Separation properties for MW -fractals

Wenxia Li ^{∗} Department of Mathematics, Central China Normal University Wuhan 430079, P.R. China

Abstract

Corresponding to the irreducible 0 − 1 matrix (a_{ij})_{n×n}, take similitude contraction mappings φ_{ij} for each a_{ij} = 1, in R^{d} with ratio 0 < r_{ij} < 1. There exist unique nonempty compact sets F_{1}, ⋯, F_{n} satisfying for each 1 ≤ i ≤ n

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

We prove that open set condition holds if and only if F_{i} is an s-set for some 1 ≤ i ≤ n, where s is such that the spectral radius of matrix (r_{ij}^{s})_{n×n} is 1. Keywords: Open set condition; s-set; MW -fractal.

1 Introduction

The self-similar fractal is defined as the unique compact set F such that F = ⋃_{i=1}^{n} φ_{i}(F) where, for each 1 ≤ i ≤ n, φ_{i} is a similitude contraction mapping in R^{d} with similitude ratio 0 < r_{i} < 1. For self-similar fractal, A.Schief [1] has proved the following three conditions are equivalent:

(i) {φ_{i} : 1 ≤ i ≤ n} satisfies the open set condition;

(ii) {φ_{i} : 1 ≤ i ≤ n} satisfies the strong open set condition;

(iii) F is a t-set, where t is such that ∑_{i=1}^{n} r_{i}^{t} = 1.

Now let us consider the generalization of self-similar fractal, sometimes called MW - fractal. Let A_{n×n} = (a_{ij})_{n×n} be an irreducible 0 − 1 matrix. Let F = {φ_{ij} : 1 ≤ i, j ≤ n, a_{ij} = 1} be a collection of similitude contraction mappings in R^{d} with 0 < r_{ij} < 1 as the similitude ratio of φ_{ij}. Then there exist unique nonempty compact sets F_{1}, ⋯, F_{n} such that for each 1 ≤ i ≤ n

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

∗Supported by the National Science Foundation of China and Natural Science Foundation of Hubei Province. AMS Subject Classifications: 28A80.

We say F satisfies the open set condition if there are nonempty bounded open sets O_{1}, ⋯, O_{n} such that for each 1 ≤ i ≤ n

O_{i} ⊃ ⋃_{1≤j≤n,a_{ij}=1} φ_{ij}(O_{j}) and φ_{ij_{1}}(O_{j_{1}}) ⋂ φ_{ij_{2}}(O_{j_{2}}) = ∅ if j_{1} ≠ j_{2}

and F satisfies the strong open set condition if furthermore F_{i} ⋂ O_{i} ≠ ∅ for some 1 ≤ i ≤ n. Obviously when F satisfies the strong open set condition, F_{i} ⋂ O_{i} ≠ ∅ for all 1 ≤ i ≤ n by (1).

It is known that if F satisfies the open set condition, then F_{i} are s-set for all 1 ≤ i ≤ n, where s is such that the spectral radius of matrix (r_{ij}^{s})_{n×n} is 1 (we adopt the convention that the (i, j)-th entry of matrix (r_{ij}^{s})_{n×n} is 0 if a_{ij} = 0) (see [2][3]). It is worth to notice that from the irreducibility of A_{n×n} it follows that number 1 is the eigenvalue with the largest absolute value of (r_{ij}^{s})_{n×n} which has strictly positive left and right eigenvectors (see Theorem 0.16 in [4]).

In this paper, we will show that the similar results hold for MW -fractals, i.e. the following three conditions are equivalent:

(i) F satisfies the open set condition;

(ii) F satisfies the strong open set condition;

(iii) F_{i} is an s-set for some 1 ≤ i ≤ n.

Now let us introduce some notations and definitions as follows.

