Hausdorff dimension of subsets of Moran

fractals with prescribed group frequency of

their codings

Wenxia Li ^{∗} Department of Mathematics East China Normal University Shanghai 200062, P.R. China

F.M. Dekking ^{†}

Thomas Stieltjes Institute of Mathematics

Delft University of Technology

ITS (SSOR)

Mekelweg 4, 2628 CD Delft, The Netherlands

∗Supported by the National Science Foundation of China 10071027. Email: wxli@math.ecnu.edu.cn †Email: f.m.dekking@its.tudelft.nl

Abstract

A well known result states that the set of numbers in base r in which the digits i occur with relative frequency p_{i} for i = 0, …, r − 1 is a set of Hausdorff dimension −\frac{1}{log r} ∑_{i=0}^{r−1} p_{i} log p_{i}. For instance, decimal numbers in which only the digits 1 and 6 occur, both with relative frequencies \frac{1}{2}, have Hausdorff dimension log 2/ log 10. In this paper we generalize this result to the situation where one prescribes the relative frequencies of groups of digits in the expansion. E.g., suppose we require that in the decimal expansion digits from \{0, 1, 2\} occur with relative frequency \frac{1}{2}, and also that digits from {3, 4, …, 9} occur with this relative frequency. Our result shows that the Hausdorff dimension of this set is (log 2 + \frac{1}{2} log 3 + \frac{1}{2} log 7)/ log 10. Actually we take a much more general geometric viewpoint, considering subsets of Moran fractals specified by prescribing the relative frequencies of groups of symbols in their codings. We determine the Hausdorff dimension of such sets, and moreover give necessary and sufficient conditions for such a set to have positive Hausdorff measure in its dimension. Key Words: Hausdorff dimension; Moran fractals; Multifractal decomposition. AMS Subject Classifications: 28A80, 28A78.

1 Introduction

One of the first non-compact fractals sets to be studied is that of the numbers in the unit interval having an anomalous distribution of their digits in a fixed base r. According to Borel’s theorem, almost all numbers will have their digits distributed according to the probability vector (\frac{1}{r}, \frac{1}{r}, ⋯, \frac{1}{r}). So the set M(p_{0}, ⋯, p_{r−1}) of numbers for which the digits 0, 1, ⋯, r −1 are distributed according to the probability vector (p_{0}, ⋯, p_{r−1})—not equal to uniform distribution— has measure zero, and it is interesting to obtain an idea of its size by the value of its Hausdorff dimension. Eggleston [2] proved that

dim_{H} M(p_{0}, ⋯, p_{r−1}) = −\frac{1}{log r} ∑_{i=0}^{r−1} p_{i} log p_{i}.

In this paper we will generalize Eggleston’s result by weakening the requirements on the relative frequency of the digits: partition the set of digits in k groups Γ_{1}, …, Γ_{k}, i.e., the Γ_{j} are disjoint sets with union {0, ⋯, r − 1}. Now prescribe the (overall) relative frequency c_{j} with which the digits from the jth group occur, and denote this set by M(Γ_{1}, ⋯, Γ_{k}). As a special case of our result it will appear that

dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = \frac{1}{log r} ∑_{j=1}^{k} c_{j}(log #Γ_{j} − log c_{j}).

Here we recall that the Hausdorff dimension of a set E in d-dimensional Euclidean space is obtained by considering the s−dimensional Hausdorff measure H^{s}(E) of E defined by

H^{s}(E) = lim_{δ↓0} H_{δ}^{s}(E),

where H_{δ}^{s}(E) is obtained by covering E with sets D_{i} of diameter at most δ:

H_{δ}^{s}(E) = inf {∑_{i=1}^{∞} |D_{i}|^{s} : |D_{i}| ≤ δ and ⋃_{i=1}^{∞} D_{i} ⊇ E} .

The Hausdorff dimension of E is now the unique number in [0, d], denoted by dim_{H} E, such that

H^{s}(E) = 0 if s > dim_{H} E and H^{s}(E) = +∞ if s < dim_{H} E.

Our goal is to generalize Eggleston’s result even further. The sets M(p_{0}, ⋯, p_{r−1}) can be considered as subsets of a one dimensional self similar set with constant scaling factor \frac{1}{r} (the set [0, 1] (!)). We will consider so called Moran fractals in d-dimensional space which admit different scalings a_{1}, ⋯, a_{r}. Various properties of these sets have been studied, e.g., in [1], [2], [4], [7], [8], [10], [12], [13], [14], and [17]. Typically, the tool to describe and analyze Moran fractals is their coding in sequence space: each point in the set is coded via a finite to one coding map. E.g., for the examples above in the unit interval, the coding is just the expansion of a point in digits in base r. One can therefore specify subsets E of the Moran fractal by prescribing relative (group-)frequencies of their codings. Our main result (Theorem 1) gives the Hausdorff dimension of the sets thus obtained. Our second result concerns the Hausdorff measure H^{t}(E) of E, when t = dim_{H} E. This is called the Hausdorff measure of E in its dimension, and can be 0, finite and positive, or infinite. The value of H^{t}(E) is a well known way to assess the regularity of E. We shall give necessary and sufficient conditions for H^{t}(E) to be positive and finite (Theorem 2). This result permits us (at the end of the paper), to solve a problem posed by Cawley and Mauldin in [1].

2 Notations and main results

A Moran fractal can be constructed as follows. Denote Ω = {1, 2, ⋯, r}, where r ≥ 2. The following notations will be used in this paper.

[i] Ω^{∗} = ⋃_{m=1}^{∞}Ω^{m} with Ω^{m} = {σ = (σ(1), σ(2), ⋯, σ(m)) : σ(j) ∈ Ω} for m ∈ N; while Ω^{ω} = {σ = (σ(1), σ(2), ⋯) : σ(j) ∈ Ω};

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

[iii] σ|m = (σ(1), σ(2), ⋯, σ(m)) for σ ∈ Ω^{ω} and m ∈ N;

[iv] For σ ∈ Ω^{m}, the cylinder set C(σ) is defined as C(σ) = {τ ∈ Ω^{ω} : τ|m = σ};

[v] When f_{i}(x) : R^{d} ⟶ R^{d}, are maps for 1 ≤ i ≤ r, we denote f_{σ}(x) = f_{σ(1)}∘⋯∘f_{σ(m)}(x) for σ ∈ Ω^{m} and x ∈ R^{d}.

