Dimensions in Infinite Iterated Function Systems Consisting of Bi-Lipschitz Mapping

DIMENSIONS IN INFINITE ITERATED FUNCTION SYSTEMS CONSISTING OF BI-LIPSCHITZ MAPPING

CHIH-YUNG CHU# AND SZE-MAN NGAI*

Abstract. We study inﬁnite iterated functions systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorﬀ, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.

1. Introduction

The study of infinite iterated function systems form an important branch of the theory of IFSs. Mauldin and Urbanski [10] showed that fractal phenomena not exhibited by finite IFSs can appear in IIFSs, such as the existence of dimensionless limit sets, and that the box and packing dimensions are strictly greater than the Hausdorff dimension. They also showed that IIFSs can be used to study complex continued fractions. Fernau [7] showed that IIFSs have stronger descriptive power than finite ones; in fact, any closed set in a separable metric space is the attractor of some IIFS, while there exists a closed and bounded subset of a complete metric space that is the attractor of an IIFS but not of a finite IFS.

The open set condition (OSC) is assumed in [10]. Dimensions of IIFSs consisting of conformal contractions, but not satisfying (OSC), are studied [11] by Tong and the second author. The conformality condition is relaxed in [3], but the IFSs studied are ﬁnite.

The main purpose of this paper is to weaken all three assumptions, namely, (OSC), finiteness of the IFS, and conformality. We assume, as in [3], that the IFS maps are bi- Lipschitz essential contractions. By assuming an IIFS satisfies the logarithmic distortion property and a cardinality growth condition (CGC) (see definitions in Section 2), we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set, extending a theorem in [3].

2010 Mathematics Subject Classification. Primary: 28A80, 28A78. Key words and phrases. Fractal, infinite iterated function system, Hausdorff dimension. The first author is supported in part by a Shaanxi Natural Science Foundation of China (Grant No. 2017JM1018), a State Foundation for Studying Abroad (CSC Scholarship No. 201706305020). The second author is supported in part by the National Natural Science Foundation of China, grant 11771136, and Construct Program of the Key Discipline in Hunan Province. *Corresponding author. #The first author also publishes under the name Zhiyong Zhu. 1 2 CHIH-YUNG CHU# AND SZE-MAN NGAI*

For FIFSs of contractive similitudes satisfying (WSC), Zerner proved in [12] that if the attractor K is not contained in a hyperplane, then the Hausdorﬀ dimension of K is equal to the growth dimension of the IFS. This result is extended by Deng and Ngai [2] to conformal FIFSs and by Ngai and Tong [11] to IIFSs of contractive similitudes. In this paper, we generalize this result to IFSs of essential contractions (see Theorem 2.18).

This paper is organized as follows. In Section 2, we ﬁrst introduce some basic deﬁnitions, preliminary results, and assumptions, and then state the main results of this paper. In Section 3, we study the logarithmic distortion property (LDP) and a variant called the logarithmic* distortion property (L*DP). We also study two topological pressure functions. Sections 4 is devoted to the proof of the main theorems on dimension formula. In Section 5, we study the growth dimension and prove that, under suitable assumptions, it equals the other dimensions. Finally in Section 6, we illustrate our main theorems by a number examples.

2. Preliminaries and Statements of Main Results

In this section we ﬁrst introduce some basic deﬁnitions and notation. Then we state the main theorems in the paper.

d Deﬁnition 2.1. Let X be a non-empty compact subset of R , equipped with the Euclidean metric, and let I be a countable set with at least two elements, Si : X → X, i ∈ I, be a countable family of mappings. If there exists a metric ρ on X that is topologically equivalent to the Euclidean metric and a constant η ∈ (0, 1) such that

ρ(Si(x),Si(y)) ≤ ηρ(x, y) for all i ∈ I and x, y ∈ X, (2.1) then we call {Si}i∈I an iterated function system (IFS) of essential contractions with respect to ρ. We say that {Si}i∈I is an infinite iterated function system (IIFS) of essential contractions if I is countably infinite, and a finite iterated function system (FIFS) of essential contractions if I is finite.

Let 0 k I : = {∅},I := {(i1, . . . , ik): i1, . . . , ik ∈ I} for k ≥ 1, ∞ ∗ [ k ∞ I : = I ,I := {(i1, i2,...): ij ∈ I for all j ≥ 1}. k=0 We call I∗ the space of finite words with alphabets i ∈ I, and I∞ the coding space. For k i = (i1, . . . , ik) ∈ I , we denote by |i| = k the length of i and write Si := Si1 ◦ · · · ◦ Sik (S∅ is defined to be the identity). We often denote i = (i1, . . . , ik) simply by i = i1 . . . ik and let −m i := i1 . . . ik−m be the word obtained from i by deleting its last m(1 ≤ m ≤ k) alphabets. ∗ S ∞ If i ∈ I I and n ≥ 1 does not exceed the length of i, we denote by i|n the word i1 ··· in consisting of the first n alphabets of i. INFINITE ITERATED FUNCTION SYSTEMS 3

Since ρ is topologically equivalent to the Euclidean metric, (X, ρ) is a compact metric d d space in R and {Si}i∈I is a family of contraction mapping on (X, ρ). For E ⊂ R , we ∞ denote the diameter of E by diam(E) = supx,y∈E ρ(x, y). For each i = i1i2 · · · ∈ I , the compact sets Si|n (X), n ≥ 1, are decreasing and their diameters converge to zero. In fact, by (2.1), n diam(Si|n (X)) ≤ η diam(X). (2.2) This implies that the set ∞ \ π(i) = Si|n (X) n=1 is a singleton and therefore this formula deﬁnes a map π : I∞ → X, called the coding map, which, in view of (2.2), is continuous. Following [10] and [7], we deﬁne the limit set and attractor of an IFS as follows.

Definition 2.2. Let {Si}i∈I be an IFS described as above. Define the limit set as ∞ [ \ K := Si|n (X). i∈I∞ n=1 If K is closed, it is called the attractor (or fixed point) of the IFS.

