The dimension of subsets of Moran sets

determined by the success run behaviour of

their codings

Wenxia Li ^{∗} Department of Mathematics, Central China Normal University, Wuhan 430079, P.R. China F.M. Dekking ^{†} Thomas Stieltjes Institute of Mathematics and Delft University of Technology, ITS (SSOR) Mekelweg 4, 2628 CD Delft, The Netherlands

Abstract

Let F ⊂ R^{n} be a Moran set associated with the set {0 < a_{j} < 1, j = 0, 1, ⋯, r}. Let Γ be a non-empty subset of {0, 1, 2, ⋯, r} with non-empty complement. Associated with the behaviour of success run of symbols from Γ in the coding space {0, 1, ⋯, r}^{N} is a decomposition of F such that

F = ⋃_{t∈[0,+∞]} F_{t}.

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

Depending on F this might be a partition of F or almost a partition of F in the sense that sup_{x∈F} #{t : x ∈ F_{t}} < +∞. We prove that each F_{t} is dense in F, and dim_{H} F_{t} = dim_{P} F_{t} = dim_{B} F_{t} = dim_{H} F = dim_{P} F = dim_{B} F = s with ∑_{j=0}^{r} a_{j}^{s} = 1 . For L^{1}-a.e. t ∈ [0, +∞], H^{s}(F_{t}) = 0 and F_{t} is an s-set when t = −(log ∑_{j∈Γ} a_{j}^{s})^{−1}. Moreover, associated with this decomposition {F_{t} : t ∈ [0, +∞]} of F is a measurable function Y such that each F_{t} is a level set of Y. The fractal dimensions of the graph of Y are also determined.

AMS Subject Classifications: 28A80, 28A78.

1 Introduction

A basic task in Fractal geometry is to determine or estimate the various dimensions of fractal sets. Fractal dimensions are introduced to measure the sizes of fractal sets and are employed in many different disciplines. Unfortunately, it is very difficult to determine the exact fractal dimensions of general fractal sets. Many results on fractal dimensions are obtained for fractal sets with a special structure. Among them is a typical fractal structure called Moran set or Moran fractal. In order to describe Moran fractals let us introduce the following notations. Denote Ω = {0, 1, ⋯, r}. Here r is a positive integer.

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

(ii) Ω^{k} = {σ = (σ(1), σ(2), ⋯, σ(k)) : σ(i) ∈ 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|)). For any τ ∈ Ω^{∗}, σ ∈ Ω^{ω} write τ ∗ σ = (τ(1), ⋯, τ(|τ|), σ(1), σ(2), ⋯);

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

(v) For σ ∈ Ω^{k}, the cylinder set C(σ) with base σ is defined as C(σ) = {τ ∈ Ω^{ω} : τ|k = σ} for k ∈ N; (vi) Let h_{j} : R^{n} ⟶ R^{n}, 0 ≤ j ≤ r. Denote h_{σ}(x) = h_{σ(1)} ∘ ⋯ ∘ h_{σ(k)}(x) for σ ∈ Ω^{k} and

x ∈ R^{n}.

Fixing a nonempty compact set J ⊂ R^{n} with \overline{intJ} = J and positive real numbers 0 < a_{j} < 1, j = 0, 1, ⋯, r, the related Moran set (or Moran fractal) is defined in the following way. Step 1. For each σ ∈ Ω^{k}, k ∈ N, construct a compact set J_{σ} ⊂ R^{n} by induction:

• A family {J_{j} : j ∈ Ω} of nonoverlapping nonempty compact subsets of J is chosen such that \overline{intJ_{j}}=J_{j} and |J_{j}| = a_{j}|J| where | · | denotes the diameter of set.

• Suppose that J_{σ} is given for some σ ∈ Ω^{k}. Take a family {J_{σ∗j} : j ∈ Ω} of nonoverlapping nonempty compact subsets of J_{σ} such that \overline{intJ_{σ∗j}} = J_{σ∗j}, |J_{σ∗j}| = a_{j}|J_{σ}| and J_{σ∗j} contains an open ball of diameter c|J_{σ∗j}| where c is a positive constant independent of σ ∗ j. Step 2. The Moran fractal F associated with {0 < a_{j} < 1, j ∈ Ω} and the J_{σ}, σ ∈ Ω^{∗} is defined as the nonempty compact set

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

We shall refer to J_{σ} as a k-th level component set of F if σ ∈ Ω^{k}. Define ϕ: Ω^{ω} ⟶ R^{n} by

{ϕ(σ)} = ⋂_{k=1}^{∞} J_{σ|k}.  (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 an injection. An important property of ϕ is that there is positive constant c_{1}, independent of x ∈ F, such that

sup_{x∈F} #{ϕ^{−1}(x)} < c_{1}  (3)

by means of Lemma 9.2 in [5]. Let ρ be the usual metric on Ω^{ω} defined by

ρ(σ, τ) = 2^{− min{i:σ(i)≠τ(i)}},

with the convention ρ(σ, σ) = 0. Let F be equiped with the Euclidean metric. Then ϕ is continuous. Thus each x ∈ F can be encoded via ϕ: σ ∈ Ω^{ω} is called a location code of x ∈ F if ϕ(σ) = x (Note that there are multiple location codes for some x ∈ F). Therefore ϕ is also called the coding map and Ω^{ω} is called the code space (or symbolic space). Some comments about Moran fractals are listed below.