Fixing a nonempty compact set J ⊂ R^{d} with \overline{intJ} = J, a constant 0 < c < 1 and positive real numbers 0 < a_{i} < 1, i = 1, 2, ⋯, r, the related Moran fractal (or Moran set) is defined according to the following two steps.

Step 1. For each σ ∈ Ω^{m}, m ∈ N, construct a compact set J_{σ} in R^{d} by induction: • A family {J_{j} : j = 1, 2, ⋯, r} of non-overlapping nonempty compact subsets of J is chosen for m = 1 such that \overline{intJ_{j}} = J_{j}, |J_{j}| = a_{j}|J|, where | · | denotes the diameter of a set, and J_{j} contains an open ball of diameter c|J_{j}|;

• Suppose that J_{σ} is given for some σ ∈ Ω^{m}. Choose a family {J_{σ∗i} : i = 1, 2, ⋯, r} of non-overlapping nonempty compact subsets of J_{σ} such that \overline{intJ_{σ∗i}} = J_{σ∗i}, |J_{σ∗i}| = a_{i}|J_{σ}| and J_{σ∗i} contains an open ball of diameter c|J_{σ∗i}|.

Step 2. The Moran fractal F associated with {0 < a_{i} < 1 : i = 1, 2, ⋯, r} and (J_{σ})_{σ∈Ω^{∗}} is defined as the nonempty compact set

F = ⋂_{m=1}^{∞} ⋃_{σ∈Ω^{m}} J_{σ}.  (1)

The compact sets J_{σ}, σ ∈ Ω^{∗} are generally referred to as component sets of F. In particular, J_{σ} is referred to as a mth level component set of F if σ ∈ Ω^{m}. Define ϕ: Ω^{ω} ⟶ R^{d} by

{ϕ(σ)} = ⋂_{m=1}^{∞} J_{σ|m}.  (2)

It is easy to see that ϕ(Ω^{ω}) = F and ϕ(C(σ)) = F ⋂ J_{σ} by (1). But ϕ may not be injective. Let ρ be the metric on Ω^{ω} such that for any σ, τ ∈ Ω^{ω}

ρ(σ, τ) = 2^{− min{i:σ(i)≠τ(i)}},

with the convention ρ(σ, σ) = 0. Let F be equipped with the Euclidean metric. Then ϕ is continuous. Thus each x ∈ F can be encoded via ϕ: a sequence σ ∈ Ω^{ω} is called a location code of x ∈ F if ϕ(σ) = x. Therefore ϕ is also called the coding map and Ω^{ω} is called the code space (or symbolic space). As a result, F is a projection of Ω^{ω} on R^{d} via ϕ. Some comments about Moran fractals are listed below.

(C1) Moran fractals are regular fractals in the sense that it has been proven that 0 < H^{s}(F) < ∞, with s = dim_{H} F, given by ∑_{i=1}^{r} a_{i}^{s} = 1 (see [9],[11], [13] & [14]).

(C2) A Moran fractal is termed as map-specified if there exist similitude contractions f_{i}, i = 1, 2, ⋯, r, such that J_{σ} = f_{σ}(J) for any σ ∈ Ω^{∗}. In this case F is actually the self-similar set determined by f_{i}, 1 ≤ i ≤ r, which satisfy the open set condition with respect to the open set O = intJ (i.e. ⋃_{i=1}^{r} f_{i}(O) ⊆ O with a disjoint union on the left) and the coding map ϕ in (2) can be changed into

\{ϕ(σ)\} = ⋂_{m=1}^{∞} f_{σ|m}(\overline{O}) = \{ lim_{m→∞} f_{σ|m}(0)\}.

(C3) Let 0 < R < |J| · min_{1≤i≤r} a_{i}. A component set J_{σ} of F is termed as a R-size component set if

|J_{σ}| ≤ R and |J_{σ|(|σ|−1)}| > R.

It is easy to see that for any 0 < R < |J|·min_{1≤i≤r} a_{i}, the set of all R-size component sets of F is a non-overlapping finite R-covering of F. Hence, by means of Lemma 9.2 in [3], the requirement that J_{σ} contains an open ball of diameter c|J_{σ}| implies an important fact: there exists a positive integer ϑ, independent of R and x ∈ R^{d}, such that any ball B_{R}(x) with radius R and center at x intersects at most ϑ of the R-size component sets of F. Many analogues of this fact appear in this paper. A direct sequel leads to an important property of ϕ:

sup_{x∈F} #{ϕ^{−1}(x)} < ϑ.  (3)

Otherwise, suppose that for some x ∈ F we have #{ϕ^{−1}(x)} > ϑ. Take ϑ+1 different elements σ_{1}, σ_{2}, ⋯, σ_{ϑ+1} from this set. Let m ∈ N be such that σ_{1}|m, σ_{2}|m, ⋯, σ_{ϑ+1}|m differ from each other. Taking R = min_{1≤i≤ϑ+1} |J_{σ_{i}|m}|, x lies in at least ϑ + 1 R-size component sets of F, which implies that the ball B_{R}(x) intersects at least ϑ + 1 of the R-size component sets of F.

(C4) A more general Moran fractal structure was proposed by Zhiying Wen (see [17] for details), where the code space Ω^{ω} = ∏_{n=1}^{∞} Ω_{n} and corresponding to different Ω_{n} there are different scaling coefficients {0 < a_{n,j} < 1, j = 1, 2, ⋯, r_{n}}. Some dimension results of these generalized Moran sets can be found in [5], [6], [7],[8] and [11]. The class of generalized Moran sets clearly contains the class of Moran sets discussed here, and in fact is far larger, since a generalized Moran set often has different fractal dimensions, and has zero or infinite Hausdorff measure in its dimension.

For any E ⊆ F there exists Λ ⊂ Ω^{ω} such that E = ϕ(Λ). For certain Λ, it should be possible to determine the dimensions of the projections E = ϕ(Λ). Some solutions, which depend on the structure of Λ, can be found in [1], [2], [4], [7], [8], [10], [12], [13], [14], etc.

Let 2 ≤ k ≤ r. Fix real numbers c_{j}, j = 1, 2, ⋯, k, such that c_{j} ≥ 0 and ∑_{j=1}^{k} c_{j} = 1. Let Γ_{j}, j = 1, ⋯, k, be disjoint nonempty subsets of Ω with ⋃_{j=1}^{k} Γ_{j} = Ω. In this paper we will consider sets Λ specified by relative frequencies:

Λ = M̂(Γ_{1}, ⋯, Γ_{k}) = \{σ ∈ Ω^{ω} : lim_{n→∞} \frac{#\{1 ≤ i ≤ n : σ(i) ∈ Γ_{j}\}}{n} = c_{j}, 1 ≤ j ≤ k\} .

