The Dimensions Of Self-Similar Sets ^{∗}

Wenxia Li, Dongmei Xiao Dept. of Math., University of California at Berkeley, CA 94720, USA.

Abstract

For self-similar set F we prove that dim_{H} F = dim_{B} F = dim_{P} F using different method from Fa[4] and give implicitly the dimension value even if the open set condition isn’t satisfied.

1 Introduction

Let ϕ_{i} be similar contraction mappings in R^{d} with ratios c_{i} ,1 ≤ i ≤ n.Hu[5] proved that there exists unique compact set F ⊂ R^{d} such that

F = ⋃_{i=1}^{n} ϕ_{i}(F).  (1)

Further dim_{H}F = dim_{B}F = dim_{P} F = s and F is an s-set where s is such that

∑_{i=1}^{n} c_{i}^{s} = 1,  (2)

if ϕ_{i}’s satisfy the open set condition, i.e., there is a bounded nonempty open set O such that

⋃_{i=1}^{n} ϕ_{i}(O) ⊂ O  (3)

with the left hand is disjoint union. Recently Sh[10] proved that F is an s-set here ∑_{i=1}^{n} c_{i}^{s} = 1 if and only if ϕ_{i}’s satisfy the open condition.

Now for ϵ > 0 write

Ω(ϵ) = {σ ∈ S^{∗} | c_{σ} ≤ ϵ and c_{σ|(|σ|−1)} > ϵ},

where S^{∗} = ⋃_{i=1}^{∞}{1, 2, ⋯, n}^{i} and c_{σ} = c_{σ(1)}c_{σ(2)} ⋯ c_{σ(k)} for σ = (σ(1), σ(2), ⋯, σ(k)) ∈ S^{∗}. And for σ ∈ S^{∗}, |σ| denotes the length of σ and σ|k = (σ(1), ⋯, σ(k)) for k ≤ |σ|. Let A ⊂ R^{d} be a bounded open set with A ⊃ F.It is easy to see that c_{0}ϵ < c_{σ} ≤ ϵ for any σ ∈ Ω(ϵ) where c_{0} = min_{1≤i≤n}c_{i}. We introduce nonnegative real numbers α_{0}(A) and β_{0}(A) as follows

∗Partially supported by a grant from the National Science Foundation of China. AMS Subject Classifications: 28A80

α_{0}(A) = sup\{α | \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α)}} = ∞\},  (4)

β_{0}(A) = sup\{β | \overline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−β)}} = ∞\},  (5)

where ϕ_{σ} = ϕ_{σ(1)} ∘ ϕ_{σ(2)} ∘ ⋯ ∘ ϕ_{σ(k)} for σ = (σ(1), σ(2), ⋯, σ(k)) ∈ S^{∗} and m_{d}(B) is the Lebesque measure of B ⊂ R^{d}.

In this paper we prove

(i) α_{0}(A) and β_{0}(A) are independent of the choice of A and α_{0}(A) = β_{0}(A).We denote the common value by α_{0};

(ii) dim_{H}F = dim_{B}F = dim_{P} F = α_{0}s; (For self-similar set F Fa[4] has proved that its Hausdorff dimension,Box dimension and Packing dimension are equal)

(iii) H^{α_{0}s}(F) < ∞ iff \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} < ∞;

(iv) If H^{α_{0}s}(F) > 0 then \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} > 0;

(v) We generalize this dimension results into the cases of MW-construction (Ma & Wi[9]) and recurrent sets (De[2],Be[1] and Wen[11]).

2 Demensions of self-similar set

It is easy to get the following

Proposition 2.1

α_{0}(A) = inf\{α | \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α)}} = 0\},

β_{0}(A) = inf\{β | \overline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−β)}} = 0\}.

Proposition 2.2 0 ≤ α_{0}(A) ≤ 1; 0 ≤ β_{0}(A) ≤ 1.