1) Ω = {σ = (σ(1), σ(2), ⋯) : a_{σ(k),σ(k+1)} = 1 for k ∈ N}; Ω_{k} = {σ = (σ(1), ⋯, σ(k)) : a_{σ(l),σ(l+1)} = 1 for 1 ≤ l ≤ k − 1} and Ω^{∗} = ⋃_{k≥2} Ω_{k}. For σ ∈ Ω^{∗}, |σ| denotes the word length and the σ(k), 1 ≤ k ≤ |σ|, the k-th component of σ. When k ≤ |σ|, we denote σ|k = (σ(1), ⋯, σ(k)). Let Ω_{k}^{∗} = {σ ∈ Ω^{∗} : σ(1) = k} for 1 ≤ k ≤ n, and Ω̃^{∗} = ∏_{k=1}^{n} Ω_{k}^{∗}. σ̃ denotes the element of Ω̃^{∗} with σ_{k} ∈ Ω_{k}^{∗}, 1 ≤ k ≤ n, being the k-th component of σ̃.

2) For any σ ∈ Ω_{k}, k ≥ 2, r_{σ} = ∏_{l=1}^{k−1} r_{σ(l),σ(l+1)}; φ_{σ} = φ_{σ(1),σ(2)} ∘ φ_{σ(2),σ(3)} ∘ ⋯ ∘ φ_{σ(k−1),σ(k)}; F_{σ} = φ_{σ}(F_{σ(k)}).

3) For any σ, τ ∈ Ω^{∗}, σ and τ are called to be connected if and only if σ(|σ|) = τ(1), and define στ = (σ(1), ⋯, σ(|σ|), τ(2), ⋯, τ(|τ|)). It is easy to see that for connected σ and τ, |στ| = |σ| + |τ| − 1, r_{στ} = r_{σ}r_{τ}, φ_{στ} = φ_{σ} ∘ φ_{τ} and F_{στ} = φ_{σ}(F_{τ}).

4) For any two different σ, τ ∈ Ω^{∗}, we say σ and τ are incomparable if there exists no ω ∈ Ω^{∗} such that σ = τω or τ = σω. We adopt the convention that σ and σ are comparable. For any σ̃, τ̃ ∈ Ω̃^{∗}, we say σ̃ and τ̃ are incomparable if σ_{k} and τ_{k} are incomparable for all 1 ≤ k ≤ n. For A ⊂ Ω̃^{∗}, A is termed as incomparable subset of Ω̃^{∗} if for any different σ̃, τ̃ ∈ A, σ̃ and τ̃ are incomparable.

5) For any σ̃, τ̃ ∈ Ω̃^{∗}, define σ̃τ̃ = ω̃ where ω_{k} = σ_{k}τ_{σ_{k}(|σ_{k}|)}. Then for any σ̃, τ̃ ∈ Ω̃^{∗}, the notation σ̃τ̃ will work, but not for any σ, τ ∈ Ω^{∗}. Define rσ̃ = (r_{σ_{1}}, ⋯, r_{σ_{n}})′ for any σ̃ ∈ Ω̃^{∗}.

6) For (a_{1}, ⋯, a_{n})′, (b_{1}, ⋯, b_{n})′ ∈ R^{n}, we say (a_{1}, ⋯, a_{n})′ ≤ (b_{1}, ⋯, b_{n})′ if and only if a_{i} ≤ b_{i} for all 1 ≤ i ≤ n, and (a_{1}, ⋯, a_{n})′ < (b_{1}, ⋯, b_{n})′ if and only if a_{i} < b_{i} for all 1 ≤ i ≤ n.

7) We use d(·, ·) to denote the Euclidean metric in R^{d}, and D(·, ·) the Hausdorff metric in the complete metric space H(R^{d}) of nonempty compact subsets of R^{d}. For positive real number r and set E, write B(r, E) = {y ∈ R^{d} : d(y, E) < r}. Thus B(r, {x}) is an open ball with centre at x and radius r.