Let

M(Γ_{1}, ⋯, Γ_{k}) = ϕ(Λ) = ϕ(M̂(Γ_{1}, ⋯, Γ_{k})),

i.e., the subset of the Moran fractal F whose elements have their codings lying in Γ_{j} with a prescribed relative frequency c_{j}. We remark that M(Γ_{1}, ⋯, Γ_{k}) is dense in F since σ ∈ M̂(Γ_{1}, ⋯, Γ_{k}) if and only if i ∗ σ ∈ M̂(Γ_{1}, ⋯, Γ_{k}), i ∈ Ω; and that M(Γ_{1}, ⋯, Γ_{k}) = ⋃_{i=1}^{r} f_{i}(M(Γ_{1}, ⋯, Γ_{k})) for the map-specified case (cf. (C2)). Let

Z(t) = ∑_{j=1}^{k} c_{j} log ∑_{i∈Γ_{j}} a_{i}^{t} − ∑_{j=1}^{k} c_{j} log c_{j},

where here, and throughout the whole paper, we adopt the convention 0 · log 0 = 0. It is easy to see that the function Z(t) has a unique zero in [0, s] where s is defined by ∑_{i=1}^{r} a_{i}^{s} = 1, since Z(t) is strictly decreasing with Z(0) ≥ 0 and Z(s) ≤ 0. In the present paper, we obtain the following results:

Theorem 1. dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = t where t is the unique nonnegative real number such that

∑_{j=1}^{k} c_{j} log c_{j} = ∑_{j=1}^{k} c_{j} log ∑_{i∈Γ_{j}} a_{i}^{t}.  (4)

Theorem 2. Suppose dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = t. Then the following four statements are equivalent:

(i) 0 < H^{t}(M(Γ_{1}, ⋯, Γ_{k})) < ∞; (ii) ∑_{i=1}^{r} a_{i}^{t} = 1, i.e., t = dim_{H} F (cf. (C1)); (iii) c_{j} = ∑_{i∈Γ_{j}} a_{i}^{t}, 1 ≤ j ≤ k; (iv) dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = dim_{H}(F \ M(Γ_{1}, ⋯, Γ_{k})).

As an application of these results we rewrite, for convenience, Ω = {0, 1, ⋯, r − 1}. Take J = [0, 1] and for each σ ∈ Ω^{ω} let J_{σ} = f_{σ}(J) (cf. (C2)), where f_{i} : [0, 1] ⟶ [0, 1] is defined by

f_{i}(x) = \frac{1}{r}(x + i), i = 0, 1, ⋯, r − 1.

Then by (1) we have that F = [0, 1] is a map-specified Moran set. In this case we have a_{i} = \frac{1}{r} for all i, and ϕ : Ω^{ω} = \{0, 1, ⋯, r − 1\}^{N} ⟶ [0, 1] is given by

ϕ(σ) = lim_{m→∞} f_{σ|m}(0) = ∑_{m=1}^{∞} \frac{σ(m)}{r^{m}}.

Hence the r-ary expansion of ϕ(σ) is ϕ(σ) = (0.σ(1) σ(2) ⋯)_{r}. Thus M(Γ_{1}, ⋯, Γ_{k}) consists of those x ∈ F = [0, 1] for which the occurrence of digits of Γ_{j} in its base-r expansion has fixed relative frequency c_{j}. Theorem 1 and 2 yield that

dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = \frac{∑_{j=1}^{k} c_{j}(log #Γ_{j} − log c_{j})}{log r},

and M(Γ_{1}, ⋯, Γ_{k}) has positive Hausdorff measure in its dimension if and only if c_{j} = #Γ_{j}/r, implying dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = 1. In particular, taking Γ_{i} = {i−1} for 1 ≤ i ≤ r, M(Γ_{1}, ⋯, Γ_{r}) just consists of those real numbers in [0, 1] in whose r-ary expansion the digit i has density c_{i}. So we recover Eggleston’s result ([2]), mentioned in the introduction.

3 Hausdorff dimension and measure property

In this section we will first determine the Hausdorff dimension of M(Γ_{1}, ⋯, Γ_{k}). The usual way is to find an appropriate probability measure μ supported on the set in order to obtain a lower bound dim_{H} μ for its Hausdorff dimension. The measure μ can be constructed as the image measure under ϕ of a probability measure μ̂ supported on M̂(Γ_{1}, ⋯, Γ_{k}). For μ̂ we take the infinite product probability measure on Ω^{ω} corresponding to probability vectors (p_{1}, p_{2}, ⋯, p_{r}) satisfying ∑_{i∈Γ_{j}} p_{i} = c_{j} for j = 1, ⋯, k. The key is to choose concrete p_{i}’s such that dim_{H} μ reaches a maximum. The estimation of the upper bound of its Hausdorff dimension will be done by choosing an efficient sequence of coverings of the set.

Proof of Theorem 1. We have to pay attention to the fact that some c_{j}’s may be zero. Without loss of generality we assume that c_{j} > 0 for 1 ≤ j ≤ k_{1} and c_{j} = 0 for j > k_{1}, where k_{1} ≤ k. Denote Ω_{1} = ⋃_{j=1}^{k_{1}} Γ_{j}.

(A) dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≥ t. This bound is trivial when t = 0. Suppose t > 0. Let

b_{i} = (a_{i}^{t}c_{j})/ ∑_{l∈Γ_{j}} a_{l}^{t}, i ∈ Γ_{j}.  (5)

Thus we have ∑_{i∈Ω_{1}} b_{i} = 1 and b_{i} = 0 for i ∈ Ω \ Ω_{1}. Construct a probability measure μ̂ on Ω^{ω} by defining for any σ ∈ Ω^{m}

μ̂(C(σ)) = ∏_{i=1}^{m} b_{σ(i)},

where C(σ) = {τ ∈ Ω^{ω} : τ|m = σ} is the cylinder set determined by σ. Let μ on F be the image measure of μ̂ under ϕ. Let

M̂ = \{σ ∈ Ω^{ω} : σ ∈ Ω_{1}^{ω} and lim_{n→∞} \frac{#\{1 ≤ l ≤ n : σ(l) = i\}}{n} = b_{i}, 1 ≤ i ≤ r\} .  (6)

Then M̂ ⊆ M̂(Γ_{1}, ⋯, Γ_{k}) and M = ϕ(M̂) ⊆ M(Γ_{1}, ⋯, Γ_{k}). By Birkhoff’s Ergodic Theorem (ref. [16]) we have for μ̂-a.e. σ

lim_{n→∞} \frac{#\{1 ≤ l ≤ n : σ(l) = i\}}{n} = b_{i}, 1 ≤ i ≤ r.