Proof Note that ∑_{σ∈Ω(ϵ)} c_{σ}^{s} = 1. Taking α = 0 then

\underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s}} = \underline{lim}_{ϵ→0}ϵ^{−d}m_{d}( ⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A)) ≥ c

for some positive constant c. Thus α_{0}(A) ≥ 0. On the other hand, taking α = 1, we have

\underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{CardΩ(ϵ)} ≤ c

for some constant c. Thus α_{0}(A) ≤ 1.

0 ≤ β_{0}(A) ≤ 1 can be proved by the same method. QED

Theorem 2.3

(i) α_{0}(A) and β_{0}(A) are independent of the choice of A and α_{0}(A) = β_{0}(A),denoting the common value by α_{0};

(ii) dim_{H}F = dim_{B}F = dim_{P} F = α_{0}s;

(iii) H^{α_{0}s}(F) < ∞ iff \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} < ∞;

(iv) If H^{α_{0}s}(F) > 0 then \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} > 0.

Proof (i) For B ⊂ R^{d} and ϵ > 0 let

B^{ϵ} = {x ∈ R^{d} : there exists y ∈ B such that ρ(x, y) < ϵ}

where ρ(x, y) is the Euclidean distance between x and y . Since A is a bounded open set containing set F, there are positive number δ_{1} and δ_{2} such that F^{δ_{1}} ⊂ A ⊂ F^{δ_{2}} which means α_{0}(F^{δ_{1}}) ≤ α_{0}(A) ≤ α_{0}(F^{δ_{2}}) and β_{0}(F^{δ_{1}}) ≤ β_{0}(A) ≤ β_{0}(F^{δ_{2}}). Thus it suffices to prove α_{0}(F^{δ}) and β_{0}(F^{δ}) are independent of the choice of positive number δ and α_{0}(F^{δ}) = β_{0}(F^{δ}), which follows from the proof of (ii).

(ii) Fixing x ∈ F and denoting the diameter of A by |A| we choose subfamily Ω^{∗}(ϵ) from Ω(ϵ) such that

(1)for any different σ, τ ∈ Ω^{∗}(ϵ), ρ(ϕ_{σ}(x), ϕ_{τ}(x)) > 4|A|ϵ;

(2) if σ ∈ Ω(ϵ) \ Ω^{∗}(ϵ) there exiets τ ∈ Ω^{∗}(ϵ) such that ρ(ϕ_{σ}(x), ϕ_{τ}(x)) ≤ 4|A|ϵ.

Let J(ϵ) = CardΩ^{∗}(ϵ). Thus

⋃_{σ∈Ω^{∗}(ϵ)} B(ϕ_{σ}(x), 5|A|ϵ) ⊃ ⋃_{σ∈Ω(ϵ)} B(ϕ_{σ}(x), |A|ϵ) ⊃ ⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A)

where B(x, r) denotes a ball in R^{d} with center at x and radiu r. Thus

J(ϵ)m_{d}B(ϕ_{σ}(x), 5|A|ϵ) ≥ m_{d} ⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A).

Therefore for any nonnegative real number α

J(ϵ)ϵ^{αs} ≥ \frac{c|A|^{−d}ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α)}},  (6)

where c is a positive constant. First we prove dim_{H}F ≥ α_{0}(A)s. It is clear when α_{0}(A) = 0. Suppose α_{0}(A) > 0 and take 0 < α < α_{0}(A). Thus by the definition of α_{0}(A) and (6) we can take ϵ_{1} > 0 such that

J(ϵ_{1})ϵ_{1}^{αs} ≥ 2c_{0}^{−αs}.  (7)

Considering any finite open c_{0}ϵ_{1}|A|-covering {V_{i}} of F , we have

(a) if there exists some V_{i} such that |V_{i}| ≥ (c_{0}ϵ_{1})^{2}|A| then

∑_{i} |V_{i}|^{αs} ≥ (c_{0}ϵ_{1})^{2αs}|A|^{αs};  (8)