The limit set K is a Souslin set and is hence measurable with respect to Hausdorff measure (see, [4]). K is independent of the metric ρ. Moreover, K is the attractor if and only if it S ∞ satisfies K = i∈I Si(K). Since Sj(π(i)) = π(ji) for any j ∈ I and i ∈ I , we have ∞ [ [ [ [ [ [ K = π(I ) = π ji = π(ji) = Sj(π(i)) j∈I i∈I∞ j∈I i∈I∞ j∈I i∈I∞ [ [ [ ∞ [ = Sj π(i) = Sj(π(I )) = Sj(K). j∈I i∈I∞ j∈I j∈I Thus the limit set K satisfies [ K = Si(K). i∈I Notice that if I is finite, then K is compact. Unlike FIFSs, K is not necessarily compact if I is countably infinite. Also, if K is not compact, it is not necessarily the unique non-empty d S bounded subset of R satisfying K = i∈I Si(K) (see, e.g., [11, Example 2.2]). It follows from Banach’s fixed point theorem and uniform contractivity that {Si}i∈I has a unique S fixed point K, and if U ⊂ X is a nonempty invariant set (i.e., i∈I Si(U) ⊂ U), then K ⊂ K ⊂ U. It is well known that if I is finite, then ∞ \ [ K = Si(X), (2.3) n=1 i∈In and thus K is a Borel set. When I is countably infinite, Mauldin and Urbańskiin [10] showed that (2.3) also holds under the assumption that the system {Si : i ∈ I} is pointwise finite (meaning that each element of X belongs to at most finitely many elements of Si(X)).

In this case, K is an Fσδ and is thus a Borel set. As pointed out in [10], K may even have a 4 CHIH-YUNG CHU# AND SZE-MAN NGAI* much more complicated descriptive set-theoretic structure when the system is not assumed to be pointwise ﬁnite. In this paper, we will consider such IFSs of essential contractions.

Let |·| denote the Euclidean metric. We write

|Si(x) − Si(y)| |Si(x) − Si(y)| ∗ ri := inf ,Ri := sup , i ∈ I , (2.4) x6=y∈X |x − y| x6=y∈X |x − y|

Rij ≤ RiRj, rij ≤ Rirj, rij ≤ riRj, (2.6) rij ≥ rirj,Rij ≥ riRj,Rij ≥ Rirj, which will be used repeatedly in the rest of this paper.

Assumption A. Throughout this paper we assume that ri > 0 for each i, equivalently, Si, i ∈ I, are bi-Lipschitz mappings with respect to Euclidean metric. In particular, r > 0 when I is ﬁnite.

Remark 2.3. It is possible that R ≥ 1. Since Si, i ∈ I, are essential contractions, Ri converges uniformly to 0 as |i| tends to inﬁnity. As a consequence, we also have 0 ≤ r < 1 and R < +∞.

d s d ◦ For any E ⊂ R , we use dimH(E), dimP(E), dimB(E), H (E), L , |E|, and E to denote, respectively, the Hausdorﬀ dimension, packing dimension, box dimension, s-dimensional Hausdorﬀ measure, d-dimensional Lebesgue measure, Euclidean diameter, and interior of E. For any set A, we let #A denote its cardinality. A set U ⊂ X is said to be open if it is open in the relative Euclidean topology of X.

Fix an invariant set U ⊂ X and let 0 < b < 1. Define ∗ SI,b := {i = (i1, . . . , in) ∈ I : Ri ≤ b < Ri− }, ∗ ∗ d d d d SI,b(U) := {i = (i1, . . . , in) ∈ I : L (Si(U)) ≤ b L (U) < L (Si− (U)}, (2.7) AI,b := {Si : i ∈ SI,b}, ∗ ∗ AI,b(U) := {Si : i ∈ SI,b(U)}. ∞ Remark 2.4. Since R can be greater than 1, for (i1, i2,...) ∈ I , there could be more than ∗ ∗ one prefix i = (i1, . . . , in) ∈ I such that i ∈ SI,b (resp. SI,b(U)). However, by Remark 2.3, the number of such prefixes must be finite.

0 ∗ Remark 2.5. It is possible that Si = Si0 for distinct i, i ∈ I ; we identify such Si and ∗ ∗ Si0 . For IFSs of contractive similitudes, SI,b = SI,b(U) and so AI,b = AI,b(U). In general, ∗ however, they need not be the same. For an IIFS, both AI,b and AI,b(U) are countably inﬁnite sets. INFINITE ITERATED FUNCTION SYSTEMS 5

d ◦ Deﬁnition 2.6. Let X ⊂ R be nonempty compact subset with X 6= ∅ and let Si : X → X, i ∈ I, be bi-Lipschitz essential contractions.

(a) We say that {Si}i∈I has the logarithmic distortion property (LDP) if there is a constant ξ > 0 such that b lim sup = 0. + ξ b→0 i∈SI,b ri |ln b|

(b) Let U ⊂ X be an invariant set. We say that {Si}i∈I has the logarithmic* distortion property (L*DP) if there is a constant ζ > 0 such that

b Ri lim sup ζ = 0 and lim sup ζ = 0. (2.8) b→0+ ∗ b→0+ ∗ i∈SI,b(U) ri |ln b| i∈SI,b(U) ri |ln b|

Mauldin and Urbański[10] defined the bounded distortion property (BDP) of an IFS of injective C1 conformal contractions. We extend this definition directly to IFSs of essential contractions as follows.

Definition 2.7. An IFS {Si}i∈I of essential contractions as in Definition 2.1 is said to have the bounded distortion property (BDP) if there exists a constant c > 0 such that R i ≤ c for all i ∈ I∗. ri Remark 2.8. In the above definitions, we do not assume that the IFS maps are differen- tiable.

Definition 2.9. Let X and {Si}i∈I satisfy the hypotheses of Definition 2.6, U ⊂ X be a bounded invariant set that is open in the relative topology of X with Ld(U) > 0, and Φ be ∗ a subset of {Si : i ∈ I }. We call a subcollection {ϕj}j∈J ⊂ Φ a packing family of Φ with respect to U if it satisfies the following two conditions:

(i) ϕj(U), j ∈ J, are pairwise disjoint;

(ii) for any ϕ ∈ Φ, ϕ(U) intersects at least one ϕj(U).

We denote by PI,U (b) the class of all packing families of AI,b with respect to U, and ∗ ∗ denote by PI,U (b) the class of all packing families of AI,b(U). Remark 2.10. The deﬁnition of packing familiy here is a generalization of that in [3] to the case I is countably inﬁnite.

