t) = μ((t, x_{k}]) ≤ μ(h_{t̃|(z(t,n_{k}+1)−1)}([0, 1])) = a_{t̃|(z(t,n_{k}+1)−1)}^{ξ},

and

x_{k} − t ≥ λ([u_{k}, x_{k}]) = a_{t̃|(z(t,n_{k})−1)}β_{t(z(t,n_{k}))}.

Denote \underline{β} = min_{j∈\{0,1,···,r−1\}} β_{j}. Then we obtain with a similar reasoning which led to (4)

\frac{F (x_{k}) − F (t)}{x_{k} − t} ≤ \frac{a_{t̃|(z(t,n_{k}+1)−1)}^{ξ}}{a_{t̃|(z(t,n_{k})−1)}β_{t̃(z(t,n_{k}))}} ≤ \frac{(∏_{i=1}^{z(t,n_{k}+1)−1} a_{t(i)})^{ξ}}{\underline{β} ∏_{i=1}^{z(t,n_{k})−1} a_{t(i)}}
≤ \frac{ā^{ξ}\underline{a}^{1−ξ}}{a_{r}^{ξ}\underline{β}} [(\frac{\underline{a}}{a_{r}} )^{\frac{n_{k}(ξ−1)}{z(t,n_{k})}} a_{r}^{\frac{z(t,n_{k}+1)}{z(t,n_{k})} ξ−1}]^{z(t,n_{k})} .  (7)

Let

Q = (\frac{\underline{a}}{a_{r}} )^{\frac{n_{k}(ξ−1)}{z(t,n_{k})}} a_{r}^{\frac{z(t,n_{k}+1)}{z(t,n_{k})} ξ−1} .

Taking logs and using (6) we obtain

log Q = ξ log a_{r} [\frac{n_{k}}{z(t, n_{k})}(1 − ξ^{−1}) (\frac{log \underline{a}}{log a_{r}} − 1) + \frac{z(t, n_{k} + 1)}{z(t, n_{k})} − ξ^{−1}]
≤ ξc log a_{r} < 0.  (8)

From (7) and (8) it follows that the right lower derivative at t is finite by letting k ⟶ ∞. QED

3 Dimensions of the set of non-differentiability points

In this section, we determine the dimensions of S. The following lemma is a special case of a result in [4].

Lemma 3.1 Let Γ = {0, 1, ⋯, r−1} and z(t, n) denote the position of the n-th occurrence of elements of Γ in t̃. For given 0 < p ≤ 1, let

C(p) = \{t ∈ C \ \{right endpoints of C\} : lim sup_{n→∞} \frac{z(t, n + 1)}{z(t, n)} ≥ p^{−1}\} .  (9)

Let η = η(p) be such that

p log ∑_{j∈Ω} a_{j}^{η} + (1 − p) log a_{r}^{η} = 0.  (10)

Then we have dim_{H} C(p) = η and dim_{P} C(p) = dim_{B} C(p) = dim_{H} C = ξ where ξ is defined in (2).

It is easy to verify that η(p) is strictly increasing and continuous in p and that η(0+) < η(p) ≤ η(1) = ξ. We also consider for 0 < p ≤ 1 and with the same Γ

C^{∗}(p) = \{t ∈ C \ \{right endpoints of C\} : lim sup_{n→∞} \frac{z(t, n + 1)}{z(t, n)} > p^{−1}\} .  (11)

Directly from Lemma 3.1 it follows that dim_{P} C^{∗}(p) = dim_{B} C^{∗}(p) = dim_{H} C = ξ. Moreover, dim_{H} C^{∗}(p) = η(p). To see that this follows from Lemma 3.1, approximate C^{∗}(p) by a union of C(p_{k})’s, where p_{k} ↑ p.

Lemma 3.2 Let Γ = \{0, 1, ⋯, r − 1\}, \underline{a} = min_{j∈Γ} a_{j}, ā = max_{j∈Γ} a_{j}, and let

p_{1}^{−1} = max \{ξ^{−1}, (ξ^{−1} − 1)\frac{log \underline{a}}{log a_{r}} + 1\}; p_{2}^{−1} = min \{ξ^{−1}, (ξ^{−1} − 1)\frac{log ā}{log a_{r}} + 1\} .

Then C^{∗}(p_{1}) ⊆ N^{+} ⊆ C(p_{2}) and η(p_{1}) ≤ dim_{H}(N^{+}) ≤ η(p_{2}).

Proof By Lemma 3.1 it suffices to prove C^{∗}(p_{1}) ⊆ N^{+} ⊆ C(p_{2}). Let t ∈ N^{+}. By Proposition 2.2 we have

lim sup_{n→∞} [\frac{n}{z(t, n)}(1 − ξ^{−1}) (\frac{log ā}{log a_{r}} − 1) + \frac{z(t, n + 1)}{z(t, n)} − ξ^{−1}] ≥ 0.