(b) otherwise for each σ ∈ Ω^{∗}(ϵ_{1}) let V_{σ} = {V_{i} : V_{i} ∩ B(ϕ_{σ}(x), ϵ_{1}|A|) ≠ ∅}. Then V_{σ} is a covering of ϕ_{σ}(F) and any different σ, τ ∈ Ω^{∗}(ϵ_{1}), V_{σ} ⋂ V_{τ} = ∅. Take λ_{1} ∈ Ω^{∗}(ϵ_{1}) such that

∑_{V_{i}∈V_{λ_{1}}} |V_{i}|^{αs} = min_{σ∈Ω^{∗}(ϵ_{1})} ∑_{V_{i}∈V_{σ}} |V_{i}|^{αs}.

Therefore by (7) we have

∑_{i} |V_{i}|^{αs} ≥ J(ϵ_{1}) ∑_{V_{i}∈V_{λ_{1}}} |V_{i}|^{αs} ≥ 2c_{0}^{−αs}ϵ_{1}^{−αs} ∑_{V_{i}∈V_{λ_{1}}} |V_{i}|^{αs}
= 2(c_{λ_{1}}c_{0}^{−1}ϵ_{1}^{−1})^{αs} ∑_{V_{i}∈V_{λ_{1}}} |ϕ_{λ_{1}}^{−1}V_{i}|^{αs} ≥ 2 ∑_{V_{i}∈V_{λ_{1}}} |ϕ_{λ_{1}}^{−1}V_{i}|^{αs}.  (9)

Since V_{λ_{1}} is a covering of ϕ_{λ_{1}}(F), ϕ_{λ_{1}}^{−1}V_{λ_{1}} = {ϕ_{λ_{1}}^{−1}(V_{i}) : V_{i} ∈ V_{λ_{1}}} is a finite open c_{0}ϵ_{1}|A|-covering of F. As above we have

(a’) if there exists ϕ_{λ_{1}}^{−1}(V_{i}) ∈ ϕ_{λ_{1}}^{−1}V_{λ_{1}} such that |ϕ_{λ_{1}}^{−1}(V_{i})| ≥ (c_{0}ϵ_{1})^{2}|A| then (8) holds by (9);

(b’) othewise denote ϕ_{λ_{1}}^{−1}V_{λ_{1}} by {U_{i}}. Repeating the above step for the covering {U_{i}} of F and noticing that Card{V_{i}} is finite, thus (8) holds after finite steps. Consequently dim_{H}F ≥ αs which means dim_{H}F ≥ α_{0}(A)s.

Now taking δ_{1} > 0 we prove that dim_{H}F ≤ \underline{dim}_{B}F ≤ α_{0}(F^{δ_{1}})s. Letting α > α_{0}(F^{δ_{1}}) there exists sequence ϵ_{n} ↘ 0 such that

\frac{ϵ_{n}^{−d}m_{d}(⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(F^{δ_{1}}))}{∑_{σ∈Ω(ϵ_{n})} c_{σ}^{s(1−α)}} ≤ 1.

Thus

ϵ_{n}^{−d}m_{d}( ⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(F^{δ_{1}})) ≤ ∑_{σ∈Ω(ϵ_{n})} c_{σ}^{s(1−α)} ≤ (c_{0}ϵ_{n})^{−sα},

(c_{0}ϵ_{n})^{d−sα} ≥ c_{0}^{d}m_{d}( ⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(F^{δ_{1}})) ≥ c_{0}^{d}m_{d}(F^{c_{0}ϵ_{n}δ_{1}}),

d − sα ≤ \frac{log(m_{d}(F^{c_{0}ϵ_{n}δ_{1}})c_{0}^{d})}{log(c_{0}ϵ_{n})},

d − sα ≤ \overline{lim}_{n→0} \frac{log[c_{0}^{d}m_{d}(F^{c_{0}ϵ_{n}δ_{1}})]}{log(c_{0}ϵ_{n})} ≤ \overline{lim}_{ϵ→0} \frac{log[m_{d}(F^{c_{0}ϵδ_{1}})]}{log(c_{0}ϵδ_{1})},

which implies \underline{dim}_{B}F ≤ sα by the Proposition 3.2 of Fa[3]. Therefore \underline{dim}_{B}F ≤ sα_{0}(F^{δ_{1}}).