For an index set I, we deﬁne, throughout the rest of this paper, F = F(I) := {J ⊂ I : J is ﬁnite}. (2.9)

∗ Let J ∈ F and {Si}i∈J be the associated FIFS. For any 0 < b < 1, let PJ,U (b) and PJ,U (b) be the corresponding packing families as in Deﬁnition 2.9. 6 CHIH-YUNG CHU# AND SZE-MAN NGAI*

Definition 2.11. Let X, {Si}i∈I and U satisfy the hypotheses of Definition 2.9. We say that {Si}i∈I satisfies cardinality growth condition (CGC) with respect to U if

ln(supΦ∈ (b) #Φ) γ := sup lim PJ,U < ∞. (2.10) J∈F b→0+ − ln b

We generalize the deﬁnition of topological pressure functions in [3] as follows.

Definition 2.12. Let X, {Si}i∈I and U satisfy the hypotheses of Definition 2.9 and fix λ ∈ (0, 1). Define 1 X Q (s) := sup lim ln inf Rs , s ∈ , λ n ϕ R n→∞ n Φ∈PJ,U (λ ) J∈F ϕ∈Φ 1 X s Qλ(s) := sup lim ln sup Rϕ , s ∈ R. n→∞ n n J∈F Φ∈PJ,U (λ ) ϕ∈Φ

We call Qλ (resp. Qλ) the lower (resp. upper) topological pressure function with scale λ. If Qλ = Qλ, we denote by Qλ the common function and call it a topological pressure function with scale λ. Note that λ is ﬁxed and s is the variable of the functions Qλ(s) and Qλ(s).

Similar to Deﬁnition 2.12, we deﬁne ∗ 1 X d s/d Qλ(s) := sup lim ln inf [L (ϕ(U))] , s ∈ R, n→∞ n Φ∈P∗ (λn) J∈F J,U ϕ∈Φ ∗ 1 X d s/d Qλ(s) := sup lim ln sup [L (ϕ(U))] , s ∈ R. J∈F n→∞ n ∗ n Φ∈PJ,U (λ ) ϕ∈Φ

Definition 2.13. Let X and {Si}i∈I satisfy the hypotheses of Definition 2.6. We say that {Si}i∈I satisfies the weak separation condition (WSC) if there exist an invariant subset ◦ D ⊂ X with D 6= ∅, called a WSC-region, and a constant κ ∈ N such that sup # ϕ ∈ AI,b : x ∈ ϕ(D) ≤ κ for all b ∈ (0, 1). (2.11) x∈X If E ⊂ X is an invariant subset and (2.11) holds, we call E a WSC-set.

J For any J ∈ F, we denote by K the attractor of the FIFS {Si}i∈J and write αJ := J dimH K .

Deﬁnition 2.14. Let X and {Si}i∈I satisfy the hypotheses of Deﬁnition 2.6. We call K a quasi s-set if HαJ (KJ ) < ∞ for any J ∈ F.

Remark 2.15. If {Si}i∈I satisﬁes (BDP) and (WSC), then it follows from [9, Theorem 3.2] that K is a quasi s-set.

We now state the ﬁrst two main results of this paper. INFINITE ITERATED FUNCTION SYSTEMS 7

Theorem 2.16. Let X, {Si}i∈I ,U satisfy the hypotheses of Deﬁnition 2.9. Assume that (LDP) and (CGC) hold. Then for any λ ∈ (0, 1) and any sequence of packing families J kn n {Sin,j }j=1 ∈ PJ,U (λ ), n ∈ N, the following hold:

(a) for all s ∈ R, J kn 1 X s Q (s) = Qλ(s) = sup lim ln R = (s − dimH K) ln λ; λ n→∞ n in,j J∈F j=1 (b) let γ be deﬁned in Deﬁnition 2.11, then J ln kn dimH K = dimP K = dimB K = γ = sup lim . J∈F n→∞ −n ln λ

Theorem 2.17. Let X, {Si}i∈I ,U satisfy the hypotheses of Deﬁnition 2.9. Assume that (LDP), (L*DP) and (CGC) hold. Then for any λ ∈ (0, 1) and any sequence of packing J kn ∗ n families {Sin,j }j=1 ∈ PJ,U (λ ), n ∈ N, we have J kn ∗ ∗ 1 X d s/d Q (s) = Q (s) = sup lim ln [L (Si (U)] = Qλ(s) λ λ n→∞ n n,j J∈F j=1 = s − dimH K ln λ for all s ∈ R.

d ◦ Let X ⊂ R be nonempty compact subset with X 6= ∅, and {Si}i∈I be a IFS of essential contractions on X (I is finite or countably infinite). Following Zerner [12], we define the growth dimension of an FIFS {Si}i∈I as

ln #AI,b dG = lim . b→0+ − ln b J For each ﬁnite set J ⊂ I, we denote by dG the growth dimension of the associated FIFS {Si}i∈J . Following [11], we deﬁne the growth dimension of the IIFS {Si(x)}i∈I (In this case,

#(AI,b) = ∞ for any b ∈ (0, 1)) as J dG = sup{dG : J ⊂ I is ﬁnite}. (2.12)

Theorem 2.18. Let X, {Si}i∈I ,U satisfy the hypotheses of Deﬁnition 2.9 and let K be the associated limit set. Assume that (LDP), (WSC) and (CGC) hold, and K is both a WSC-set and a quasi s-set. Then

dimH K = dimP K = dimB K = γ = dG, where dG is the growth dimension of K.

3. (LDP), (L*DP), and Topological Pressure Functions

3.1. Properties of (LDP) and (L*DP). For any E ⊂ X and any i ∈ I∗, the following inequalities will be used repeatedly,

ri |E| ≤ |Si(E)| ≤ Ri |E| , d 1/d d 1/d d 1/d (3.1) ri(L (E)) ≤ (L (Si(E))) ≤ Ri(L (E)) . 8 CHIH-YUNG CHU# AND SZE-MAN NGAI*

Lemma 3.1. Assume the same hypotheses on X and {Si}i∈I as in Definition 2.9, and assume that (LDP) holds. Let Ri and ri be defined as in (2.4), and ξ > 0 be defined as in Definition 2.6(a). Then for all b0 > 0 sufficiently small, there exists a constant c1 = c1(b0) > 0 such that b ξ ≤ ri ≤ Ri ≤ b for all i ∈ SI,b and b ∈ (0, b0). (3.2) c1 |ln b|

Moreover, c1(b0) → 0 as b0 → 0.