Now if \frac{log ā}{log a_{r}} > 1, then lim sup_{n→∞} \frac{z(t,n+1)}{z(t,n)} ≥ ξ^{−1}. If not, we use that \frac{n}{z(t,n)} ≤ 1 and so lim sup_{n→∞} \frac{z(t,n+1)}{z(t,n)} ≥ ξ^{−1} − (1 − ξ^{−1}) (\frac{log ā}{log a_{r}} − 1). So we find that

lim sup_{n→∞} \frac{z(t, n + 1)}{z(t, n)} ≥ p_{2}^{−1},

i.e., t ∈ C(p_{2}). On the other hand, t ∈ C^{∗}(p_{1}) implies in a similar way that t ∈ N^{+}. QED

Theorem 3.3 Let C = ⋃_{j=0}^{r} h_{j}(C) be the Cantor set determined by {h_{j}(x) = a_{j}x + b_{j} : 0 ≤ j ≤ r}, and let ξ be its Hausdorff dimension. If S is the set of non-differentiability points of the Cantor distribution determined by a_{0}, ⋯, a_{r} on C, then dim_{H} S = ξ^{2} = (dim_{H} C)^{2} and dim_{B} S = dim_{P} S = dim_{H} C = ξ.

Proof Since S = N^{+} ⋃ N^{−} ⋃{the endpoints of C} and because of the symmetry between N^{+} and N^{−}, it suffices to determine the dimensions of N^{+}. By Lemma 3.2 and Lemma 3.1, dim_{B} N^{+} = dim_{P} N^{+} = dim_{H} C = ξ is trivial. We define a = min_{j∈Ω} a_{j}. Let 0 < δ < a^{2}, and let

Ω_{δ} = {σ ∈ Ω^{∗} : a_{σ} ≤ δ and a_{σ|(|σ|−1)} > δ}.

Note that for each σ ∈ Ω_{δ} we have aδ < a_{σ} ≤ δ. Thus for any σ, τ ∈ Ω_{δ}

\frac{log δ}{log a + log δ} ≤ \frac{log a_{σ}}{log a_{τ}} ≤ \frac{log a + log δ}{log δ} .  (12)

Here we would like to relate Ω_{δ} to ξ by showing that

lim_{δ→0} \frac{log #Ω_{δ}}{− log δ} = ξ,  (13)

where ξ is defined in (2) and #Ω_{δ} denotes the number of elements of Ω_{δ}. By N_{δ}(C) we denote the smallest number of sets of diameter at most δ that cover C. By Lemma 9.2 in [2] there exists a positive constant q independent of δ such that

q#Ω_{δ} ≤ N_{δ}(C) ≤ #Ω_{δ}.  (14)

Hence (13) holds by (14) and the fact lim_{δ→0} \frac{log N_{δ}(C)}{− log δ} = ξ. Now let H_{δ} = \{h_{σ} : σ ∈ Ω_{δ}\}. Note that h_{σ} is a similitude mapping with ratio 0 < a_{σ} < 1 for each σ ∈ H_{δ}, that the family H_{δ} of similitude mappings still satisfies the open set condition, and that the unique self-similar set determined by H_{δ} equals C:

C = ⋃_{σ∈H_{δ}} h_{σ}(C).

We also have that ξ defined in (2) satisfies ∑_{σ∈Ω_{δ}} a_{σ}^{ξ} = 1. If we denote μ_{δ} the self-similar probability measure on C corresponding to the probability vector (a_{σ}^{ξ} : σ ∈ Ω_{δ}), then μ_{δ} = μ since μ_{δ}(h_{τ}([0, 1])) = μ(h_{τ}([0, 1])) for any τ ∈ Ω_{δ}^{k} and k ∈ N. Hence for the corresponding non-differentiability points N_{δ}^{+} we have N_{δ}^{+} = N^{+} for all δ > 0. There exists a unique σ ∈ Ω_{δ} with σ(j) = r for j = 1, 2, ⋯, |σ|. We denote this special element by σ_{δ}. Note that σ_{δ} plays the same role in Ω_{δ} as r in Ω, in the sense that h_{σ_{δ}}([0, 1]) is the right most interval in [0, 1] of the intervals (h_{σ}([0, 1]))_{σ∈Ω_{δ}}. Let

Γ_{δ} := Ω_{δ} \ {σ_{δ}} and a_{δ} := a_{σ_{δ}} = a_{r}^{|σ_{δ}|}.

We will use the notations \underline{a}_{δ} = min_{σ∈Γ_{δ}} a_{σ}, ā_{δ} = max_{σ∈Γ_{δ}} a_{σ},

p_{1}(δ) = (max \{ξ^{−1}, (ξ^{−1} − 1)\frac{log \underline{a}_{δ}}{log a_{δ}} + 1\})^{−1}  (15)

and

p_{2}(δ) = (min \{ξ^{−1}, (ξ^{−1} − 1)\frac{log ā_{δ}}{log a_{δ}} + 1\})^{−1} .