(C1) The two crucial requirements in Step 1 (nonoverlapping and open ball condition – an analogue of the open set condition for self-similar sets) make the structure of the Moran set simpler, close to that of self-similar sets and lead to the exclusion of the non-self-similar self-affine sets (ref. [2],[7],[13],[16],[17] and [18],etc.).

(C2) The Moran fractals considered here are more general than those defined in [3] where J_{σ∗i} is geometrically similar to J_{σ} and [19] where the open balls are required to be centered at the points on F.

(C3) A Moran fractal is termed map-specified if there exist similitude contractions h_{j}, j = 0, 1, ⋯, r, such that J_{σ} = h_{σ}(J) for any σ ∈ Ω^{∗}. In this case F is actually the selfsimilar set determined by {h_{j} : j ∈ Ω}, which satisfies the open set condition with respect to the open set O = intJ (i.e. ⋃_{j=0}^{r} h_{j}(O) ⊂ O with a disjoint union on the left) and the coding map ϕ in (2) can be changed into

\{ϕ(σ)\} = ⋂_{k=1}^{∞} h_{σ|k}(\overline{O}) = \{ lim_{k→∞} h_{σ|k}(0)\}.  (4)

(C4) Moran fractals F are regular fractals in the sense that it has been proven that dim_{H} F = dim_{P} F = dim_{B} F = s and F is an s−set with ∑_{j∈Ω} a_{j}^{s} = 1 (see [12], [14], [15] & [19]).

(C5) The more general Moran fractal structure proposed by Zhiying Wen can be produced by a similar method where the code space Ω^{ω} = ∏_{i=1}^{∞} Ω_{i} and corresponding to different Ω_{i} there are different scaling coefficients {0 < a_{i,j} < 1, j = 0, 1, ⋯, r_{i}}.

Some fractal dimension results of this generalized Moran sets can be found in [8],[9], [10],[11] and [14], etc. The class of generalized Moran sets clearly contains the class of Moran sets, and in fact is far larger than the class of Moran sets, since a generalized Moran set often has different fractal dimensions and is not an s−set.

For any E ⊆ F there exists Λ ⊂ Ω^{ω} such that E = ϕ(Λ). For certain Λ, it is possible to determine the fractal dimensions of the projections E = ϕ(Λ). Up to now solutions which depend on the structure of Λ are obtained at least in the following cases:

(S1) Let D = (d_{i,j}) be a (r + 1) × (r + 1) matrix with entries 0 and 1, and Λ = {σ ∈ Ω^{ω} : d_{σ(k),σ(k+1)} = 1 for all k ∈ N} (see [1], [15] & [19]);

(S2) Λ = ∏_{i=1}^{∞} Ω_{i}, where Ω_{i} is a non-empty subset of Ω for i = 1, 2, ⋯ (see [10] & [11]);

(S3) Let F be map-specified and Λ compact and invariant under the left-shift (see [7]).

Now let Γ ⊆ Ω = {0, 1, ⋯, r} be non-empty such that Γ^{c} ≠ ∅. For n ∈ N and σ ∈ Ω^{ω}, we define N_{n}(σ), the length of the Γ-run starting at n, by

N_{n}(σ) = \{ \begin{matrix}0, & if σ(n) ∉ Γ \\ k, & if σ(n) ∈ Γ, σ(n + 1) ∈ Γ, ⋯, σ(n + k − 1) ∈ Γ and σ(n + k) ∉ Γ \\ +∞, & if σ(k) ∈ Γ for k ≥ n. \end{matrix}

Let (Q(n)) be a sequence of real numbers such that Q(n) ↑ +∞ and Q(n) = o(n) as n ⟶ +∞. Let

Λ = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{N_{n}(σ)}{Q(n)} = 1\} .

The Hausdorff dimension dim_{H}(·), box dimension dim_{B}(·) and packing dimension dim_{P} (·) of the projection M = ϕ(Λ) will be determined in Section 2. Throughout this paper, log denotes the logarithm to base e. Now let us recall a probability result attributed to D.J. Newman (see [4], p.61):

Proposition In a sequence of independent Bernoulli trials {X_{n}} with success probability p ∈ (0, 1), define Ñ_{n} to be the length of the maximal success run commencing at trial n, that is

{Ñ_{n} = j} = {X_{n+j} = 0, X_{i} = 1, n ≤ i < n + j}, j ≥ 0.

Then

P \{lim sup_{n→∞} \frac{log p}{log n}Ñ_{n} = −1\} = 1.