Repeating the above procedure of proof with β_{0}(A) instead of α_{0}(A) we can attain dim_{H}F ≥ β_{0}(A)s and dim_{H}F ≤ \overline{dim}_{B}F ≤ β_{0}(F^{δ_{1}})s for any given δ_{1} > 0. As a result,we get dim_{H} F = dim_{P} F = dim_{B} F = α_{0}(F^{δ_{1}})s = β_{0}(F^{δ_{1}})s for any given δ_{1} > 0 which indicates α_{0}(F^{δ_{1}}) and β_{0}(F^{δ_{1}}) are independent of the choice of δ_{1} > 0 and α_{0}(F^{δ_{1}}) = β_{0}(F^{δ_{1}}). Furthermore α_{0}(A) = β_{0}(A) and they are independent of the choice of open set A by (i).

(iii) Now we prove H^{α_{0}s}(F) < ∞ iff \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} < ∞.

Suppose that \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} = ∞. Then we can take ϵ_{1} > 0 such that (7)

holds with α_{0} instead of α. For any k ∈ N and for any finite open (c_{0}ϵ_{1})^{k}|A|-covering {V_{i}} of F, repeating k − 1 time steps of proof of the above we can get

∑_{i} |V_{i}|^{α_{0}s} ≥ 2^{k−1} ∑_{j} |U_{j}|^{α_{0}s},

where {U_{j}} is a finite open c_{0}ϵ_{1}|A|-covering of F. According to the same mahtod of (ii) after finite steps, saying l steps, we get

∑_{j} |U_{j}|^{α_{0}s} ≥ 2^{l}(c_{0}ϵ_{1})^{2α_{0}s}|A|^{α_{0}s}, ∑_{j} |V_{j}|^{α_{0}s} ≥ 2^{l+k−1}(c_{0}ϵ_{1})^{2α_{0}s}|A|^{α_{0}s},

which means H^{α_{0}s}(F) = ∞ if letting k tends to ∞.

Suppose H^{α_{0}s}(F) = ∞. Thus for any M > 0 there exists ϵ_{0} such that for any ϵ_{0}- covering {V_{i}} of F

∑_{i} |V_{i}|^{α_{0}s} > M.

On the other hand, for any ϵ > 0

J(ϵ)(ϵδ_{1})^{d} ≤ const.m_{d}(∪_{σ∈Ω(ϵ)}ϕ_{σ}(A)),

since ⋃_{σ∈Ω^{∗}(ϵ)} B(ϕ_{σ}(x), c_{0}ϵδ_{1}) ⊂ ⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A) where δ_{1} is such that F^{δ_{1}} ⊂ A. Thus

J(ϵ)ϵ^{α_{0}s} ≤ const.\frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}}.

Now taking ϵ such that 10ϵ|A| < ϵ_{0} and considering the covering {B(ϕ_{σ}(x), 5|A|ϵ), σ ∈ Ω^{∗}(ϵ)} of F which is a ϵ_{0}-covering of F, we have

∑_{σ∈Ω^{∗}(ϵ)} (10|A|ϵ)^{α_{0}s} = const.J(ϵ)ϵ^{α_{0}s} ≥ M.

Therefore

\frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} ≥ const.M,

for ϵ < (10|A|)^{−1}ϵ_{0} which indicates

lim_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} = ∞.