8) In the following, denote h = min_{1≤i≤n} H^{s}(F_{i})/ max_{1≤i≤n} H^{s}(F_{i}) and r_{0} = min_{a_{ij}=1} r_{ij}.

2 Main results

Proposition 2.1 If H^{s}(F_{i}) > 0 for some 1 ≤ i ≤ n, then H^{s}(F_{i}) > 0 for all 1 ≤ i ≤ n and

(r_{ij}^{s})_{n×n} ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) = ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) .  (2)

Proof It is easy to get that H^{s}(F_{i}) > 0 for all 1 ≤ i ≤ n and

( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) ≤ (r_{ij}^{s})_{n×n} ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) )  (3)

by (1). Now let (β_{1}, ⋯, β_{n}) be a strictly positive left eigenvector of matrix (r_{ij}^{s})_{n×n} relative to the eigenvalue 1. Therefore

(β_{1}, ⋯, β_{n}) ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) ≤ (β_{1}, ⋯, β_{n})(r_{ij}^{s})_{n×n} ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) = (β_{1}, ⋯, β_{n}) ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) )

which implies (3) takes equality. QED

By Proposition 2.1 we can get the following Corollary 2.2 immediately.

Corollary 2.2 For any incomparable σ, τ ∈ Ω_{k}^{∗}, 1 ≤ k ≤ n, we have H^{s}(F_{σ} ⋂ F_{τ}) = 0. QED

Now for any B ⊂ R^{d}, let

μ(B) = inf{∑_{i∈I} |U_{i}|^{s} : U_{i} open set and ⋃_{i∈I} U_{i} ⊃ B},

where |A| denotes the diameter of set A. The μ(·) is an outer measure (see [5]). Similar to the Proposition 3 of [6], we have

Proposition 2.3 For any Borel measurable set B_{i} ⊂ F_{i}, 1 ≤ i ≤ n, we have H^{s}(B_{i}) = μ(B_{i}).

Proof By the definition of μ(·) and H^{s}(·), it is clear that

μ(B_{i}) ≤ H^{s}(B_{i}).  (4)

Now let us consider the case B_{i} = F_{i}, 1 ≤ i ≤ n. We suffice to show that for each open covering {U_{j}(i)}_{j} of F_{i} and each δ > 0, there is another covering {V_{j}(i)}_{j} of F_{i} with |V_{j}(i)| < δ for each j and ∑_{j} |V_{j}(i)|^{s} = ∑_{j} |U_{j}(i)|^{s}. To do this, we choose k large enough such that r_{σ} < δ| ⋃_{i=1}^{n} ⋃_{j} U_{j}(i)|^{−1} for each σ ∈ Ω_{k}. We replace each open set family {U_{j}(i)}_{j} with ⋃_{σ∈Ω_{k},σ(1)=i} {φ_{σ}(U_{j}(σ(k)))}_{j}, denoted by {V_{j}(i)}_{j}, which covers the F_{i}. So

( ∑_{j} |V_{j}(1)|^{s} ; ⋮ ; ∑_{j} |V_{j}(n)|^{s} ) = (r_{ij}^{s})_{n×n}^{k} ( ∑_{j} |U_{j}(1)|^{s} ; ⋮ ; ∑_{j} |U_{j}(n)|^{s} ) ,

which leads that

(β_{1}, ⋯, β_{n}) ( ∑_{j} |V_{j}(1)|^{s} ; ⋮ ; ∑_{j} |V_{j}(n)|^{s} ) = (β_{1}, ⋯, β_{n}) ( ∑_{j} |U_{j}(1)|^{s} ; ⋮ ; ∑_{j} |U_{j}(n)|^{s} ) .

Therefore

(β_{1}, ⋯, β_{n}) ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) ≤ (β_{1}, ⋯, β_{n}) ( μ(F_{1}) ; ⋮ ; μ(F_{n}) ) .

Thus we have μ(F_{i}) = H^{s}(F_{i}) for 1 ≤ i ≤ n by (4).