Therefore μ̂(M̂) = μ(M) = 1. Now for σ ∈Ω^{∗} write a(σ) = ∏_{i=1}^{|σ|} a_{σ(i)} and b(σ) = ∏_{i=1}^{|σ|} b_{σ(i)}. Then for any σ ∈ M̂ it is easy to verify that

lim_{l→∞} \frac{log b(σ|l)}{log a(σ|l)} = lim_{l→∞} \frac{log ∏_{i=1}^{l} b_{σ(i)}}{log ∏_{i=1}^{l} a_{σ(i)}} = \frac{∑_{i∈Ω_{1}} b_{i} log b_{i}}{∑_{i∈Ω_{1}} b_{i} log a_{i}} ,  (7)

using (6). Note that

∑_{i∈Ω_{1}} b_{i} log b_{i} = ∑_{j=1}^{k_{1}} ∑_{i∈Γ_{j}} b_{i} log b_{i} = ∑_{j=1}^{k_{1}} ∑_{i∈Γ_{j}} b_{i}(t log a_{i} + log c_{j} − log ∑_{l∈Γ_{j}} a_{l}^{t})
= t ∑_{j=1}^{k_{1}} ∑_{i∈Γ_{j}} b_{i} log a_{i} + ∑_{j=1}^{k_{1}}[( ∑_{i∈Γ_{j}} b_{i})(log c_{j} − log ∑_{l∈Γ_{j}} a_{l}^{t})]
= t ∑_{j=1}^{k_{1}} ∑_{i∈Γ_{j}} b_{i} log a_{i} + ∑_{j=1}^{k_{1}} c_{j}(log c_{j} − log ∑_{l∈Γ_{j}} a_{l}^{t}) = t ∑_{i∈Ω_{1}} b_{i} log a_{i}, (8)

by (5) and (4). From (7) and (8) it follows that for any σ ∈ M̂

lim_{l→∞} \frac{log b(σ|l)}{log a(σ|l)} = t.

Fix 0 < ϵ < t. Let M^{(m)} = ϕ(M̂^{(m)}) where

M̂^{(m)} = \{σ ∈ M̂ : \frac{log b(σ|l)}{log a(σ|l)} > t − ϵ for all l ≥ m\} .  (9)

Then

1 = μ̂(M̂) = lim_{m→∞} μ̂(M̂^{(m)}), 1 = μ(M) = lim_{m→∞} μ(M^{(m)}).

Now fix m such that μ̂(M̂^{(m)}) > 0. Let μ̂_{m} be the restriction of μ̂ to M̂^{(m)} and let μ_{m} be the induced measure on M^{(m)} of μ̂_{m} by ϕ, i.e., for any Borel set A ⊆ M^{(m)}

μ_{m}(A) = μ̂_{m}(ϕ^{−1}(A)) = μ̂(ϕ^{−1}(A) ∩ M̂^{(m)}).

By B_{R}(x) we denote the closed ball with center at x and radius R. Let 0 < R < (min_{1≤i≤r} a_{i})^{m}. For each σ ∈ M̂^{(m)} there exists a positive integer h(σ, R) such that

R · min_{1≤i≤r} a_{i} ≤ a(σ|h(σ, R)) ≤ R.  (10)

Note that h(σ, R) > m and write W = {σ|h(σ, R) : σ ∈ M̂^{(m)}}. For any fixed x ∈ M^{(m)} let W^{∗} = {τ ∈ W : J_{τ} ⋂ B_{R}(x) ⋂ M^{(m)} ≠ ∅}. Then there exists a finite positive constant ξ independent of R and x such that # W^{∗} ≤ ξ by Lemma 9.2 in [3]. So

μ_{m}(B_{R}(x)) ≤ μ̂_{m} ( ⋃_{τ∈W^{∗}} C(τ)) ≤ ∑_{τ∈W^{∗}} μ̂(C(τ)) = ∑_{τ∈W^{∗}} b(τ) ≤ ξ · R^{t−ϵ},

by (9) and (10). So we get

lim inf_{R→0} \frac{log μ_{m}(B_{R}(x))}{log R} ≥ t − ϵ.

Theorem 1 in [15] tells us

dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≥ dim_{H}M ≥ dim_{H}M^{(m)} ≥ t − ϵ,

which implies, letting ϵ → 0, dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≥ t. (B) dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≤ t. This part of the proof is similar to that in [2]. For ϵ > 0 and m ∈ N, let

Ω(m, ϵ) = {σ ∈ Ω^{m} : (c_{j} − ϵ)m ≤ #{1 ≤ i ≤ m : σ(i) ∈ Γ_{j}} ≤ (c_{j} + ϵ)m, 1 ≤ j ≤ k}.

Note that for any given ϵ > 0 and any given σ ∈ M̂(Γ_{1}, ⋯, Γ_{k}) there exists an m_{0} ∈ N such that

|\frac{#\{1 ≤ i ≤ m : σ(i) ∈ Γ_{j}\}}{m} − c_{j}| < ϵ,

for all m ≥ m_{0}, i.e. σ ∈ Ω(m, ϵ) for all m ≥ m_{0}. Then for any ϵ > 0 we have

⋃_{m_{0}=1}^{∞} ⋂_{m=m_{0}}^{∞} ⋃_{σ∈Ω(m,ϵ)} J_{σ} ⊃ M(Γ_{1}, ⋯, Γ_{k}).

Thus, in order to prove that dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≤ t, it is sufficient to show that, for every integer m_{0} and every η > 0, there is an ϵ > 0 (which is a function of η only) such that dim_{H}(⋂_{m=m_{0}}^{∞} ⋃_{σ∈Ω(m,ϵ)} J_{σ}) ≤ t + η. Since ⋂_{m=m_{0}}^{∞} ⋃_{σ∈Ω(m,ϵ)} J_{σ} is covered by {J_{σ} : σ ∈ Ω(m, ϵ)} for m ≥ m_{0}, it is sufficient to show that given η > 0, there is an ϵ > 0 such that

∑_{σ∈Ω(m,ϵ)} |J_{σ}|^{t+η} → 0 (m → ∞).