(iv) Suppose \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} = 0. Then for any h > 0 there exist sequence ϵ_{n} ↘ 0 such that

ϵ_{n}^{−d}m_{d}( ⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(A)) < h ∑_{σ∈Ω(ϵ_{n})} c_{σ}^{s(1−α_{0})} ≤ hc_{0}^{−α_{0}s}ϵ_{n}^{−α_{0}s}.

We consider the covering {B(ϕ_{σ}(x), 5ϵ_{n}|A|), σ ∈ Ω^{∗}(ϵ_{n})} of F. Since

⋃_{σ∈Ω^{∗}(ϵ_{n})} B(ϕ_{σ}(x), c_{0}ϵ_{n}δ_{1}) ⊂ ⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(A)

where δ_{1} is such that F^{δ_{1}} ⊂ A, then

J(ϵ_{n})m_{d}(B(ϕ_{σ}(x), c_{0}ϵ_{n}δ_{1})) ≤ m_{d}( ⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(A)),  (10)

J(ϵ_{n}) ≤ const.ϵ_{n}^{−d}m_{d}( ⋃_{σ∈Ω(ϵ_{n})} ϕ_{σ}(A)) ≤ const.hϵ_{n}^{−α_{0}s}.

Therefore we have

∑_{σ∈Ω^{∗}(ϵ_{n})} |B(ϕ_{σ}(x), 5ϵ_{n}|A|)|^{α_{0}s} = J(ϵ_{n})(10|A|ϵ_{n})^{α_{0}s} ≤ const.h,

which indicates H^{α_{0}s}(F) = 0. As a result we get H^{α_{0}s}(F) > 0 implies \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} > 0. QED

Conjecture: If \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} > 0 then H^{α_{0}s}(F) > 0.

Corollary 2.4 If ϕ_{i}’s satisfy the open set condition then dim_{H}F = dim_{B}F = dim_{P} F = s. Proof Let bounded nonempty open set O make ϕ_{i}’s satisfy the open set condition. Taking A = O^{1} thus

const. ≥ \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(O^{1}))}{CardΩ(ϵ)} ≥ \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(\overline{O}))}{CardΩ(ϵ)} ≥ const. > 0,

which means α_{0} = 1. Therefore dim_{H}F = dim_{B}F = dim_{P} F = s by Theorem 2.3. QED

Remark 2.5 If the above Conjecture holds then it is easy to get

(a) F is a α_{0}s-set iff 0 < \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{∑_{σ∈Ω(ϵ)} c_{σ}^{s(1−α_{0})}} < ∞;

(b) ϕ_{i}’s satisfy the open set condition iff \underline{lim}_{ϵ→0}^{\frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{CardΩ(ϵ)}} > 0;

(c) H^{s}(F) = 0 iff \underline{lim}_{ϵ→0}^{\frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω(ϵ)} ϕ_{σ}(A))}{CardΩ(ϵ)}} = 0.

3 Generalization to MW-construction and generalized recurrent set

Let A_{n×n} = (a_{ij})_{n×n} be an irreducible 0 − 1 matrix. {ϕ_{ij} : a_{ij} = 1} is a family of similar maps in R^{d} with the ratio c_{ij} for ϕ_{ij}. Let s be such that the spectral radius of (a_{ij}c_{ij}^{s})_{n×n} is 1 where we take a_{ij}c_{ij}^{s} = 0 when a_{ij} = 0.Write

Ω_{A} = {σ ∈ ∏_{1}^{∞}{1, 2, ⋯, n} : σ = (σ(1), σ(2), ⋯), a_{σ(l),σ(l+1)} = 1, l ∈ N},

Ω_{A}^{∗} = {σ ∈ ⋃_{i=2}^{∞} {1, 2, ⋯, n}^{i} : σ = (σ(1), ⋯, σ(k)), a_{σ(l),σ(l+1)} = 1, 1 ≤ l ≤ k − 1}.