Associated with this proposition a class of special subsets F_{t}, t ∈ [0, +∞], of F can be defined by letting

Λ_{t} = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{N_{n}(σ)}{log n} = t\} and F_{t} = ϕ(Λ_{t}).  (5)

The following properties of F_{t} will be proved in Section 2:

(P1) F = ⋃_{t∈[0,+∞]} F_{t} with sup_{x∈F} #{t : x ∈ F_{t}} < +∞;

(P2) Each F_{t} is dense in F;

(P3) dim_{H} F_{t} = dim_{P} F_{t} = dim_{B} F_{t} = s where s is defined by ∑_{j∈Ω} a_{j}^{s} = 1;

(P4) Moreover, if F is map-specified with similitudes h_{j}, j = 0, 1, ⋯, r, i.e., a self-similar set, then each F_{t} is invariant under {h_{j} : j = 0, 1, ⋯, r}, i.e.,

F_{t} = ⋃_{j=0}^{r} h_{j}(F_{t}).

On the other hand, the F_{t} can be considered as the level sets of the function Y (σ) = lim sup_{n→∞} \frac{N_{n}(σ)}{log n} defined on the probability space (Ω^{ω}, G^{ω}, P). This will be discussed in Section 3 and the following results will be obtained (let s be as in P3):

(P5) For L^{1}-a.e. t ∈ [0, +∞], H^{s}(F_{t}) = 0, but when t = t̂ = −(log ∑_{j∈Γ} a_{j}^{s})^{−1}, H^{s}(Ft̂) > 0, from this it follows directly that Ft̂ is an s-set;

(P6) Let G be the graph of Y. Then dim_{H} G = dim_{P} G = dim_{B} G = 1 + s.

2 Dimension of subsets of Moran sets

In this section, a general dimension result on a class of subsets of Moran sets is first obtained. Then this result will be applied to give a decomposition of Moran sets. The following proposition will be employed.

Proposition 2.1 Let M = ϕ(∏_{i=1}^{∞} Ω_{i}) where the Ω_{i} are non-empty subsets of Ω = {0, 1, ⋯, r}, i ∈ N. Let d(k) be such that

∏_{i=1}^{k}( ∑_{j∈Ω_{i}} a_{j}^{d(k)}) = 1.

Then dim_{H} M = lim inf_{k→∞} d(k) =^{Δ} \underline{d} (see [10]).

Proof This result can be found in [10] for the more general Moran fractal structure (see (C5)). Here a simplified proof is given for this special case. For any d > \underline{d}, there exists a sequence {n_{k} : k = 1, 2, ⋯} such that d(n_{k}) < d. For any δ > 0, we can take k large enough such that {J_{σ} : σ ∈ ∏_{i=1}^{n_{k}} Ω_{i}} is a δ-covering of M. Then

H_{δ}^{d}(M) ≤ _{σ∈∏_{i=1}^{n_{k}} Ω_{i}}^{∑} |J_{σ}|^{d} = |J|^{d} ∏_{i=1}^{n_{k}}( ∑_{j∈Ω_{i}} a_{j}^{d}) ≤ |J|^{d} ∏_{i=1}^{n_{k}}( ∑_{j∈Ω_{i}} a_{j}^{d(n_{k})}) = |J|^{d},

which implies that dim_{H} M ≤ d. So we have dim_{H} M ≤ \underline{d}. Now we turn to prove that dim_{H} M ≥ \underline{d}. Let us suppose without loss of generality that \underline{d} > 0. Then we only need to prove that for any fixed 0 < d < \underline{d}, dim_{H} M ≥ d. Let us construct a probability measure μ̃ on S^{ω} =^{Δ} ∏_{i=1}^{∞} Ω_{i} such that for any σ ∈ S^{k} =^{Δ} ∏_{i=1}^{k} Ω_{i}, k = 1, 2, ⋯

μ̃(C^{∗}(σ)) = \frac{∏_{i=1}^{k} a_{σ(i)}^{d}}{∏_{i=1}^{k}(∑_{j∈Ω_{i}} a_{j}^{d})},

where C^{∗}(σ) =^{Δ} {θ ∈ S^{ω} : θ|k = σ} is a cylinder in S^{ω} with base σ. Let μ be the image measure under ϕ defined in (2) and restricted to S^{ω} here. For ϵ > 0, write S_{ϵ} = {σ ∈

⋃_{k=1}^{∞} S^{k} : a_{σ} =^{Δ} ∏_{i=1}^{|σ|} a_{σ(i)} ≤ ϵ and a_{σ|(|σ|−1)} > ϵ}. For any x ∈ M, ϵ > 0, by B(x, ϵ) we

denote the closed ball with center at x and radius ϵ. Let L_{x,ϵ} =^{Δ} {σ ∈ S_{ϵ} : J_{σ} ⋂ B(x, ϵ) ≠ ∅}. Since M = ⋃_{σ∈S_{ϵ}}(M ⋂ J_{σ}) and J_{σ}, σ ∈ S_{ϵ} are pairwise nonoverlapping, there exists a finite constant c independent of x, ϵ such that 1 ≤ #L_{x,ϵ} ≤ c. Thus