Proof. By Definition 2.6, for sufficiently small b0 > 0, b 0 < c1 := sup sup ξ < ∞. b∈(0,b0) i∈SI,b ri |ln b| b This implies that ξ ≤ c1 for any b ∈ (0, b0). Thus the conclusion follows from the ri|ln b| definition of SI,b and the fact ri ≤ Ri.

Lemma 3.2. Assume the same hypotheses on X, {Si}i∈I and U as in Definition 2.9, and assume that (L*DP) holds. Let Ri and ri be as in (2.4), and ζ > 0 be defined as in Definition 2.6(b). Then for all b0 > 0 sufficiently small, there is a constant c2 = c2(b0) > 0 such that

b ζ ∗ ζ ≤ ri ≤ Ri ≤ c2 |ln b| b for all i ∈ SI,b(U) and b ∈ (0, b0). (3.3) c2 |ln b|

Moreover, c2(b0) → 0 as b0 → 0.

Proof. By Deﬁnition 2.6(b), there exists a constant c3 > 0 such that for suﬃciently small b0 > 0, b 0 < c3 = sup sup < ∞ ∗ ζ b∈(0,b0) i∈SI,b(U) ri |ln b| and a constant c4 > 0 such that Ri 0 < c4 = sup sup < ∞. ∗ ζ b∈(0,b0) i∈SI,b(U) ri |ln b|

This implies that for any b ∈ (0, b0),

b Ri ζ ≤ c3 and ζ ≤ c4. (3.4) ri |ln b| ri |ln b| ∗ ∗ By the deﬁnition of SI,b(U) and (3.1), i ∈ SI,b(U) implies that Ld(S (U))1/d r ≤ i ≤ b. (3.5) i Ld(U)

By letting c2 := max{c3, c4} and combining (3.4), (3.5), and the fact that ri ≤ Ri, we obtain (3.3).

Remark 3.3. If {Si}i∈I satisﬁes (BDP), then there is a constant c > 0 such that Ri/ri ≤ c ∗ for all i ∈ SI,b ∪ SI,b(U). Thus from (3.2) and (3.3), respectively, (LDP) and (L*DP) hold. Examples of IFSs satisfying (LDP) and (L*DP) but not (BDP) will be given in Section 6. INFINITE ITERATED FUNCTION SYSTEMS 9

∞ ∞ ∞ Let σ : I → I denote the left shift map on I , i.e., σ(i) = i2i3 ··· for i = i1i2 ··· . ∗ Since it is possible that Rij ≥ Ri for some i, j ∈ I (see Remark 2.4), we have the following property.

Proposition 3.4. Let r < 1 and SI,Ri be deﬁned as in (2.5) and (2.7) respectively. Assume ∗ that r > 0. Then there exists an integer k0 > 0 such that for any i ∈ I with |i| > k0, there is a decomposition i = i1i2 with |i2| ≤ k0 such that i1 ∈ SI,Ri .

Proof. As mentioned in Remark 2.3, Ri → 0 uniformly as |i| → ∞. Thus there exists an ∗ ∗ integer k0 > 0 such that Ri ≤ r for all i ∈ I with |i| ≥ k0. Let i ∈ I with |i| = n > k0. First we check Ri−1 . If Ri < Ri−1 , then we take i1 = i and i2 = ∅, and thus the conclusion holds. If not, we check Ri−2 . If Ri < Ri−2 , then Ri−1 ≤ Ri < Ri−2 . Thus the conclusion holds by −1 n−1 letting i1 = i and i2 = σ (i). Continue. Note that there exists some 1 ≤ k ≤ n−1 such −k that R2−k+1 ≤ Ri < Ri , for otherwise, one would get r ≤ Rin−1 ≤ Ri < r, a contradiction. We further claim that there exists some k ∈ N with 1 ≤ k ≤ k0 such Ri−k+1 ≤ Ri < Ri−k . To see this we suppose on the contrary that there exists some ` ∈ N satisfying k0 +1 ≤ ` ≤ n−1 such that Ri−(`−1) ≤ Ri < Ri−` . Then one would get

Ri−(`−1) ≤ Ri ≤ Ri−(`−1) Rσn−(`−1)(i) ≤ Ri−(`−1) r < Ri−(`−1) , n−(`−1) where the third inequality is because |σ (i)| = ` − 1 ≥ k0. This contradiction proofs −k n−k the claim. The asserted result now follows by letting i1 = i and i2 = σ (i).

Lemma 3.5. Assume the same hypotheses on X, {Si}i∈I and U as in Deﬁnition 2.9 and r be deﬁned as in (2.5). If r > 0, then (LDP) implies (L*DP).

∗ Proof. Let k0 be as in Proposition 3.4. For t ∈ I with |t| = n ≥ k0, let n = lk0 + t with n l 0 ≤ t < k0. Then r ≤ rt ≤ Rt ≤ r . Taking logarithm, we have n ln r n ln r ≤ ln rt ≤ ln Rt ≤ l ln r ≤ , |t| ≥ k0. k0 This implies that ln R 1 t ≥ > 0. (3.6) ln rt k0

Let ξ > 0 be as in Definition 2.6(a) and b0 ∈ (0, 1) be sufficiently small. Since (LDP) holds, we have from Lemma 3.1 that there is a constant c1 > 0 such that b ξ ≤ ri ≤ Ri ≤ b for all i ∈ SI,b and b ∈ (0, b0). (3.7) c1 |ln b| ∗ To prove (2.8), we assume, without loss of generality, that |j| > k0 for any j ∈ SI,b(U) and ∗ Rj < b0, since we need only consider sufficiently small b > 0. By the definition of SI,b(U), we have d 1/d d 1/d L (S (U)) L (S −1 (U)) j ≤ b < j , Ld(U) Ld(U) and thus rj ≤ b < Rj−1 . As Rj ≥ Rj−1 r, we get −1 rj ≤ b < r Rj. (3.8) 10 CHIH-YUNG CHU# AND SZE-MAN NGAI*