Let

K = 4\frac{max_{1≤j≤k_{1}} c_{j}}{min_{1≤j≤k_{1}} c_{j}} max \{\frac{∑_{i∈Γ_{j_{2}}} a_{i}^{t+η}}{∑_{i∈Γ_{j_{1}}} a_{i}^{t+η}} , 1 ≤ j_{1}, j_{2} ≤ k\} .  (11)

Let ϵ satisfy

0 < ϵ < \frac{1}{r} min_{1≤j≤k_{1}} c_{j}  (12)

and

∏_{j=1}^{k_{1}} (\frac{∑_{i∈Γ_{j}} a_{i}^{t+η}}{∑_{i∈Γ_{j}} a_{i}^{t}} )^{c_{j}} · ∏_{j=k_{1}+1}^{k} (\frac{∑_{i∈Γ_{j}} a_{i}^{t+η}}{ϵ} )^{ϵ} · K^{2kϵ} < 1.  (13)

Now we have

∑_{σ∈Ω(m,ϵ)} |J_{σ}|^{t+η} = |J|^{t+η} ∑ \frac{m!}{∏_{j=1}^{k} d_{j}!} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}  (14)

where the summation is over sets of non-negative integers d_{j}, 1 ≤ j ≤ k such that

∑_{j=1}^{k} d_{j} = m and m(c_{j} − ϵ) ≤ d_{j} ≤ m(c_{j} + ϵ), 1 ≤ j ≤ k.  (15)

Hence for i ∈ Ω and j ∈ Ω_{1}, if mϵ > 1

\frac{d_{i} + 1}{d_{j}} ≤ \frac{m(c_{i} + ϵ) + 1}{m(c_{j} − ϵ)} ≤ 4\frac{max_{1≤j≤k_{1}} c_{j}}{min_{1≤j≤k_{1}} c_{j}} ,  (16)

by (12) and (15). Let

Q_{1} = \frac{m!}{∏_{j=1}^{k} d_{j}!} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}  (17)

and Q_{2} be the term of the sum (14) for which in (17) the d_{j_{1}} and d_{j_{2}} , for some j_{1} ≠ j_{2} with j_{1} ∈ Ω and j_{2} ∈ Ω_{1}, are replaced by d_{j_{1}} + 1 and d_{j_{2}} − 1 respectively and the other d_{j}’s are kept fixed. Thus when mϵ > 1

\frac{Q_{1}}{Q_{2}} = \frac{d_{j_{1}} + 1}{d_{j_{2}}} · \frac{∑_{i∈Γ_{j_{2}}} a_{i}^{t+η}}{∑_{i∈Γ_{j_{1}}} a_{i}^{t+η}} ≤ 4\frac{max_{1≤j≤k_{1}} c_{j}}{min_{1≤j≤k_{1}} c_{j}} · \frac{∑_{i∈Γ_{j_{2}}} a_{i}^{t+η}}{∑_{i∈Γ_{j_{1}}} a_{i}^{t+η}} ≤ K,  (18)

by (11) and (16). Let Q_{0} be the term of (14) for which d_{j} is the integer part of mc_{j} for 1 ≤ j ≤ k_{1} −1, the integer part of mϵ for k_{1} +1 ≤ j ≤ k. Therefore there are real numbers 0 ≤ δ_{j} < 1 with j ≠ k_{1} such that d_{j} = mc_{j} − δ_{j} for 1 ≤ j ≤ k_{1} − 1, d_{j} = mϵ − δ_{j} for k_{1} + 1 ≤ j ≤ k, and d_{k_{1}} = m − ∑_{j≠k_{1}} d_{j} = mc_{k_{1}} − (k − k_{1})mϵ + ∑_{j≠k_{1}} δ_{j} = mc_{k_{1}} − δ_{k_{1}} with δ_{k_{1}} = (k − k_{1})mϵ − ∑_{j≠k_{1}} δ_{j}. Then by means of Stirling formula n! = \sqrt{2πn}(n/e)^{n}e^{\frac{θ}{12n}} (0 < θ < 1) and (4) we have

Q_{0} = \frac{m!}{∏_{j=1}^{k} d_{j}!} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}} = \frac{\sqrt{2π}m^{m+ \frac{1}{2}} e^{−m}e^{\frac{θ}{12m}}}{∏_{j=1}^{k} \sqrt{2π}d_{j}^{d_{j}+ \frac{1}{2}} e^{−d_{j}}e^{\frac{θ_{j}}{12d_{j}}}} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}
≤ K_{1} \frac{m^{m+ \frac{1}{2}}}{∏_{j=1}^{k} d_{j}^{d_{j}+ \frac{1}{2}}} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}
= K_{1} \frac{m^{m+ \frac{1}{2}}}{∏_{j=1}^{k_{1}}(mc_{j} − δ_{j})^{mc_{j}−δ_{j}+ \frac{1}{2}} ∏_{j=k_{1}+1}^{k}(mϵ − δ_{j})^{mϵ−δ_{j}+ \frac{1}{2}}} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}
= K_{1} \frac{m^{− \frac{1}{2} (k−1)}}{∏_{j=1}^{k_{1}}(c_{j} − \frac{δ_{j}}{m})^{mc_{j}−δ_{j}+ \frac{1}{2}} ∏_{j=k_{1}+1}^{k}(ϵ − \frac{δ_{j}}{m})^{mϵ−δ_{j}+ \frac{1}{2}}} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}
≤ K_{2} \frac{m^{− \frac{1}{2} (k−1)}}{∏_{j=1}^{k_{1}} c_{j}^{mc_{j}} ∏_{j=k_{1}+1}^{k} ϵ^{mϵ}} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}}
≤ K_{3} · m^{− \frac{1}{2} (k−1)} · ∏_{j=1}^{k_{1}} (\frac{∑_{i∈Γ_{j}} a_{i}^{t+η}}{∑_{i∈Γ_{j}} a_{i}^{t}} )^{mc_{j}} · ∏_{j=k_{1}+1}^{k} (\frac{∑_{i∈Γ_{j}} a_{i}^{t+η}}{ϵ} )^{mϵ},  (19)

where K_{1}, K_{2} and K_{3} are appropriate positive constants independent of m. Since (18) implies that for any term Q of (14)