μ(B(x, ϵ)) ≤ ∑_{σ∈L_{x,ϵ}} μ̃(C^{∗}(σ)) = ∑_{σ∈L_{x,ϵ}} \frac{∏_{k=1}^{|σ|} a_{σ(k)}^{d}}{∏_{i=1}^{|σ|}(∑_{j∈Ω_{i}} a_{j}^{d})} ≤ cϵ^{d},

where for the last inequality we use that ∏_{i=1}^{|σ|}(∑_{j∈Ω_{i}} a_{j}^{d}) ≥ 1 when ϵ is small enough, since d < \underline{d}. Then by Frostman’s lemma (see [5]) we obtain H^{d}(M) > 0 which implies that dim_{H} M ≥ d. QED

Remark 2.2 One can show furthermore that \overline{dim}_{B}M = dim_{P} M = lim sup_{k→∞} d(k) (see [11]), and that 0 < H\underline{d}(M) < +∞ if and only if 0 < lim inf_{k→∞} ∏_{i=1}^{k}(∑_{j∈Ω_{i}} a\underline{d} ; j) < +∞ (see [10] & [14]).

Theorem 2.3 Suppose Q(n) ↑ +∞ and Q(n) = o(n) as n ⟶ +∞. Let s be such that ∑_{j∈Ω} a_{j}^{s} = 1. Let M = ϕ(Λ) where

Λ = \{σ ∈ Ω^{ω} : lim sup_{n→∞} \frac{N_{n}(σ)}{Q(n)} = 1\} .

Then dim_{H} M = dim_{P} M = dim_{B} M = s, and M is dense in F.

Proof It suffices to prove dim_{H} M ≥ s since dim_{H} M ≤ dim_{P} M ≤ s and dim_{H} M ≤ \underline{dim}_{B}M ≤ \overline{dim}_{B}M ≤ s always hold. In the following, we shall complete our proof by proving that for any given 0 < d < s there exists a subset E = E_{d} of M such that dim_{H} E ≥ d. Let the constant c be defined by

c =^{Δ} max{log #Γ, log #Γ^{c}, | log ∑_{j∈Γ} a_{j}^{s}|, | log ∑_{j∈Γ^{c}} a_{j}^{s}|}.  (6)

Consider the non-negative strictly decreasing function

G(x) = log ∑_{j∈Ω} a_{j}^{x}, 0 ≤ x ≤ s.

Clearly G(x) ↓ 0 as x ↑ s. Now let ϵ be defined by \frac{6cϵ}{1−ϵ} = G(d). Then the solutions of the inequality

0 ≤ G(x) = log ∑_{j∈Ω} a_{j}^{x} ≤ \frac{6cϵ}{1 − ϵ}

will lie in [d, s]. Given any sequence of integers 0 < k_{1} < u_{1} < u_{1,1} < ⋯ < u_{1,n_{1}} < k_{2} < u_{2} < ⋯ < k_{i} < u_{i} < u_{i,1} < u_{i,2} < ⋯ < u_{i,n_{i}} < k_{i+1} < u_{i+1} < ⋯, we construct a set E as follows:

E = {x ∈ F : x(k_{i}), x(u_{i}), x(u_{i,j}) ∈ Γ^{c} and x(k) ∈ Γ for k_{i} < k < u_{i}, i ≥ 1, 1 ≤ j ≤ n_{i}}.

Here for convenience we use x(k) to denote the k-th component of a location code of x ∈ F. The set E is a closed subset of F. In the definition of E, when k_{i} < k < u_{i}, we call x(k) a Γ-choice; when k = k_{i}, u_{i} and u_{i,j}, x(k) is called a Γ^{c}-choice; and the rest of the x(k) are called Ω-choices. By respectively N_{Ω}(k), N_{Γ}(k) and N_{Γ^{c}}(k) we denote the total number of Ω-, Γ- and Γ^{c}- choices among the first k entries in a location code of a point in E . Thus we have that N_{Ω}(k) + N_{Γ}(k) + N_{Γ^{c}}(k) = k for k ∈ N. Note that for k_{i} − 1 ≤ k ≤ u_{i}, N_{Ω}(k) = N_{Ω}(k_{i} − 1). For convenience we put

f_{i} = N_{Ω}(k_{i} − 1).

Now define a sequence of positive integers b_{i}, i ∈ N, by

b_{i+1} = b_{i} + [Q(b_{i} + 1)],  (7)

where [Q(b_{i} +1)] is the largest integer less than Q(b_{i} +1). Here we take b_{1} large enough to ensure Q(b_{1} + 1) ≥ 2. So the b_{i} increase strictly and tend to +∞. We take the sequence of positive integers k_{1}, u_{1}, u_{1,1}, ⋯, u_{1,n_{1}}, k_{2}, u_{2}, ⋯ in the definition of E as the sequence b_{1}, b_{2}, · · ·. Now for any n ∈ N there is an i with b_{i} ≤ n < b_{i+1}. Thus for any σ ∈ Ω^{ω} with ϕ(σ) ∈ E, because N_{n}(σ) ≤ b_{i+1} − b_{i} − 1, N_{b_{i}}(σ) = 0 and Q is non-decreasing we have