Combining (3.6) and (3.8), we see that exists some constantc ˜ ≥ 1 such that ln R c˜−1 ≤ j ≤ c.˜ (3.9) ln b As |j| > k0, Proposition 3.4 implies that there exists a decomposition j = j1j2 with |j2| ≤ k0 such that j1 ∈ SI,Rj . As Rj < b0, substituting b and i in (3.7) by Rj and j1 respectively yields Rj ξ ≤ c1. rj1 |ln Rj|

Using (2.6) and the fact that j1 ∈ SRj and |j2| ≤ k0, we have

k0 |j2| r rj1 ≤ rj1 rj2 ≤ rj ≤ rj1 R . (3.10) Combining (3.8)–(3.10), we get −1 ξ ξ b r Rjc˜ c1c˜ ξ ≤ ξ ≤ k +1 =: c3 < ∞ (3.11) k0 r 0 rj |ln b| r rj1 |ln Rj| and ξ ξ ξ Rj Rj Rjc˜ c1c˜ c1c˜ ξ ≤ ξ ≤ ξ ≤ k ≤ k +1 = c3 < ∞. (3.12) k0 r 0 r 0 b |ln b| rj |ln b| r rj1 |ln Rj| Furthermore, we can see from the proof of Lemma 3.1 that c1 → 0 as b0 → 0. Therefore c3 → 0 as b0 → 0, and thus the (L*DP) holds from the (3.11), (3.12) and the fact b ∈ (0, b0).

3.2. Properties of topological pressures.

Proposition 3.6. Let X, {Si}i∈I and U satisfy the hypotheses of Deﬁnition 2.9. Let λ ∈

(0, 1) and r be deﬁned as in (2.5). Assume that r > 0 and (CGC) holds. Then both Qλ(s) and Qλ(s) are real-valued, strictly decreasing, and continuous functions on R that tend to

−∞ and ∞ as s tends to ∞ and −∞, respectively. Moreover, Qλ(0) ≥ Qλ(0) ≥ 0 and Qλ(s) is convex on R.

Proof. Let k0 be described as in Proposition 3.4. Write C := sup{Ri : |i| < k0}. Then n C < ∞. For any J ∈ F, let ϕ = Si1···ik ∈ Φ ∈ PJ,U (λ ). Let k − 1 = lk0 + m with l 0 k/k0 0 ≤ m < k0, where l ∈ N ∪ {0}. Then we have Ri1···ik−1 ≤ Cr ≤ C r for some constant C0. Hence

k n 0 k/k0 r ≤ rϕ ≤ Rϕ ≤ λ < Ri1···ik−1 ≤ C r . (3.13) It follows that 0 k < nk0 logr λ − k0 logr C , (3.14) and thus 0 nk0 log λ−k0 log C Rϕ ≥ r r r . (3.15) It follows that from (3.13) and (3.15) that when s ≥ 0,

0 (nk0 logr λ−k0 logr C )s X s ns r ≤ Rϕ ≤ #Φλ , ϕ∈Φ INFINITE ITERATED FUNCTION SYSTEMS 11 and when s < 0,

0 ns X s (nk0 logr λ−k0 logr C )s λ ≤ Rϕ ≤ #Φr . ϕ∈Φ Thus if s ≥ 0,

0 P s sk ln C ln(infΦ∈ (λn) R ) sk ln λ − 0 ≤ PJ,U ϕ∈Φ ϕ 0 n n P s ln(supΦ∈ (λn) Rϕ) ≤ PJ,U ϕ∈Φ n ln(supΦ∈ (λn) #Φ) ≤ PJ,U + s ln λ n ln(supΦ∈ (λn) #Φ) ≤ PJ,U (− ln λ) + s ln λ; − ln λn if s < 0, P s ln(infΦ∈ (λn) R ) s ln λ ≤ PJ,U ϕ∈Φ ϕ n P s ln(supΦ∈ (λn) Rϕ) ≤ PJ,U ϕ∈Φ n 0 ln(supΦ∈ (λn) #Φ) sk log C ln r ≤ PJ,U + sk log λ ln r − 0 r n 0 r n 0 ln(supΦ∈ (λn) #Φ) sk ln C = PJ,U (− ln λ) + sk ln λ − 0 . − ln λn 0 n

Letting n → ∞ and using (2.10) and the deﬁnitions of Qλ(s) and Qλ(s), we get

sk0 ln λ ≤ Qλ(s) ≤ Qλ(s) ≤ (s − γ) ln λ, if s ≥ 0;

s ln λ ≤ Qλ(s) ≤ Qλ(s) ≤ (sk0 − γ) ln λ, if s < 0.

Hence Qλ(s) and Qλ(s) are real-valued, Qλ(0) ≥ Qλ(0) ≥ 0. Moreover, since 0 < λ < 1, we have lims→∞ Qλ(s) = lims→∞ Qλ(s) = −∞ and lims→−∞ Qλ(s) = lims→−∞ Qλ(s) = ∞.

Next, since Qλ(s) and Qλ(s) are real-valued, for any δ > 0, we have 1 X s nδ Qλ(s + δ) ≤ sup lim ln sup Rϕλ n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ

= Qλ(s) + δ ln λ < Qλ(s), (3.16) and by (3.15), 1 0 X s δ(nk0 logr λ−k0 logr C ) Qλ(s + δ) ≥ sup lim ln sup Rϕr n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ

= Qλ(s) + δk0 ln λ, that is,

Qλ(s) + δk0 ln λ ≤ Qλ(s + δ) ≤ Qλ(s) + δ ln λ < Qλ(s). (3.17) 12 CHIH-YUNG CHU# AND SZE-MAN NGAI*

Moreover, for any δ > 0, we also have 1 0 X s −δ(nk0 logr λ−k0 logr C ) Qλ(s − δ) ≤ sup lim ln sup Rϕr n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ

≤ Qλ(s)) − δk0 ln λ,

1 X s −nδ Qλ(s − δ) ≥ sup lim ln sup Rϕλ n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ

≥ Qλ(s) − δ ln λ > Qλ(s), that is,

Qλ(s) < Qλ(s) − δ ln λ ≤ Qλ(s − δ) ≤ Qλ(s)) − δk0 ln λ. (3.18)

From (3.17) and (3.18), we see that Qλ(s) is strictly decreasing and continuous on R. We can get analogous results for Qλ(s) in the same way.