Now for Borel measurable sets B_{i} ⊂ F_{i}, 1 ≤ i ≤ n, we have

( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) = ( H^{s}(B_{1}) ; ⋮ ; H^{s}(B_{n}) ) + ( H^{s}(F_{1}\B_{1}) ; ⋮ ; H^{s}(F_{n}\B_{n}) ) ≥ ( μ(B_{1}) ; ⋮ ; μ(B_{n}) ) + ( μ(F_{1}\B_{1}) ; ⋮ ; μ(F_{n}\B_{n}) )

≥ ( μ(F_{1}) ; ⋮ ; μ(F_{n}) ) = ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) )

which complete our proof by (4). QED

Theorem 2.4 If H^{s}(F_{i}) > 0 for some 1 ≤ i ≤ n, then F satisfies the strong open set condition.

Proof First it is easy to see that H^{s}(F_{i}) > 0 for some 1 ≤ i ≤ n implies that 0 < H^{s}(F_{i}) < ∞ for all 1 ≤ i ≤ n. On the other hand, it is not difficult to show that s = 0 if and only if F_{i} is singleton for all 1 ≤ i ≤ n. Then the strong open set condition holds when s = 0. In the following, we assume s > 0. Given 0 < p < 1, let

I_{p} = {σ ∈ Ω^{∗} : r_{σ} < p ≤ r_{σ|(|σ|−1)}}.

It is easy to see that any two different elements of I_{p} are incomparable. We divide the proof into the following four steps.

(a) Given x > 0, there exists open set families {U_{j}(i)}_{j}, 1 ≤ i ≤ n , such that ⋃_{j} U_{j}(i) ⊃ F_{i} and

( ∑_{j} |U_{j}(1)|^{s} ; ⋮ ; ∑_{j} |U_{j}(n)|^{s} ) ≤ (1 + x^{s}) ( H^{s}(F_{1}) ; ⋮ ; H^{s}(F_{n}) ) .  (5)

Writing V_{i} = ⋃_{j} U_{j}(i), 1 ≤ i ≤ n, let δ = min_{1≤i≤n} d(F_{i}, V_{i}^{c}). At first, we will show that if σ̃ and τ̃ are incomparable and h^{\frac{1}{s}} rτ̃ > xrσ̃, then

( D(F_{σ_{1}}, F_{τ_{1}}) ; ⋮ ; D(F_{σ_{n}}, F_{τ_{n}}) ) ≥ δ ( r_{σ_{1}} ; ⋮ ; r_{σ_{n}} ) .  (6)

Otherwise, without loss of generality we assume D(F_{σ_{1}}, F_{τ_{1}}) < δr_{σ_{1}}. Since

( d(F_{σ_{1}}, φ_{σ_{1}}(V_{σ_{1}(|σ_{1}|)}^{c})) ; ⋮ ; d(F_{σ_{n}}, φ_{σ_{n}}(V_{σ_{n}(|σ_{n}|)}^{c})) ) ≥ δ ( r_{σ_{1}} ; ⋮ ; r_{σ_{n}} ) ,

we have F_{τ_{1}} ⊂ φ_{σ_{1}}(V_{σ_{1}(|σ_{1}|)}). Thus by Corollary 2.2, Proposition 2.3 and (5)

(1 + x^{s})r_{σ_{1}}^{s}H^{s}(F_{σ_{1}(|σ_{1}|)}) < r_{σ_{1}}^{s}H^{s}(F_{σ_{1}(|σ_{1}|)}) + r_{τ_{1}}^{s}H^{s}(F_{τ_{1}(|τ_{1}|)}) = H^{s}(F_{σ_{1}}) + H^{s}(F_{τ_{1}})
= H^{s}(F_{σ_{1}} ⋃ F_{τ_{1}}) ≤ ∑_{j} |φ_{σ_{1}}(U_{j}(σ_{1}(|σ_{1}|)))|^{s}
= r_{σ_{1}}^{s} ∑_{j} |U_{j}(σ_{1}(|σ_{1}|))|^{s} ≤ (1 + x^{s})r_{σ_{1}}^{s}H^{s}(F_{σ_{1}(|σ_{1}|)})

which leads a contradiction.