lim sup_{n→∞} \frac{N_{n}(σ)}{Q(n)} ≤ lim sup_{i→∞} \frac{b_{i+1} − b_{i} − 1}{Q(b_{i} + 1)} = 1

by (7). On the other hand, for any σ ∈ Ω^{ω} with ϕ(σ) ∈ E we have

lim sup_{i→∞} \frac{N_{k_{i}+1}(σ)}{Q(k_{i} + 1)} = lim sup_{i→∞} \frac{u_{i} − k_{i} − 1}{Q(k_{i} + 1)} = 1

by (7). Therefore E ⊆ M. Note that this holds for any choice of n_{1}, n_{2}, · · · . We shall now make a choice for this sequence, based on the previously defined ϵ. Suppose that the n_{ℓ} are defined for ℓ = 1, 2, ⋯, i − 1, then also k_{i} and u_{i} are determined. Letting n_{i} vary, we have

lim_{n_{i}→∞} \frac{N_{Ω}(u_{i,n_{i}})}{u_{i,n_{i}}} = lim_{n_{i}→∞} \frac{N_{Ω}(u_{i}) + u_{i,n_{i}} − u_{i} − n_{i}}{u_{i,n_{i}}}
= 1 − lim_{n_{i}→∞} \frac{n_{i}}{u_{i,n_{i}}} = 1 − lim_{n_{i}→∞} \frac{n_{i} − (n_{i} − 1)}{u_{i,n_{i}} − u_{i,n_{i}−1}}
= 1 − lim_{n_{i}→∞} \frac{1}{[Q(u_{i,n_{i}−1} + 1)]} = 1.

Here we use Stolz’s theorem for the third equality above. Therefore we can choose n_{i} such that

f_{i+1} = N_{Ω}(k_{i+1} − 1) ≥ (1 − ϵ)k_{i+1}.  (8)

According to Proposition 2.1 we have dim_{H} E = lim inf_{k→∞} d(k) where d(k) satisfies

(∑_{j∈Ω} a_{j}^{d(k)})^{N_{Ω}(k)}( ∑_{j∈Γ^{c}} a_{j}^{d(k)})^{N_{Γ^{c}}(k)}(∑_{j∈Γ} a_{j}^{d(k)})^{N_{Γ}(k)} = 1.  (9)

Taking logs in (9), and using N_{Ω}(k) + N_{Γ}(k) + N_{Γ^{c}}(k) = k we get

log ∑_{j∈Ω} a_{j}^{d(k)} = −\frac{N_{Γ^{c}}(k)}{N_{Ω}(k)} log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − \frac{k − N_{Ω}(k) − N_{Γ^{c}}(k)}{N_{Ω}(k)} log ∑_{j∈Γ} a_{j}^{d(k)}.  (10)

We shall show that there exist an i^{∗} such that d(k) ≥ d when k ≥ k_{i^{∗}}. Then dim_{H} E ≥ d by Proposition 2.1. Now by the properties of (Q(n)) we can take i^{∗} such that for all i ≥ i^{∗}

\frac{[Q(k_{i} + 1)]}{k_{i}} ≤ ϵ and [Q(u_{i} + 1)] − 1 > \frac{− log ∑_{j∈Γ^{c}} a_{j}^{d}}{log ∑_{j∈Ω} a_{j}^{d}} .  (11)

We shall consider two cases for the k with k ≥ k_{i^{∗}}. CASE 1. For some i ≥ i^{∗} one has k_{i} ≤ k ≤ u_{i}. In this case N_{Ω}(k) = N_{Ω}(u_{i}) = f_{i} and hence the equality (10) can be written as

log ∑_{j∈Ω} a_{j}^{d(k)} = −\frac{N_{Γ^{c}}(k)}{f_{i}}(log ∑_{j∈Γ^{c}} a_{j}^{d(k)} − log ∑_{j∈Γ} a_{j}^{d(k)}) − \frac{k − f_{i}}{f_{i}} log ∑_{j∈Γ} a_{j}^{d(k)}.  (12)

Note that

0 ≤ \frac{N_{Γ^{c}}(k)}{f_{i}} = \frac{k − N_{Ω}(k) − N_{Γ}(k)}{f_{i}} ≤ \frac{k − f_{i}}{f_{i}} ,  (13)

and

0 ≤ \frac{k − f_{i}}{f_{i}} ≤ \frac{u_{i}}{f_{i}} − 1 = \frac{k_{i} + [Q(k_{i} + 1)]}{f_{i}} − 1 = \frac{k_{i}}{f_{i}} (1 + \frac{[Q(k_{i} + 1)]}{k_{i}}) − 1
≤ \frac{1}{1 − ϵ} (1 + \frac{[Q(k_{i} + 1)]}{k_{i}}) − 1 = \frac{ϵ}{1 − ϵ} + \frac{1}{1 − ϵ} · \frac{[Q(k_{i} + 1)]}{k_{i}}, (14)