Finally, for any 0 < t < 1, s1, s2 ∈ R, we have by using Young’s inequality and the fact that ln x is convex, 1 X ts1 (1−t)s2 Qλ(ts1 + (1 − t)s2) = sup lim ln sup Rϕ Rϕ n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ 1 X s1 t s2 1−t = sup lim ln sup (Rϕ ) (Rϕ ) n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ 1 X ≤ t sup lim ln sup Rs1 n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ 1 X + (1 − t) sup lim ln sup Rs2 n→∞ n n J∈F Φ∈PJ,U λ ϕ∈Φ

= tQλ(s1) + (1 − t)Qλ(s2). (3.19)

Thus Qλ(s) is convexity on R. This completes the proof.

We deﬁne the eﬀective domains of Qλ as

domQλ := {s ∈ R : −∞ < Qλ(s) < ∞}.

The eﬀective domain of Qλ is deﬁned analogously, denoted by dom(Qλ). We have the following property without assuming r > 0.

Proposition 3.7. Let X, {Si}i∈I and U satisfy the hypotheses of Deﬁnition 2.9. Assume that λ ∈ (0, 1) and (CGC) holds. Then Qλ(s) (resp. Qλ(s)) is strictly decreasing on dom Qλ (resp. dom Qλ) that tends to −∞ as s tends to ∞. Moreover, Qλ(s) is convex on dom Qλ, ◦ and hence continuous on (dom Qλ) , 0 ≤ Qλ(0) ≤ Qλ(0) = −γ ln λ < ∞, where γ is deﬁned as in (2.10). INFINITE ITERATED FUNCTION SYSTEMS 13

Proof. It follows from (CGC) holds and the deﬁnitions of Qλ(s) and Qλ(s) that

ln(sup n #Φ) Φ∈PJ,U (λ ) 0 ≤ Qλ(0) ≤ Qλ(0) = sup lim J∈F n→∞ n

ln(sup n #Φ) Φ∈PJ,U (λ ) = sup lim n (− ln λ) = −γ ln λ < ∞. J∈F n→∞ − ln λ

Next, for any s ∈ dom Qλ and δ > 0 with s + δ ∈ dom Qλ, we can see from the proof of Proposition 3.6 that (3.16) holds for Qλ, and exactly the same inequality as (3.16) holds for

Qλ. Moreover, we have from the proof of Proposition 3.6 that (3.19) holds for Qλ. Hence the rest of conclusions hold.

The next Lemma follows from [3, Proposition 18] and Assumption A.

Lemma 3.8. Let X and {Si}i∈I satisfy the hypotheses of Theorem 2.16. Then for any nonempty invariant open set U ⊂ X with Ld(U) > 0 and sequence of packing families J kn {Sin,j }j of AJ,λn , we have

kJ kJ 1 Xn 1 Xn Q (s) = sup lim ln Rs = sup lim ln rs , s ∈ λ in,j in,j R n→∞ n n→∞ n J∈F j=1 J∈F j=1 J J kn kn 1 X s 1 X s Qλ(s) = sup lim ln Ri = sup lim ln ri , s ∈ R. n→∞ n n,j n→∞ n n,j J∈F j=1 J∈F j=1

The above Lemma implies that for IFSs satisfying (LDP) and (CGC), the deﬁnitions of the topological pressures are independent of the choice of the invariant open set U and the packing families of AJ,λn .

4. Proof of Theorem 2.16 and Theorem 2.17

Let X, {Si}i∈I and ρ be described as in Deﬁnition 2.1. We assume that there exists a set P p = {pi : i ∈ I} with 0 < pi < 1 and i∈I pi = 1. Let S = {Si}i∈I , M be the set of Borel regular probability measures having bounded support on X. Let

BC(X) = {f : X → R : f is continuous and bounded on bounded subsets}. For µ ∈ M, φ ∈ BC(X), deﬁne µ(φ) = R φdµ. For ν ∈ M, we deﬁne S : M → M by

X −1 S(ν)(A) = piν(Si (A)), for any Borel set A ⊂ X. i∈I We say that ν is variant with respect to (S, p) if S(ν) = ν. Using a proof similar to that of [8, Theorem 4.4(1)], we have

Lemma 4.1. Let X, {Si}i∈I and ρ be described as in Deﬁnition 2.1. Then there exists a P −1 unique Borel regular probability measure µ such that µ = i∈I piµ ◦ Si . 14 CHIH-YUNG CHU# AND SZE-MAN NGAI*

Let X and {Si}i∈I be described as in Deﬁnition 2.1. We say that {Si}i∈I satisﬁes the open set condition (OSC) if there exists a nonempty bounded invariant open set O ⊂ X (in the relative Euclidean topology of X), called an OSC-set, such that Si(O) ∩ Sj(O) = ∅ for all i 6= j.

Using Lemma 4.1 and a similar proof as that of [3, Proposition 20], we have the following result.

Proposition 4.2. Let K be the attractor of an FIFS {Si}i∈I satisfying the hypotheses of Deﬁnition 2.6, and assume that (LDP) holds. If (OSC) holds with an OSC-set U ⊇ K and P s j∈I rj > 1 with rj < 1, then dimH K ≥ s.

Proposition 4.3. Let K be the limit set of the IFS {Si}i∈I , and AI,b be deﬁned as in (2.7). Then K = S ϕ(K). ϕ∈AI,b

S Proof. For any i ∈ I, we have Si(K) ⊂ K since K = i∈I Si(K). This implies that for any ϕ = S ∈ , ϕ(K) ⊂ K. Thus S ϕ(K) ⊂ K. Notice that the following i1···in AI,b ϕ∈AI,b consequences: [ [ [ [ K = Si(K) = Si( Sj(K)) = Sij(K) = ··· i∈I i∈I j∈I i,j∈I [ = Si1···ip (K). i1,...,ip∈I Similarly, [ [ Si1···ip (K) = Si1···ip Sip+1 (K) = Si1···ipip+1 (K). ip+1∈I ip+1∈I

∞ Thus if x ∈ K, then there exists i = i1i2 · · · ∈ I such that K ⊃ Si1 (K) ⊃ Si1i2 (K) ⊃

· · · ⊃ {x}. Since there exists n such that i|n ∈ SI,b, we have x ∈ Si|n (K), and thus K ⊂ S ϕ(K). This completes this proof. ϕ∈AI,b

Proposition 4.4. Let K be the limit set of the IFS {Si}i∈I and F be deﬁned as in (2.9). S J Then K = J∈F K .