(b) Let 0 < ε < \frac{1}{3}. For any σ̃ ∈ Ω̃^{∗}, writing p_{k} = |B(εr_{σ_{k}}, F_{σ_{k}})| and let

I^{∗}(σ̃) = {τ̃ ∈ Ω̃^{∗} : τ_{k} ∈ I_{p_{k}}, 1 ≤ k ≤ n},

I_{1}(σ̃) = {τ̃ ∈ Ω̃^{∗} : τ_{k} ∈ I_{p_{k}}, F_{τ_{k}} ⋂ B(εr_{σ_{k}}, F_{σ_{k}}) ≠ ∅, 1 ≤ k ≤ n}.

Note that for any τ̃, ω̃ ∈ I^{∗}(σ̃), τ̃ and ω̃ are incomparable if and only if τ_{k} ≠ ω_{k} for all 1 ≤ k ≤ n. Let I(σ̃) be the incomparable subset of I_{1}(σ̃) which has the largest cardinality. Thus there exists no element of I_{1}(σ̃) which is incomparable with each element of I(σ̃). Writing γ = sup_{σ̃∈Ω̃^{∗}} Card I(σ̃), we will show that γ < ∞. Now fix σ̃ ∈ Ω̃^{∗}. Then for any incomparable τ̃, ω̃ ∈ I^{∗}(σ̃), we have

h^{\frac{1}{s}} rτ̃ > h^{\frac{1}{s}} r_{0}rω̃.

Taking x = h^{\frac{1}{s}} r_{0} in (a), we can get the relating δ and

(D(F_{ω_{1}}, F_{τ_{1}}), ⋯, D(F_{ω_{n}}, F_{τ_{n}}))′ ≥ δrω̃

by (6). Therefore there exist y_{ω_{k}} ∈ F_{ω_{k}(|ω_{k}|)} and y_{τ_{k}}^{∗} ∈ F_{τ_{k}(|τ_{k}|)}, 1 ≤ k ≤ n, such that

d(φ_{ω_{k}}(y_{ω_{k}}), φ_{τ_{k}}(y_{τ_{k}}^{∗})) ≥ δr_{ω_{k}}.  (7)

Take z_{ω_{k}} ∈ F_{ω_{k}(|ω_{k}|)} and z_{τ_{k}}^{∗} ∈ F_{τ_{k}(|τ_{k}|)}, 1 ≤ k ≤ n, such that

d(y_{ω_{k}}, z_{ω_{k}}) < \frac{1}{3}δr_{0}, d(y_{τ_{k}}^{∗}, z_{τ_{k}}^{∗}) < \frac{1}{3}δr_{0}.  (8)

Then from (7) and (8) it follows that for 1 ≤ k ≤ n

d(φ_{ω_{k}}(z_{ω_{k}}), φ_{τ_{k}}(z_{τ_{k}}^{∗})) ≥ \frac{δ}{3}r_{ω_{k}}.

Now for each 1 ≤ k ≤ n we fix a finite set Z_{k} ⊂ F_{k} whose \frac{1}{3}δr_{0} neighborhood covers F_{k}. Therefore for any incomparable τ̃, ω̃ ∈ I^{∗}(σ̃), there exist z_{k} ∈ Z_{k}, 1 ≤ k ≤ n, such that

d(φ_{τ_{k}}(z_{τ_{k}(|τ_{k}|)}), φ_{ω_{k}}(z_{ω_{k}(|ω_{k}|)})) ≥ \frac{δ}{3}r_{τ_{k}} ≥ \frac{δ}{3}p_{k}r_{0}.