by (8). Note that | log ∑_{j∈Γ^{c}} a_{j}^{d(k)}| ≤ c, | log ∑_{j∈Γ} a_{j}^{d(k)}| ≤ c by (6). Therefore when i ≥ i^{∗} and k_{i} ≤ k ≤ u_{i}

log ∑_{j∈Ω} a_{j}^{d(k)} ≤ \frac{2ϵ}{1 − ϵ} · 2c + \frac{2ϵ}{1 − ϵ} · c = \frac{6cϵ}{1 − ϵ},

by (12), (13), (14) and the first inequality of (11), which means that d(k) ≥ d. CASE 2. For some i ≥ i^{∗} one has u_{i} < k < k_{i+1}. First note that if d(k) ≥ d and x(k + 1) is a Ω-choice, then N_{Ω}(k + 1) = N_{Ω}(k) + 1, and hence we see directly from (9) that also d(k + 1) ≥ d since ∑_{j∈Ω} a_{j}^{d(k)} ≥ 1. To finish Case 2, it suffices therefore to show that for each k = u_{i,ℓ}, ℓ = 1, ⋯, n_{i} we have d(u_{i,ℓ}) ≥ d. Putting u_{i,0} = u_{i}, this will be done by showing that d(u_{i,ℓ}) ≥ d implies d(u_{i,ℓ+1}) ≥ d for ℓ = 0, ⋯, n_{i} − 1 (by Case 1 we have already u_{i,0} ≥ d). In fact, taking k = u_{i,ℓ} and k = u_{i,ℓ+1} in (9), we obtain

(∑_{j∈Ω} a_{j}^{d(u_{i,ℓ})})^{N_{Ω}(u_{i,ℓ})}( ∑_{j∈Γ^{c}} a_{j}^{d(u_{i,ℓ})})^{N_{Γ^{c}}(u_{i,ℓ})}(∑_{j∈Γ} a_{j}^{d(u_{i,ℓ})})^{N_{Γ}(u_{i,ℓ})} = 1,  (15)

and

(∑_{j∈Ω} a_{j}^{d(u_{i,ℓ+1})})^{N_{Ω}(u_{i,ℓ+1})}( ∑_{j∈Γ^{c}} a_{j}^{d(u_{i,ℓ+1})})^{N_{Γ^{c}}(u_{i,ℓ+1})}(∑_{j∈Γ} a_{j}^{d(u_{i,ℓ+1})})^{N_{Γ}(u_{i,ℓ+1})} = 1.  (16)

Now suppose that d(u_{i,ℓ+1}) < d. Then, since N_{Γ}(u_{i,ℓ+1}) = N_{Γ}(u_{i,ℓ}), N_{Γ^{c}}(u_{i,ℓ+1}) = N_{Γ^{c}}(u_{i,ℓ}) + 1 and N_{Ω}(u_{i,ℓ+1}) = N_{Ω}(u_{i,ℓ}) + [Q(u_{i,ℓ} + 1)] − 1, it follows from (15) and (16) that

(∑_{j∈Ω} a_{j}^{d(u_{i,ℓ+1})})^{[Q(u_{i,ℓ}+1)]−1}( ∑_{j∈Γ^{c}} a_{j}^{d(u_{i,ℓ+1})}) < 1.  (17)

But since Q(n) is increasing, and ∑_{j∈Ω} a_{j}^{d(u_{i,ℓ+1})} > 1, this implies

[Q(u_{i} + 1)] − 1 ≤ [Q(u_{i,ℓ} + 1)] − 1 ≤ \frac{− log ∑_{j∈Γ^{c}} a_{j}^{d(u_{i,ℓ+1})}}{log ∑_{j∈Ω} a_{j}^{d(u_{i,ℓ+1})}} ≤ \frac{− log ∑_{j∈Γ^{c}} a_{j}^{d}}{log ∑_{j∈Ω} a_{j}^{d}} ,

which is impossible by the second inequality of (11). So we must have d(u_{i,ℓ+1}) ≥ d, and we have finished the proof of Case 2. Hence we have obtained that dim_{H} E ≥ d, and completed the proof of the dimension result on M. Finally, the density result is derived directly from the fact that if σ ∈ Λ then for any k ∈ N those τ ∈ Ω^{ω} with τ(i) = σ(i), i ≥ k will lie in Λ. QED

As a Corollary to Theorem 2.3 we can now prove (P1), (P2), (P3) and (P4) of the previous section. Proof (of (P1)–(P4)). By (3) and the definition (5) of Λ_{t} we have ⋃_{t∈[0,+∞]} Λ_{t} = Ω^{ω} which leads to (P1). (P2) and (P3) can be directly derived from Theorem 2.3 by taking Q(n) = t log n for t ∈ (0, +∞), Q(n) = log log n for t = 0 and Q(n) = (log n)^{2} for t = +∞. Note that if σ ∈ Λ_{t} then for any j ∈ Ω we have j ∗ σ ∈ Λ_{t} since

lim sup_{n→∞} \frac{N_{n}(j ∗ σ)}{log n} = lim sup_{n→∞} [\frac{N_{n−1}(σ)}{log(n − 1)} · \frac{log(n − 1)}{log n}] = t.