\frac{Q}{Q_{0}} < K^{2kmϵ},  (20)

it follows that

∑_{σ∈Ω(m,ϵ)} |J_{σ}|^{t+η} = |J|^{t+η} ∑ \frac{m!}{∏_{j=1}^{k} d_{j}!} · ∏_{j=1}^{k} ( ∑_{i∈Γ_{j}} a_{i}^{t+η})^{d_{j}} < Q_{0} · K^{2kmϵ} · (2mϵ)^{k} · |J|^{t+η}
< K_{4} · m^{\frac{1}{2} (k+1)} · [∏_{j=1}^{k_{1}} (\frac{∑_{i∈Γ_{j}} a_{i}^{t+η}}{∑_{i∈Γ_{j}} a_{i}^{t}} )^{c_{j}} · ∏_{j=k_{1}+1}^{k} (\frac{∑_{i∈Γ_{j}} a_{i}^{t+η}}{ϵ} )^{ϵ} · K^{2kϵ}]^{m} → 0

as m → ∞, by (19), (20) and (13), where K_{4} is a appropriate positive constant independent of m. This completes the proof. QED

In Theorem 1 we show dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = t by proving that for any positive real numbers ϵ and η dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≥ t − ϵ and dim_{H} M(Γ_{1}, ⋯, Γ_{k}) ≤ t + η. So we do not get any information about the value of H^{t}(M(Γ_{1}, ⋯, Γ_{k})). In the following we will get the sufficient and necessary conditions for M(Γ_{1}, ⋯, Γ_{k}) to have positive Hausdorff measure in its dimension. Let Q = {J_{σ} : σ ∈ Ω^{∗}} and Q_{σ} = {J_{σ∗τ} : τ ∈ Ω^{∗}} for σ ∈ Ω^{∗}. For any 0 ≤ α < ∞ and any subset E of Moran set F define

H_{Q}^{α}(E) = lim inf_{δ→0}{∑_{i=1}^{∞} |V_{i}|^{α} : {V_{i}} is a nonoverlapping δ − covering of E, V_{i} ∈ Q}. (21)

Similarly, for 0 ≤ α < ∞ and E ⊆ F ⋂ J_{σ} define

H_{Q_{σ}}^{α}(E) = lim inf_{δ→0}{∑_{i=1}^{∞} |V_{i}|^{α} : {V_{i}} is a nonoverlapping δ − covering of E , V_{i} ∈ Q_{σ}}.

Lemma 3. (I) If E ⊆ F ⋂ J_{σ}, then hH_{Q_{σ}}^{α}(E) ≤ H_{Q}^{α}(E) ≤ H_{Q_{σ}}^{α}(E), where the positive real number h is independent of σ and E. (II) If E ⊆ F, then

4^{−d}(c min_{1≤i≤r} a_{i})^{d}H_{Q}^{α}(E) ≤ H^{α}(E) ≤ H_{Q}^{α}(E),  (22)

where the positive constant c can be found in the definition of the Moran set F. In particular, the covering {V_{i}} of E in (21) can be taken finite when E is compact.

Proof. (I) It is clear that H_{Q}^{α}(E) ≤ H_{Q_{σ}}^{α}(E). Let δ < a(σ)|J|. For any non-overlapping δ-covering {V_{i}} of E with V_{i} ∈ Q, write B_{i} = {V_{i}} if V_{i} ⊆ J_{σ} and

B_{i} = {J_{σ∗τ} : |J_{σ∗τ}| ≤ |V_{i}| < |J_{σ∗τ|(|σ∗τ|−1)}| and J_{σ∗τ} ⋂ V_{i} ⋂ E ≠ ∅},

if V_{i} ⊆ J_{σ} does not hold. Then there exists a finite positive constant h^{−1} independent of σ such that #B_{i} ≤ h^{−1} by the Lemma 9.2 in [3]. Hence B = ⋃_{i} B_{i} ⊆ Q_{σ} is a δ-covering of E. So we have

∑_{U∈B} |U|^{α} ≤ h^{−1} ∑_{i} |V_{i}|^{α}.

Since for any a, b ∈ Ω^{∗} either J_{a} and J_{b} are non-overlapping or J_{a} ⊆ J_{b} (or J_{b} ⊆ J_{a}), we can assume that B is a non-overlapping covering (we can delete some elements from B if necessary). Therefore we get H_{Q}^{α}(E) ≥ hH_{Q_{σ}}^{α}(E). (II) The right inequality is clear. We only need to prove the left inequality. Let {W_{i} : 1 ≤ i < p ≤ ∞} be an open δ-covering of E (where p < ∞ if E is compact). For any 1 ≤ i < p let

A_{i} = {J_{σ} : |J_{σ}| ≤ |W_{i}| < |J_{σ|(|σ|−1)}| and J_{σ} ∩ W_{i} ∩ E ≠ ∅}.

Hence ⋃_{1≤i<p} A_{i} is a δ-covering of E with each of its elements in Q. Similarly as in (I), we can assume the ⋃_{1≤i<p} A_{i} is a non-overlapping covering of E (we can delete some J_{σ} from some A_{i} if necessary). Again # A_{i} is bounded by a constant by Lemma 9.2 in [3]. But this time, we like to give an argument. In fact, let B_{i} be an open ball with center in W_{i} and radius 2|W_{i}| and recall that for each σ ∈ Ω^{∗}, J_{σ} contains an open ball with diameter c|J_{σ}|. Then

H^{d}(B_{i}) = (4|W_{i}|)^{d} ≥ H^{d}( ⋃_{J_{σ}∈A_{i}} J_{σ}) ≥ ∑_{J_{σ}∈A_{i}} (c|J_{σ}|)^{d} ≥ (c|W_{i}| min_{1≤i≤r} a_{i})^{d} · # A_{i},

which implies

# A_{i} ≤ 4^{d}(c min_{1≤i≤r} a_{i})^{−d}.  (23)

Consequently for any an open δ-covering {W_{i} : 1 ≤ i < p ≤ ∞} of E we can get a δ-covering ⋃_{1≤i<p} A_{i} of E, with each of its elements in Q (this covering is finite if p < ∞), such that

_{J_{σ}∈⋃_{1≤i<p} A_{i}}^{∑} |J_{σ}|^{α} = ∑_{1≤i<p} ∑_{J_{σ}∈A_{i}} |J_{σ}|^{α} ≤ ∑_{1≤i<p}(|W_{i}|^{α} · # A_{i})
≤ 4^{d}(c min_{1≤i≤r} a_{i})^{−d} ∑_{1≤i<p} |W_{i}|^{α}  (24)

by (23). Thus our result (22) is obtained by (24) and (21), taking the limit as δ → 0. QED