There exist unqiue compact sets F_{1}, F_{2}, ⋯, F_{n} which sometimes is called MW-construction such that

F_{i} = ⋃_{{j:a_{ij}=1}} ϕ_{ij}(F_{j}), 1 ≤ i ≤ n.  (11)

It is well-known that when {ϕ_{ij} : a_{ij} = 1} satisfy the open condition, i.e., there is nonempty bounded open sets O_{1}, O_{2}, ⋯, O_{n} such that

O_{i} ⊃ ⋃_{{j:a_{ij}=1}} ϕ_{ij}(O_{j}), 1 ≤ i ≤ n,

with the right hand being disjoint union,we have

dim_{H}F_{i} = dim_{B}F_{i} = dim_{P} F_{i} = s, 1 ≤ i ≤ n,

and F_{i} are all s-set.

Furthermore in Li[6] we prove that

Proposition 3.1 {ϕ_{ij} : a_{ij} = 1} satisfies the open set condition iff F_{i} is an s-set for some 1 ≤ i ≤ n where s is given above. QED

Now for 1 ≤ i ≤ n let

α_{i} = sup\{α : \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)}))}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α)}} = ∞\},  (12)

β_{i} = sup\{β : \overline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)}))}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−β)}} = ∞\},

where A_{i} ⊃ F_{i} are bounded open sets; |σ| denotes the lengh of σ; Ω_{i}(ϵ) = {σ ∈ Ω_{A}^{∗} : σ(1) = i, c_{σ} ≤ ϵ and c_{σ|(|σ|−1)} > ϵ}; c_{σ} = c_{σ(1),σ(2)}c_{σ(2),σ(3)} ⋯ c_{σ(|σ|−1),σ(|σ|)}; ϕ_{σ} = ϕ_{σ(1),σ(2)} ∘ ϕ_{σ(2),σ(3)} ∘ ⋯ ∘ ϕ_{σ(|σ|−1),σ(|σ|)}. Write c_{0} = min_{a_{ij}=1}c_{ij}.

In usual, we always take some bounded open set A with A ⊃ ⋃_{i} F_{i} instead of A_{i}’s in (12). Similarly it is easy to get Proposition 3.2 (1) 0 ≤ α_{i} ≤ β_{i} ≤ 1 for 1 ≤ i ≤ n; (2) When {ϕ_{ij} : a_{ij} = 1} satisfies the open set condition ,we have α_{i} = β_{i} = 1 for all 1 ≤ i ≤ n. QED

Similar to Theorem 2.3 we have

Theorem 3.3 (I) All α_{i} and β_{i} are equal, denoting by α_{0} the common value. And dim_{H}F_{i} = dim_{B}F_{i} = dim_{P} F_{i} = α_{0}s for 1 ≤ i ≤ n.

(II) H^{α_{0}s}(F_{i}) < ∞ for some 1 ≤ i ≤ n iff \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)})}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α_{0})}} < ∞ for some

1 ≤ i ≤ n. And if H^{α_{0}s}(F_{i}) > 0 for some 1 ≤ i ≤ n then \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)})}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α_{0})}} > 0

for all 1 ≤ i ≤ n.

Proof (I)Without loss of generality we suppose that α_{1} = min_{1≤i≤n} α_{i}, β_{1} = min_{1≤i≤n} β_{i}, β_{n} = max_{1≤i≤n} β_{i}.

Fix some j, 1 ≤ j ≤ n. First step we prove dim_{H}F_{j} ≥ α_{1}s. Taking x_{i} ∈ F_{i} and writing δ = max_{i}|F_{i}|. We choose the subfamily Ω_{i}^{∗}(ϵ) from Ω_{i}(ϵ) such that

(1) for any σ, τ ∈ Ω_{i}^{∗}(ϵ) and σ ≠ τ

ρ(ϕ_{σ}(x_{σ(|σ|)}), ϕ_{τ}(x_{τ(|τ|)})) > 4δϵ;

(2) if σ ∈ Ω_{i}(ϵ) \ Ω_{i}^{∗}(ϵ) there exists τ ∈ Ω_{i}^{∗}(ϵ) such that

ρ(ϕ_{σ}(x_{σ(|σ|)}), ϕ_{τ}(x_{τ(|τ|)})) ≤ 4δϵ.