Now fix z_{k} ∈ Z_{k}, 1 ≤ k ≤ n, choose I ⊂ I(σ̃) such that for any different τ̃, ω̃ ∈ I

d(φ_{τ_{k}}(z_{τ_{k}(|τ_{k}|)}), φ_{ω_{k}}(z_{ω_{k}(|ω_{k}|)})) ≥ \frac{δ}{3}r_{τ_{k}} ≥ \frac{δ}{3}p_{k}r_{0}, 1 ≤ k ≤ n.

We observe that, for each 1 ≤ k ≤ n, the collection H_{k} = \{B(\frac{δ}{6}p_{k}r_{0}, φ_{τ_{k}}(z_{τ_{k}(|τ_{k}|)})), τ̃ ∈ I\} of open balls are pairwise disjoint and contained in B(p_{k} max_{1≤i≤n} |F_{i}|+ \frac{δ}{6}p_{k}r_{0}, B(εr_{σ_{k}}, F_{σ_{k}})). Thus

(\frac{1}{3}δp_{k}r_{0})^{d}CardH_{k} ≤ [2(p_{k} max_{1≤i≤n} |F_{i}| + \frac{δ}{6}p_{k}r_{0}) + p_{k}]^{d}.

Hence

CardH_{k} ≤ (\frac{6 max_{1≤k≤n} |F_{k}| + δr_{0} + 3}{δr_{0}})^{d},

which implies CardI ≤ ∏_{k=1}^{n} CardH_{k} is finite. Thus γ < ∞.

(c) Choose σ̃ such that γ = CardI(σ̃). For any τ̃ ∈ Ω̃^{∗}, it is not difficult to find that we can choose

I(τ̃σ̃) = {τ̃ω̃ : ω̃ ∈ I(σ̃)}  (9)

by maximality of I(σ̃). In the following the set I(τ̃σ̃) is always chosen as (9) for any τ̃ ∈ Ω̃^{∗}.

(d) For each 1 ≤ k ≤ n, letting

V_{k} = ⋃_{τ_{k}∈Ω_{k}^{∗}} B(\frac{ε}{2}r_{τ_{k}σ_{τ_{k}(|τ_{k}|)}}, F_{τ_{k}σ_{τ_{k}(|τ_{k}|)}}) = ⋃_{τ_{k}∈Ω_{k}^{∗}} φ_{τ_{k}}(B(\frac{ε}{2}r_{σ_{τ_{k}(|τ_{k}|)}}, F_{σ_{τ_{k}(|τ_{k}|)}})),

we will show that F satisfies strong open set condition with respect to these V_{k}. Obviously V_{k} ⋂ F_{k} ≠ ∅ for each 1 ≤ k ≤ n. Furthermore

φ_{ik}(V_{k}) = ⋃_{τ_{k}∈Ω_{k}^{∗}} B(\frac{ε}{2}r_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}, F_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}) ⊂ V_{i}.

In addition,

φ_{ik}(V_{k}) ⋂ φ_{ij}(V_{j}) = ∅

for each 1 ≤ i ≤ n and j ≠ k with a_{ik} = a_{ij} = 1. Otherwise there exist some 1 ≤ i ≤ n and τ_{k} ∈ Ω_{k}^{∗}, ω_{j} ∈ Ω_{j}^{∗} with k ≠ j such that

G = B(\frac{ε}{2}r_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}, F_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}) ⋂ B(\frac{ε}{2}r_{(i,j)ω_{j}σ_{ω_{j} (|ω_{j} |)}}, F_{(i,j)ω_{j}σ_{ω_{j} (|ω_{j} |)}}) ≠ ∅.