Proof of Theorem 2. (iii)⇒(ii). This is clear. (ii)⇒(iii). Let M(Γ_{j}, Ω \ Γ_{j}) = ϕ(M̂(Γ_{j}, Ω \ Γ_{j})), where M̂(Γ_{j}, Ω \ Γ_{j}) is defined corresponding to the relative groupfrequencies c_{j} and 1−c_{j}. For any fixed 1 ≤ j ≤ k, since M(Γ_{1}, ⋯, Γ_{k}) ⊆ M(Γ_{j}, Ω\Γ_{j}) ⊆ F by the definition of M(Γ_{1}, ⋯, Γ_{k}) and M(Γ_{j}, Ω \ Γ_{j}), we have dim_{H} M(Γ_{j}, Ω \ Γ_{j}) = t. Hence t is also such that

c_{j} log c_{j} + (1 − c_{j}) log(1 − c_{j}) = c_{j} log( ∑_{i∈Γ_{j}} a_{i}^{t}) + (1 − c_{j}) log(1 − ∑_{i∈Γ_{j}} a_{i}^{t})  (25)

by (4) in Theorem 1 and (ii). Note that at this moment it must be true that 0 < c_{j} < 1. It is easy to check that the function g(x) = c_{j}(log c_{j} −log x)+(1−c_{j})(log(1−c_{j})−log(1−x)) has a unique zero in (0, 1). Consequently we get c_{j} = ∑_{i∈Γ_{j}} a_{i}^{t} from (25). (ii)⇒(i). Construct a probability measure μ̂ on Ω^{ω} by requiring that for any σ ∈ Ω^{m}

μ̂(C(σ)) = ∏_{i=1}^{m} a_{σ(i)}^{t}

where C(σ) is the cylinder determined by σ. Let μ on F be the image measure of μ̂ under ϕ. Let

M̂ = \{σ ∈ Ω^{ω} : lim_{n→∞} \frac{#\{1 ≤ l ≤ n : σ(l) = i\}}{n} = a_{i}^{t}, 1 ≤ i ≤ r\} .

Note that by the above we know (iii) also holds. Then M̂ ⊆ M̂(Γ_{1}, ⋯, Γ_{k}), M = ϕ(M̂) ⊆ M(Γ_{1}, ⋯, Γ_{k}) and μ(M) = μ̂(M̂) = 1 by Birkhoff’s Ergodic Theorem. By B_{R}(x) we denote the closed ball with center at x and radius R. Let R small enough be given. Then for each σ ∈ M̂ there exists a positive integer h(σ, R) such that

R · min_{1≤i≤r} a_{i} ≤ a(σ|h(σ, R)) ≤ R.  (26)

Let W = {σ|h(σ, R) : σ ∈ M̂}. For any fixed x ∈ M let W^{∗} = {τ ∈ W : J_{τ} ⋂ B_{R}(x) ⋂ M ≠ ∅}. Then there exists a finite positive constant ξ independent of R and x such that # W^{∗} ≤ ξ by Lemma 9.2 in [3]. Hence

μ(B_{R}(x)) ≤ μ̂ ( ⋃_{τ∈W^{∗}} C(τ)) ≤ ∑_{τ∈W^{∗}} μ̂(C(τ)) = ∑_{τ∈W^{∗}} (a(τ))^{t} ≤ ξR^{t}

by (26). So we get

H^{t}(M(Γ_{1}, ⋯, Γ_{k})) ≥ H^{t}(M) ≥ μ(M) = 1

by Proposition 4.9(a) in [3]. On the other hand, we have

H^{t}(M(Γ_{1}, ⋯, Γ_{k})) ≤ H^{t}(F) < ∞.

(i)⇒(ii). It is easy to check, that for any m ∈ N

M̂(Γ_{1}, ⋯, Γ_{k}) = ⋃_{σ∈Ω^{m}} σ ∗ M̂(Γ_{1}, ⋯, Γ_{k})

where σ∗M̂(Γ_{1}, ⋯, Γ_{k}) = {σ∗τ : τ ∈ M̂(Γ_{1}, ⋯, Γ_{k})}. Denote M̂(σ) = σ∗M̂(Γ_{1}, ⋯, Γ_{k}) and M(σ) = ϕ(M̂(σ)). Then M(Γ_{1}, ⋯, Γ_{k}) = ⋃_{σ∈Ω^{m}} M(σ). Note that {J_{σ∗τ} : τ ∈ A ⊆ Ω^{∗}} is a non-overlapping covering of M(σ) if and only if {J_{τ} : τ ∈ A ⊆ Ω^{∗}} is a nonoverlapping covering of M(Γ_{1}, ⋯, Γ_{k}) and |J_{σ∗τ}| = a(σ)|J_{τ}|. Thus

H_{Q_{σ}}^{t}(M(σ)) = (a(σ))^{t}H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})).  (27)

Then we have

H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})) = H_{Q}^{t}( ⋃_{σ∈Ω^{m}} M(σ)) ≤ ∑_{σ∈Ω^{m}} H_{Q}^{t}(M(σ))
≤ ∑_{σ∈Ω^{m}} H_{Q_{σ}}^{t}(M(σ)) = ∑_{σ∈Ω^{m}} (a(σ))^{t}H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k}))
= (∑_{i=1}^{r} a_{i}^{t})^{m}H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})),  (28)

by Lemma 3(I) and (27). In addition, for any x ∈ M(Γ_{1}, ⋯, Γ_{k}), x must lie in at least one, but in at most ϑ of the M(σ), σ ∈ Ω^{m} by (3). We claim that

∑_{σ∈Ω^{m}} H_{Q}^{t}(M(σ)) ≤ ϑH_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})).  (29)