Let J_{i}(ϵ) = CardΩ_{i}^{∗}(ϵ). Thus

⋃_{σ∈Ω_{i}^{∗}(ϵ)} B(ϕ_{σ}(x_{σ(|σ|)}), 5δϵ) ⊇ ⋃_{σ∈Ω_{i}(ϵ)} B(ϕ_{σ}(x_{σ(|σ|)}), δϵ) ⊇ ⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)}).

Therefore we have

J_{i}(ϵ)m_{d}B(ϕ_{σ}(x_{σ(|σ|)}), 5δϵ) ≥ m_{d}( ⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)})),

J_{i}(ϵ)ϵ^{αs} ≥ \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)})}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α)}}( ∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α)})δ^{−d}const.ϵ^{αs}.

Now let (m_{1}, ⋯, m_{n}) be the strictly positive right eigenvector responding to the eigenvalue 1. Then

(c_{ij}^{s}a_{ij})_{n×n} ( m_{1} ; ⋮ ; m_{n} ) = ( m_{1} ; ⋮ ; m_{n} ) .

Therefore

[\frac{minm_{i}}{maxm_{i}} ]^{2} ≤ ∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s} ≤ [\frac{maxm_{i}}{minm_{i}} ]^{2} .

In addition

1 ≤ (ϵc_{σ}^{−1})^{αs} ≤ c_{0}^{−αs}.

Therefore

J_{i}(ϵ)ϵ^{αs} ≥ \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)})}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α)}}δ^{−d}const.  (13)

If α_{1} = 0, it is trival. We assume α_{1} > 0 and take 0 < α < α_{1}. Thus we have

\underline{lim}_{ϵ→0}J_{i}(ϵ)ϵ^{αs} = ∞,

by (13) for 1 ≤ i ≤ n. Take ϵ_{1} > 0 such that J_{i}(ϵ_{1})ϵ_{1}^{αs}c_{0}^{αs} ≥ 2 for all 1 ≤ i ≤ n. Considering the arbitrary finite open c_{0}ϵ_{1}δ-covering {V_{i}} of F_{j}, thus

(a) if there exists some V_{i} with |V_{i}| ≥ (c_{0}ϵ_{1})^{2}δ then

∑_{i} |V_{i}|^{αs} ≥ (c_{0}ϵ_{1})^{2αs}δ^{αs};  (14)

(b) otherwise we have

∑_{i} |V_{i}|^{αs} = ϵ_{1}^{αs} ∑_{i} |ϵ_{1}^{−1}V_{i}|^{αs}.

For each σ ∈ Ω_{j}^{∗}(ϵ_{1}), let V_{σ} = {V_{i} : V_{i} ∩ B(ϕ_{σ}(x_{σ(|σ|)}), ϵ_{1}δ) ≠ ∅}. Thus V_{σ} is a covering of ϕ_{σ}(F_{σ(|σ|)}) and for any σ, τ ∈ Ω_{j}^{∗}(ϵ_{1}), σ ≠ τ, we have V_{σ} ∩ V_{τ} = ∅. Take λ_{1} ∈ Ω_{j}^{∗}(ϵ_{1}) such that

∑_{V_{i}∈V_{λ_{1}}} |V_{i}|^{αs} = min_{σ∈Ω_{j}^{∗}(ϵ_{1})} ∑_{V_{i}∈V_{σ}} |V_{i}|^{αs}.