Then by Lemma 3.1 and Lemma 3.2 with Ω replaced by Ω_{δ}, Γ replaced by Γ_{δ}, a_{r} replaced by a_{δ} and p_{1}, p_{2} by p_{1}(δ) respectively p_{2}(δ), we have

η(p_{1}(δ)) ≤ dim_{H} N_{δ}^{+} ≤ η(p_{2}(δ)),  (16)

where η(p_{1}(δ)) and η(p_{2}(δ)) are defined by formula (10):

p_{1}(δ) log ∑_{σ∈Ω_{δ}} a_{σ}^{η(p_{1}(δ))} + (1 − p_{1}(δ)) log a_{δ}^{η(p_{1}(δ))} = 0,  (17)

and

p_{2}(δ) log ∑_{σ∈Ω_{δ}} a_{σ}^{η(p_{2}(δ))} + (1 − p_{2}(δ)) log a_{δ}^{η(p_{2}(δ))} = 0.

Since N_{δ}^{+} = N^{+} for all δ > 0 it follows from (16) that dim_{H} N^{+} = ξ^{2} holds if for any ε > 0 there exists a δ^{∗} > 0 such that when 0 < δ < δ^{∗} we have |η(p_{1}(δ)) − ξ^{2}| < ε and |η(p_{2}(δ)) − ξ^{2}| < ε. Verification of this claim will be given in the following only for η(p_{1}(δ)), since the same argument can be employed for η(p_{2}(δ)). To alleviate the notation we will write p_{δ} := p_{1}(δ) and η_{δ} := η(p_{1}(δ)). For x ∈ (0, ξ] let

T_{δ}(x) = p_{δ} \frac{log ∑_{σ∈Ω_{δ}} a_{σ}^{x}}{x log a_{δ}} + 1 − p_{δ}

Note that

a^{x}δ^{x}#Ω_{δ} ≤ ∑_{σ∈Ω_{δ}} a_{σ}^{x} ≤ δ^{x}#Ω_{δ},

which implies

\frac{x log δ + log #Ω_{δ}}{x log a_{δ}} ≤ \frac{log ∑_{σ∈Ω_{δ}} a_{δ}^{x}}{x log a_{δ}} ≤ \frac{x log a + x log δ + log #Ω_{δ}}{x log a_{δ}} .  (18)

Therefore by (18) we have

0 ≤ T_{δ}(x) − [1 + p_{δ} (\frac{log δ}{log a_{δ}} − 1) + \frac{1}{x} \frac{p_{δ} log #Ω_{δ}}{log a_{δ}} ] ≤ \frac{p_{δ} log a}{log a_{δ}}.  (19)

Note that by (12) and (15) we have

lim_{δ↓0} p_{δ} = ξ.  (20)

Since aδ < a_{δ} ≤ δ, and by (13) and (20) we have

lim_{δ↓0} p_{δ} (\frac{log δ}{log a_{δ}} − 1) = lim_{δ↓0} \frac{p_{δ} log a}{log a_{δ}} = 0 and lim_{δ↓0} \frac{p_{δ} log #Ω_{δ}}{log a_{δ}} = −ξ^{2}.

Thus by (19) we have for all x ∈ (0, ξ]

lim_{δ↓0} T_{δ}(x) = 1 − \frac{ξ^{2}}{x} =: T_{0}(x).  (21)

Now for any given ε > 0 satisfying 0 < ξ^{2} − ε < ξ^{2} + ε < ξ < 1, we see that

T_{0}(ξ^{2} − ε) = \frac{−ε}{ξ^{2} − ε} < −ε and T_{0}(ξ^{2} + ε) = \frac{ε}{ξ^{2} + ε} > ε.

By (21) we can take δ^{∗} > 0 so that for 0 < δ < δ^{∗}

|T_{δ}(ξ^{2} − ε) − T_{0}(ξ^{2} − ε)| < \frac{ε}{2} and |T_{δ}(ξ^{2} + ε) − T_{0}(ξ^{2} + ε)| < \frac{ε}{2}.

Then necessarily for these δ

T_{δ}(ξ^{2} − ε) < −\frac{ε}{2} and T_{δ}(ξ^{2} + ε) > \frac{ε}{2}.

Then for 0 < δ < δ^{∗}

ξ^{2} − ε < η_{δ} < ξ^{2} + ε,

since T_{δ}(x) is strictly increasing in x and T_{δ}(η_{δ}) = 0. QED

References

[1] DARST R (1995) Hausdorff dimension of sets of non-differentiability points of Cantor functions. Math. Proc. Camb. Phi. Soc., bf 117: 185–191.

[2] FALCONER KJ (1990) Fractal geometry–Mathematical foundations and applications. Chichester: John Wiley & Sons Ltd.

[3] HUTCHINSON JE (1981) Fractals and self-similarity. Indiana Univ. Math. J. 30: 713–747.

[4] LI WX, XIAO DM & DEKKING FM (2000) Non-differentiability of devil’s staircases and dimensions of subsets of Moran fractals. To appear in Mathematical Proceedings of the Cambridge Philosophical Society.