This is not obvious, and needs some demonstration. Let {V_{i}} be a non-overlapping δcovering of M(Γ_{1}, ⋯, Γ_{k}), with V_{i} ∈ Q and δ < min_{σ∈Ω^{m}} |J_{σ}|. For each σ ∈ Ω^{m}, choose the covering B_{σ} of M(σ) from {V_{i}} by taking B_{σ} = {V_{i} : V_{i} ∩ M(σ) ≠ ∅}. In this way, each V_{i} is chosen at most ϑ times. Otherwise, without loss of generality, suppose V_{1} ∈ B_{σ_{i}} for i = 1, 2, ⋯, ϑ + 1. Then there exist τ_{i} ∈ C(σ_{i}), i = 1, 2, ⋯, ϑ + 1 such that ϕ(τ_{i}) ∈ V_{1} ∩ M(σ_{i}). Take R = |V_{1}| and consider the R-size component sets of F. Consequently, corresponding to each σ_{i}, i = 1, 2, ⋯, ϑ+1, there exists a R-size component set J_{ω_{i}} such that J_{ω_{i}} ⊆ J_{σ_{i}} and ϕ(τ_{i}) ∈ J_{ω_{i}}. Therefore, taking x ∈ V_{1}, we have that the ball B_{R}(x) intersects at least ϑ + 1 of the R-size component sets of F, which contradicts (C3). Thus

∑_{σ∈Ω^{m}} ∑_{V ∈B_{σ}} |V |^{t} ≤ ϑ ∑_{i} |V_{i}|^{t},

which leads to (29). From Lemma 3(I) it follows that

∑_{σ∈Ω^{m}} H_{Q}^{t}(M(σ)) ≥ h ∑_{σ∈Ω^{m}} H_{Q_{σ}}^{t}(M(σ)).  (30)

Thus by (27),(28),(29) and (30) we have for any m ∈ N

H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})) ≤ (∑_{i=1}^{r} a_{i}^{t})^{m}H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})) ≤ h^{−1}ϑH_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})). (31)

Note that 0 < H^{t}(M(Γ_{1}, ⋯, Γ_{k})) < ∞ if and only if 0 < H_{Q}^{t}(M(Γ_{1}, ⋯, Γ_{k})) < ∞ by Lemma 3(II). Hence we obtain ∑_{i=1}^{r} a_{i}^{t} = 1 by (31). (iv)⇒(ii). If dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = dim_{H}(F \ M(Γ_{1}, ⋯, Γ_{k})), then

dim_{H} F = max{dim_{H} M(Γ_{1}, ⋯, Γ_{k}), dim_{H}(F \ M(Γ_{1}, ⋯, Γ_{k}))} = t.

Thus (ii) holds. (ii)⇒(iv). On the other hand, since there exists at least one j, 1 ≤ j ≤ k, such that c_{j} > 0, without loss of generality we can assume c_{1} > 0. Fix 0 < ϵ < c_{1} and take non-negative real numbers e_{i}, 1 ≤ i ≤ k, such that

e_{1} = c_{1} − ϵ, e_{2} = c_{2} + ϵ, and e_{j} = c_{j}, 2 < j ≤ k.

By M^{∗}(Γ_{1}, ⋯, Γ_{k}) we denote the corresponding set with e_{j} in place of c_{j}, 1 ≤ j ≤ k. Write t(ϵ) = dim_{H} M^{∗}(Γ_{1}, ⋯, Γ_{k}) where t(ϵ) is such that

(c_{1} − ϵ) · log(c_{1} − ϵ) + (c_{2} + ϵ) · log(c_{2} + ϵ) + ∑_{j=3}^{k} c_{j} log c_{j}

= (c_{1} − ϵ) · log ∑_{i∈Γ_{1}} a_{i}^{t(ϵ)} + (c_{2} + ϵ) · log ∑_{i∈Γ_{2}} a_{i}^{t(ϵ)} + ∑_{j=3}^{k} c_{j} log ∑_{i∈Γ_{j}} a_{i}^{t(ϵ)},  (32)

by Theorem 1. Since M^{∗}(Γ_{1}, ⋯, Γ_{k}) ⊆ F \ M(Γ_{1}, ⋯, Γ_{k}) ⊆ F, we get

t(ϵ) = dim_{H} M^{∗}(Γ_{1}, ⋯, Γ_{k}) ≤ dim_{H}(F \ M(Γ_{1}, ⋯, Γ_{k})) ≤ dim_{H} F = t,  (33)

by (ii). Note that t(0) = t and t(ϵ) depends on ϵ continuously by (32). Then lim_{ϵ→0} t(ϵ) = t. So dim_{H}(F \ M(Γ_{1}, ⋯, Γ_{k})) = dim_{H} M(Γ_{1}, ⋯, Γ_{k}) = t by (33). QED

Finally we discuss Question 2.18 in [1] by R. Cawley & R. D. Mauldin. When k = r the sets Γ_{j}, 1 ≤ j ≤ r, are all singletons. In this case formula (4) reduces to

t = \frac{∑_{i=1}^{r} c_{i} · log c_{i}}{∑_{i=1}^{r} c_{i} · log a_{i}} .  (34)

Thus Theorem 1 yields

dim_{H} M(Γ_{1}, ⋯, Γ_{r}) = \frac{∑_{i=1}^{r} c_{i} · log c_{i}}{∑_{i=1}^{r} c_{i} · log a_{i}} .  (35)

Now take positive real numbers p_{i}, 1 ≤ i ≤ r such that ∑_{i=1}^{r} p_{i} = 1. For any q ∈ R let β(q) be the unique real number such that

∑_{i=1}^{r} p_{i}^{q}a_{i}^{β(q)} = 1.

Let α(q) = −β′(q) = [∑_{i=1}^{r} p_{i}^{q}a_{i}^{β(q)} log p_{i}]/[∑_{i=1}^{r} p_{i}^{q}a_{i}^{β(q)} log a_{i}] and f(q) = qα(q) + β(q). Taking c_{i} = p_{i}^{q}a_{i}^{β(q)}, 1 ≤ i ≤ r, it is easy to check t = f(q) by (34). Thus, denoting—with this choice of (c_{j})—M_{q} = M(Γ_{1}, ⋯, Γ_{r}), (35) shows dim_{H} M_{q} = f(q), which was given by R. Cawley & R.D. Mauldin ([1] ) for the multifractal decomposition of Moran fractals. Assume that \frac{log p_{1}}{log a_{1}} = ⋯ = \frac{log p_{r}}{log a_{r}} does not hold. In this case f(q) = s with ∑_{i=1}^{r} a_{i}^{s} = 1 if and only if q = 0 ([1]). Question 2.18 in [1] is: “If q ≠ 0, is it true that 0 < H^{f(q)}(M_{q}) < ∞?”. Here our Theorem 2 shows that 0 < H^{f(q)}(M_{q}) < ∞ if and only if q = 0.

Acknowledgement: The authors would like to thank the referees and a board member for their beneficial suggestions.

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