Therefore

∑_{i} |V_{i}|^{αs} ≥ J(ϵ_{1}) ∑_{V_{i}∈V_{λ_{1}}} |V_{i}|^{αs} ≥ J(ϵ_{1})ϵ_{1}^{αs} ∑_{V_{i}∈V_{λ_{1}}} |ϵ_{1}^{−1}V_{i}|^{αs} ≥ 2c_{0}^{−αs} ∑_{V_{i}∈V_{λ_{1}}} |ϵ_{1}^{−1}V_{i}|^{αs}
= 2(c_{λ_{1}}c_{0}^{−1}ϵ_{1}^{−1})^{αs} ∑_{V_{i}∈V_{λ_{1}}} |ϕ_{λ_{1}}^{−1}V_{i}|^{αs} ≥ 2 ∑_{V_{i}∈V_{λ_{1}}} |ϕ_{λ_{1}}^{−1}V_{i}|^{αs}.  (15)

Since V_{λ_{1}} is a covering of ϕ_{λ_{1}}(F_{λ_{1}(|λ_{1}|)}), ϕ_{λ_{1}}^{−1}V_{λ_{1}} is a finite open c_{0}ϵ_{1}δ-covering of F_{λ_{1}(|λ_{1}|)}. Denoting ϕ_{λ_{1}}^{−1}V_{λ_{1}} by {u_{i}} as above we have

(a’) if there exists u_{i} ∈ ϕ_{λ_{1}}^{−1}V_{λ_{1}} such that |u_{i}| ≥ (c_{0}ϵ_{1})^{2}δ then (14) holds by (15).

(b’) otherwise repeating the above step and considering Card{V_{i}} finite, thus (14) holds after finite steps. Therefore dim_{H}F_{j} ≥ αs for any 0 < α < α_{1} which means dim_{H}F_{j} ≥ α_{1}s.

Similar to the proof of Theorem 2.3 we also get α_{1}s ≤ dim_{H}F_{j} ≤ \underline{dim}_{B}F_{j} ≤ α_{1}s and β_{1}s ≤ dim_{H} F_{j} ≤ \overline{dim}_{B}F_{j} ≤ β_{1}s and \overline{dim}_{B}F_{j} = β_{n}s. Thus we complete the proof. In addition it is easy to find that all α_{i}’s and β_{i}’s are equal and independent of the choice of A_{i}’ s.

(II) Finally using the same method as those in proof of Theorem 2.3 (III) and (IV) we can complete the proof of (II). QED Corollary 3.4 When {ϕ_{ij} : a_{ij} = 1} satisfies the open set condition, we have for every 0 ≤ i ≤ n, dim_{H}F_{i} = dim_{B}F_{i} = dim_{P} F_{i} = s. QED

Conjecture: if

\underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)}))}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α)}} > 0

for some 1 ≤ i ≤ n, then H^{αs}(F_{i}) > 0 for all 1 ≤ i ≤ n. Remark 3.5 (1) Since the recurrent set (Dekking [2]) and the generalized recurrent set (Li [8]) are all the special cases of MW-construction (Bedford [1] & Li [7]) the Theorem 3.3 also works there. Thus our Theomem 3.3 actually improves the main results of [11] [12] which discussed the lower bound of Hausdorff dimension of recurrent sets and self-similar sets.

(2) If the above conjecture is ture, it is easy to get

(a) F_{i} is an αs-set for some 1 ≤ i ≤ n iff for some 1 ≤ i ≤ n

0 < \underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)}))}{∑_{σ∈Ω_{i}(ϵ)} c_{σ}^{s(1−α)}} < ∞;

(b) F_{i}’s satisfy the open set condition iff for some 1 ≤ i ≤ n

\underline{lim}_{ϵ→0} \frac{ϵ^{−d}m_{d}(⋃_{σ∈Ω_{i}(ϵ)} ϕ_{σ}(A_{σ(|σ|)}))}{CardΩ_{i}(ϵ)} > 0.

Acknowledgement: The authors thank Prof. M.W.Hirsch for his kind help and support.

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