Thus Λ_{t} = ⋃_{j=0}^{r} j ∗ Λ_{t} where j ∗ Λ_{t} = {j ∗ σ : σ ∈ Λ_{t}}, and we get (P4) by (4). QED

3 Level sets and Hausdorff measure

Let (p_{0}, p_{1}, ⋯, p_{r}) be a probability vector with each p_{j} > 0 and let (Ω^{ω}, G^{ω}, P) be the probability space so that G^{ω} is the σ-algebra generated by all cylinder sets of Ω^{ω} and the probability measure P is defined for any cylinder set C(τ) by

P (C(τ)) = ∏_{i=1}^{|τ|} p_{τ(i)}.  (18)

A sequence of independent Bernoulli trials {X_{n}} on the probability space (Ω^{ω}, G^{ω}, P) with success probability p can be defined by letting ∑_{j∈Γ} p_{j} = p and X_{n}(σ) = 1, if σ(n) ∈ Γ and X_{n}(σ) = 0, for σ(n) ∈ Γ^{c}. Therefore, taking t = −\frac{1}{log p} in (5) we have P (Λ_{t}) = 1 by the Proposition in Section 1. For each t ∈ (0, +∞), let (p_{0}, p_{1}, ⋯, p_{r}) be any fixed probability vector such that ∑_{j∈Γ} p_{j} = e^{−1/t} holds, then we define μ_{t}(B) = P (ϕ^{−1}(B)) for each Borel subset B ⊆ F. Then μ_{t}(F_{t}) = 1 where F_{t} is defined in (5). On the other hand, define random variables Y (σ), Y_{n}(σ), n ∈ N on (Ω^{ω}, G^{ω}, P) as follows:

Y_{n}(σ) = \frac{N_{n}(σ)}{log n} and Y (σ) = lim sup_{n→∞} Y_{n}(σ).

We have Λ_{t} = {σ ∈ Ω^{ω} : Y (σ) = t}, i.e., Λ_{t} is a level set of Y (σ), and so the projection F_{t} = ϕ(Λ_{t}) of Λ_{t} is, in some sense, a level set of Y (σ). We consider the set

G = {(ϕ(σ), Y (σ)) : σ ∈ Ω^{ω}} ⊆ R^{n} × (R ∪ {∞}).  (19)

If ϕ is invertible, then letting x = ϕ(σ) we have G = {(x, Y (ϕ^{−1}(x)) : x ∈ F }, i.e., G is the graph of the function Y (ϕ^{−1}(x)) on F. Note that G = ⋃_{t∈[0,+∞]}{(x, t) : x ∈ F_{t}} and dim_{H} F_{t} = s. We shall show that dim_{H} G = 1+s. The following proposition can be found in Proposition 7.9 of [5] where the proof is given for n = 2, which can easily be extended to the general case.

Proposition 3.1 Let H be a Borel subset of R^{n}, n ≥ 2. If 1 ≤ t ≤ n, then

∫_{−∞}^{∞} H^{t−1}(H ⋂ L_{x})dx ≤ (n − 1)^{\frac{t−1}{2}} H^{t}(H),

where L_{x} denotes the hyperplane {(x_{1}, ⋯, x_{n}) ∈ R^{n} : x_{n} = x}. QED

Theorem 3.2 Let G be as in (19). Then dim_{H} G = dim_{P} G = dim_{B} G = 1 + s.

Proof We have dim_{H} G ≥ 1 + s by Proposition 3.1 and Theorem 2.3. In fact, for any 1 ≤ t < 1 + s,

n^{\frac{t−1}{2}} H^{t}(G) ≥ ∫_{−∞}^{∞} H^{t−1}(G ⋂ L_{x})dx = ∫_{0}^{∞} H^{t−1}(F_{x})dx = +∞.

On the other hand, the product formula 7.5 in [5] gives

\overline{dim}_{B}G ≤ \overline{dim}_{B}(F × [0, +∞]) ≤ \overline{dim}_{B}F + \overline{dim}_{B}([0, +∞]) = 1 + s. QED

Since F is an s-set, H^{s}(F_{t}) ≤ H^{s}(F) < +∞, t ∈ [0, +∞]. We will consider the problem whether F_{t} is an s-set, i.e., whether or not H^{s}(F_{t}) > 0. By L^{1} we denote the onedimensional Lebesgue measure on the real line.

Theorem 3.3 For L^{1}-a.e. t ∈ [0, +∞], H^{s}(F_{t}) = 0, but when t = t̂ = −(log ∑_{j∈Γ} a_{j}^{s})^{−1}, H^{s}(Ft̂) > 0, i.e., Ft̂ is an s-set.

Proof We first prove the second part. Consider the probability space (Ω^{ω}, G^{ω}, P) where P is defined by (18) with p_{j} = a_{j}^{s}, j ∈ Ω. By the definition (5) of Ft̂ and the Proposition in Section 1, we get P (Λt̂) = 1. Let M̂t̂ = ϕ(Λ̂t̂) where

Λ̂t̂ = \{σ ∈ Λt̂: lim_{l→∞} \frac{#\{1 ≤ k ≤ l : σ(k) = j\}}{l} = a_{j}^{s}, j ∈ Ω\} .