J S J Proof. Since K ⊂ K, we have J∈F K ⊂ K. Next we prove reverse inclusion. For any ∞ x ∈ K, it follows from the definition of π and Definition 2.2 that there exists i1i2 · · · ∈ I such that ∞ \ x = Si1···ip (X). p=1 ∞ We write {x} = xi1i2···. Let i1\··· ip denote the infinite sequence i1 ··· ipi1 ··· ip · · · ∈ I with recurrent block i1 ··· ip. Since ∞ ∞ \ \ Sj1···jq (xi1i2···) ∈ Sj1···jq ( Si1···ip (X)) = Sj1···jqi1···ip (X) = xj1···jqi1···ip···, p=1 p=1 INFINITE ITERATED FUNCTION SYSTEMS 15 we have Si ···i (x ) = x . It follows that x is the unique fixed point si ···i of 1 p i\1···ip i\1···ip i\1···ip 1 p

Si1···ip . Thus both si1···ip and xi1···ip··· belong to Si1···ip (X). Since limp→∞ diam(Si1···ip (X)) =

0, we have limp→∞ si1···ip = xi1···ip···. For any p ≥ 1, we choose J ∈ F such that (i1, . . . , ip) ∈ p J S J J = {(i1, . . . , ip): ij ∈ J, 1 ≤ j ≤ p}. Then si1···ip ∈ K , and thus x ∈ J∈F K . Notice S J n that J∈F K is a close set, and for any y ∈ K, there exists a sequence {x } of points in n S J K such that limn→∞ x = y. We see that K ⊂ J∈F K . This completes the proof.

Proof of Theorem 2.16. By Lemma 3.8, we ﬁrst require K ⊂ U. By Proposition 3.7, Qλ(0) and Qλ(0) are real numbers. We ﬁrst prove that dom Q = dom Q = R, and for any s ∈ R, Q (s) = Q (0) + s ln λ = (s − α) ln λ, λ λ (4.1) Qλ(s) = Qλ(0) + s ln λ = (s − β) ln λ, n where α, β are the unique zeroes of Qλ(s) and Qλ(s), respectively. Substituting b = λ and i = in,j into (3.2) yields n λ n J ξ ≤ rin,j ≤ Rin,j ≤ λ , n = 1, 2, . . . , j = 1, . . . , kn . c1(n |ln λ|) Hence J J ns Pkn s J 1 s ns k λ ≥ R ≥ k ξ λ , if s ≥ 0; n j=1 in,j n c1(n|ln λ|) J J ns Pkn s J 1 s ns k λ ≤ R ≤ k ξ λ , if s < 0. n j=1 in,j n c1(n|ln λ|) Since 1 1 s lim ln = 0, n→∞ n c1(n |ln λ|) by using Proposition 3.8 and the fact that Qλ(0) and Qλ(0) are real numbers, we have kJ 1 Xn 1 Q (s) = sup lim ln Rs = sup lim ln(kJ λns) λ in,j n n→∞ n n→∞ n J∈F j=1 J∈F

= Qλ(0) + s ln λ J kn 1 X s 1 J ns Qλ(s) = sup lim ln R = sup lim ln(k λ ) n→∞ n in,j n→∞ n n J∈F j=1 J∈F

= Qλ(0) + s ln λ. (4.2)

This implies that dom Q = dom Q = R. Moreover, by the properties of s ln λ, Qλ(s) and Qλ(s) are strictly decreasing and continuous on R that tend to −∞ and ∞ as s tends to ∞ and −∞, respectively. So, Qλ(s) and Qλ(s) have unique zero, denote by α and β, respectively. Thus we have Qλ(0) = −α ln λ and Qλ(0) = −β ln λ, and thus (4.1) holds.

Next, we prove

β ≤ dimH K.

Suppose, on the contrary, β > dimH K. Assume dimH K < s < β. By (4.1), Qλ(s) = (s − β) ln λ > 0. This implies, by Proposition 3.8, that there exists J ∈ F such that

J kn 1 X Qλ(s) lim ln rs ≥ > 0, n→∞ n in,j 2 j=1 16 CHIH-YUNG CHU# AND SZE-MAN NGAI*

Thus there exists an integer n > 0 depending on J, such that

J kn 1 X Qλ(s) ln rs ≥ > 0. n in,j 2 j=1

J J kn Let Kn denote the attractor of the new FIFS {Sin,j }j=1. Then this FIFS satisﬁes (OSC) J with U being an OSC-set. Since U ⊃ Kn , by applying Proposition 4.2 to the new FIFS J kn J J J {Sin,j }j=1, we get dimH Kn ≥ s. Notice that Kn ⊂ K and the fact ∞ ∞ J [ \ [ \ K = Si|m (X) ⊂ Si|n (X) = K, i∈J∞ m=1 i∈I∞ n=1 J J we get dimH K ≥ dimH K ≥ dimH Kn ≥ s, a contradiction. Thus dimH K ≥ β.

Now, we prove

Qλ(s) = Qλ(s) = (s − dimH K) ln λ. (4.3)

To this end we ﬁrst prove α ≥ dimH K. Let s > α. Then by (4.1), Qλ(s) < 0, and thus J kn s > 0 by Proposition 3.7. By assumption, for each integer n > 0, {Sin,j }j=1 is a packing family of AJ,λn with respect to U, where J ∈ F. This implies from Deﬁnition 2.9 that for

n each ϕ ∈ AJ,λ , there is at least one j such that ϕ(U)∩Sin,j (U) 6= ∅. Choose xn,j ∈ Sin,j (U). n By (3.1), both |ϕ(U)| and Sin,j (U) are less than λ |U|. Thus ϕ(U) is contained in the n J ball B2λn|U|(xn,j) with radius 2λ |U|. It follows from Proposition 4.3 and the fact K ⊂ U that J kn J [ J [ [ K = ϕ(K ) ⊂ ϕ(U) ⊂ B2λn|U|(xn,j).