Without loss of generality, we assume r_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}} ≥ r_{(i,j)ω_{j}σ_{ω_{j} (|ω_{j} |)}}. Taking y ∈ G, there exist y_{1} ∈ F_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}} and y_{2} ∈ F_{(i,j)ω_{j}σ_{ω_{j} (|ω_{j} |)}} such that

d(y, y_{1}) < \frac{ε}{2}r_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}, d(y, y_{2}) < \frac{ε}{2}r_{(i,j)ω_{j}σ_{ω_{j} (|ω_{j} |)}}.

Then we have d(y_{1}, y_{2}) < εr_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}} which means

F_{(i,j)ω_{j}σ_{ω_{j} (|ω_{j} |)}} ⋂ B(εr_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}, F_{(i,k)τ_{k}σ_{τ_{k}(|τ_{k}|)}}) ≠ ∅.  (10)

For each 1 ≤ m ≤ n with m ≠ i, take α_{m} ∈ Ω_{m}^{∗} with α_{m}(|α_{m}|) = i and let ẽ, d̃ ∈ Ω̃^{∗} be such that

e_{m} = \{ \begin{matrix}(i, j)ω_{j}σ_{ω_{j}(|ω_{j}|)} & m = i \\ α_{m}(i, j)ω_{j}σ_{ω_{j}(|ω_{j}|)} & m ≠ i \end{matrix} and d_{m} = \{ \begin{matrix}(i, k)τ_{k} & m = i \\ α_{m}(i, k)τ_{k} & m ≠ i. \end{matrix}

Thus we have

F_{e_{m}} ⋂ B(εr_{d_{m}σ_{τ_{k}(|τ_{k}|)}}, F_{d_{m}σ_{τ_{k}(|τ_{k}|)}}) ≠ ∅, 1 ≤ m ≤ n

by (10). Therefore we can get a β̃ ∈ Ω̃^{∗} with e_{m} = β_{m}θ_{m} for all 1 ≤ m ≤ n such that β̃ ∈ I_{1}(d̃σ̃) and β̃ is incomparable with each element of I(d̃σ̃) by (9) which is in contradiction with the maximality of I(d̃σ̃). QED

Now for each 1 ≤ i ≤ n, we endow Ω(i) = {σ ∈ Ω : σ(1) = i} with the product σ-field and the canonical product probability measure μ̃_{i} such that for each cylinder set C(τ) = {σ ∈ Ω(i) : σ||τ| = τ} where τ ∈ Ω_{i}^{∗}

μ̃_{i}(C(τ)) = α_{i}^{−1}α_{τ(|τ|)} ∏_{k=1}^{|τ|−1} r_{τ(k),τ(k+1)}^{s},

where (α_{1}, ⋯, α_{n})′ is some strictly positive right eigenvector of matrix (r_{ij}^{s})_{n×n} corresponding to eigenvalue 1.

The natural probability measure μ_{i} of F_{i} are defined as the image measure of μ̃_{i} under the continuous surjective mapping π_{i} : Ω(i) → F_{i} with π_{i}(σ) = lim_{k→∞} φ_{σ|k}(0) ∈ F_{i}. Notice that if we omit the restriction of 0 < ε < 1/3 in step (b) of the above proof, the proof will still work although, at this time, I(σ̃) = ∅ possibly for some σ̃ ∈ Ω̃^{∗}. In addition, by the definition of open set V_{k} in step (d) it is easy to prove that μ_{k}(V_{k}) = 1. Hence we can get the following conditions are equivalent actually:

(1) F satisfies the strong open set conddition with open sets V_{k} such that μ_{k}(V_{k}) = 1, 1 ≤ k ≤ n; (2) F satisfies the strong open set conddition;

(3) F satisfies the open set conddition;

(4) F_{k} is a s-set for some 1 ≤ k ≤ n;

(5) F_{k} are s-sets for all 1 ≤ k ≤ n;

(6) For each ε > 0, γ < ∞ holds;

(7) There exists ε > 0 such that γ < ∞. Acknowlegement The author would like to thank Prof. Zhiyin Wen for his good suggestions and comments.

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