Then M̂t̂ ⊆ Ft̂. Since P (Λt̂) = 1, we have P (Λ̂t̂) = 1 by the Law of Large Numbers or the Ergodic Theorem. Let μt̂ be the image of P under ϕ as before. By B_{r}(x) we denote the closed ball with centre at x and radius r. Let r be given small enough. Then for each σ ∈ Λ̂t̂ there exists a positive integer h(σ, r) such that

r · min_{j∈Ω} a_{j} ≤ a(σ|h(σ, r)) ≤ r,  (20)

where a(σ) =^{Δ} ∏_{k=1}^{|σ|} a_{σ(k)} for σ ∈ Ω^{∗}. Let W = {σ|h(σ, r) : σ ∈ Λ̂t̂}. For any fixed x ∈ M̂t̂

let W^{∗} = {τ ∈ W : J_{τ} ⋂ B_{r}(x) ⋂ M̂t̂ ≠ ∅}. Then there exists a finite positive constant ξ independent of r and x such that # W^{∗} ≤ ξ by Lemma 9.2 in [5]. Hence

μt̂(B_{r}(x)) ≤ P ( ⋃_{τ∈W^{∗}} C(τ)) ≤ ∑_{τ∈W^{∗}} P (C(τ)) = ∑_{τ∈W^{∗}} (a(τ))^{s} ≤ ξr^{s}

by (18) and (20). So we get

H^{s}(Ft̂) ≥ H^{s}(M̂t̂) > 0

by Frostman’s lemma. Now we turn to prove the first part. Consider the product measure space (F ×[0, +∞], B_{1}× B_{2}, H^{s} × L^{1}) of the σ-finite measure spaces (F, B_{1}, H^{s}) and ([0, +∞], B_{2}, L^{1}). Write G = ⋃_{t∈[0,+∞]}{(x, t) : x ∈ F_{t}} as before. By χ we denote the indicator of G. By Fubini’s Theorem we have

∫_{F} ∫_{[0,+∞]} χ(x, t)dL^{1}(t)dH^{s}(x) = ∫_{[0,+∞]} H^{s}(F_{t})dL^{1}(t).

Note that each fixed x ∈ F, χ(x, t) ≠ 0 only for finitely many t by (3). Then the left integral equals 0. Noting that H^{s}(F_{t}) ≥ 0 for t ∈ [0, +∞], we have that for L^{1}-a.e. t ∈ [0, +∞], H^{s}(F_{t}) = 0. QED

We conjecture that F_{t} is an s-set if and only if t = −(log ∑_{j∈Γ} a_{j}^{s})^{−1}.

As an application of the preceding, we end with an example of a measurable function T : [0, 1] ⟶ [0, +∞], such that each t-level set T^{−1}(t) ⊆ [0, 1], t ∈ [0, +∞] has Hausdorff dimension 1. Example. Take r = 2, Γ = {0, 2} and J = [0, 1]. Consider the map-specified Moran fractal F with h_{j}(x) = \frac{1}{3}x + \frac{j}{3}, x ∈ R^{1} and j = 0, 1, 2. Then we have F = [0, 1], a_{j} = \frac{1}{3}, j = 0, 1, 2 and s = 1. Let F_{t}, t ∈ [0, +∞], be defined by (5). Then we get a decomposition (F_{t}) of F satisfying the properties (P1)–(P4). Note that each x ∈ F, either has unique location code or it has only two location codes in Ω^{ω} = {0, 1, 2}×{0, 1, 2}×· · ·. In the former case, the unique location code has both infinitely many components in Γ and infinitely many components 1. So the corresponding x only lies in one of the sets F_{t}. But in the latter case, one of the two location codes only has components 0 except for finitely many components, the other only has components 2 except for finitely many components. So the corresponding x lies in F_{∞}. Hence (F_{t}) is a partition of F, satisfying the properties (P1)–(P4). Define the function T : [0, 1] ⟶ [0, ∞] by

T (x) = t if x ∈ F_{t}.

Then we get a measurable function T : [0, 1] ⟶ [0, +∞], such that each t-level set T^{−1}(t)(= F_{t}) ⊆ [0, 1], t ∈ [0, +∞] has Hausdorff dimension 1 and the graph of T (x) has Hausdorff dimension 2. Let t̂ = 1/(log 3 − log 2). Then we have H^{1}(Ft̂) > 0 by Theorem 3.3. In fact, in this special case we have

H^{1}(Ft̂) ≥ H^{1}(M̂t̂) = L^{1}(M̂t̂) = μt̂(M̂t̂) = 1,

since we always have H^{1}(·) = L^{1}(·) and μt̂ is actually the one-dimensional Lebesgue measure in this special case. Thus we get H^{1}(Ft̂) = 1 and H^{1}(F_{t}) = 0 for the other t. QED

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