ϕ∈AJ,λn ϕ∈AJ,λn j=1 J kn n J This implies that {B2λn|U|(xn,j)}j=1 is a 4λ |U| cover of K . Thus J {B2λn|U|(xn,j): j = 1, 2, . . . , kn ,J ∈ F}

n S J S J is a 4λ |U| cover of J∈F K . Notice that K = J∈F K (Proposition 4.4). This implies that J {B3λn|U|(xn,j): j = 1, 2, . . . , kn ,J ∈ F} n is a 6λ |U| cover of K. By the compactness of K, there exist J1,...,Jp such that

J p k i [ [n K ⊂ B3λn|U|(xn,j). (4.4) i=1 j=1

It follows from Proposition 3.8 and (4.2) that 1 J n s sup lim ln[kn (6λ |U|) ] = Qλ(0) + s ln λ = Qλ(s) < 0. J∈F n→∞ n J n s There exists inﬁnitely many integers n such that kn (6λ |U|) < 1. Notice that K ⊂ K and there exists J ∗ ∈ F such that ∗ J1 Jp J kn + ··· + kn ≤ kn . (4.5) INFINITE ITERATED FUNCTION SYSTEMS 17

So we get s s J∗ n s H (K) = lim Hδn (K) ≤ lim (kn )(6λ |U|) < 1 < +∞, n→∞ n→∞ and thus s ≥ dimH K. Since s > α is arbitrary, we conclude that α ≥ dimH K.

Since Qλ(s) ≤ Qλ(s) implies that α ≤ β, the above discussion implies that α = β = dimH K. Now equation (4.3) follows by substituting this into (4.1). Combining (4.3) and the fact that Qλ(0) = −γ ln λ (Proposition 3.7), we get dimH K = γ.

Finally, we prove dimH K = dimB K = dimP K. For any integer n > 0, let d d Y n n \ Y n n Bn := [(mi − 1)λ , miλ ): mi ∈ Z,K [(mi − 1)λ , miλ ) 6= ∅ i=1 i=1 and let Nn := #Bn, the cardinality of Bn. According to (4.4), we deﬁne d Y n n Bn,j : = [(mi − 1)λ , miλ ): mi ∈ Z, i=1 d \ Y n n B3λn|U|(xn,j) [(mi − 1)λ , miλ ) 6= ∅ . i=1

Ji Sp Skn TQd n n Then (4.4) implies that Bn ⊂ i=1 j=1 Bn,j. Since B3λn|U|(xn,j) i=1[(mi−1)λ , miλ ) 6= Qd n n √ ∅, we have i=1[(mi − 1)λ , miλ ) is contained in the ball B3λn|U|+ dλn (xn,j). Note also Qd n n that i=1[(mi − 1)λ , miλ ) are disjoint. Hence d √ L (B n n (xn,j)) 3λ |U|+ dλ d √ #Bn,j ≤ = L (B (0)). d Qd n n 3|U|+ d L ( i=1[(mi − 1)λ , miλ )) Therefore,

Ji p kn ∗ X X J1 Jp d √ J d √ Nn ≤ #Bn,j ≤ (kn + ··· + kn )L (B3|U|+ d(0)) ≤ (kn )L (B3|U|+ d(0)), i=1 j=1 J∗ where kn is deﬁned as in (4.5). Hence J∗ ln Nn ln(kn ) dimBK = lim ≤ lim n→∞ − ln λn n→∞ −n ln λ Q (0) ≤ λ = γ = dim K. − ln λ H

Since dimH K ≤ dimBK ≤ dimBK and dimH K ≤ dimP K ≤ dimBK, the assertion follows immediately. This completes this proof.

Proof of Theorem 2.17. For any J ∈ F, in view of (3.1) , we have

J J kn kn d s/d X X [L (Sin,j (U))] rs ≤ in,j [Ld(U)]s/d j=1 j=1 kJ Xn X ≤ Rs ≤ sup Rs , s ≥ 0, (4.6) in,j ψ ∗ n j=1 Ψ∈PJ,U (λ ) ψ∈Ψ 18 CHIH-YUNG CHU# AND SZE-MAN NGAI*

J J kn kn d s/d X X [L (Sin,j (U))] rs ≥ in,j [Ld(U)]s/d j=1 j=1 kJ Xn X ≥ Rs ≥ inf Rs , s < 0. (4.7) in,j ψ Ψ∈P∗ (λn) j=1 J,U ψ∈Ψ

k m For any b ∈ (0, λ) and any two packing families {Sii }i=1 ∈ PJ,U (b) and {Sjj }j=1 ∈ ∗ i j PJ,U (b). Let Ψ = {Sjj : Sjj (U) ∩ Sii (U) 6= ∅} and Φ = {Sii : Sii (U) ∩ Sjj (U) 6= ∅}. Using (LDP) and (L*DP) (Lemmas 3.1 and 3.2), we have

d d Rj rj j Ld(U) ≤ j Ld(U) 2 ξ+2ζ ξ c1c2 |ln b| c1 |ln b| d 1 d ≤ ξ L (Sjj U) c1 |ln b| d b d ≤ ξ L (U) c1 |ln b| ≤ rd Ld(U) ≤ Ld(S (U)) ii ii ≤ Rd Ld(U). ii 2 Let c3 := c1c2. It follows that

Rjj ξ+2ζ ≤ Rii for all 1 ≤ i ≤ k, 1 ≤ j ≤ m. (4.8) c3 |ln b|

By using (3.2) and (3.3) we see that for b > 0 with |ln b| > 1, ∪ i Sj (U) is contained Sjj ∈Ψ j ζ in a ball with center in Sii (U) and radius (1 + c2) |ln b| b |U|. Hence it follows from (3.3) again that d i d b X d #Ψ L (U) ζ ≤ L (Sjj (U)) c2 |ln b| i Sjj ∈Ψ ζ d d ≤ ((1 + c2) |ln b| b |U|) L (B1(0))

Therefore, there is a constant c4 > 0 such that k X i 2dζ m ≤ #Ψ ≤ c4 |ln b| k. (4.9) i=1

By interchanging the roles of the two packing families, it can be proved in the same way that there exist constants c5 > 0 and c6 > 0 such that

Rii ζ+ξ ≤ Rjj for all i, j satisfying 1 ≤ i ≤ k and 1 ≤ j ≤ m, (4.10) c5 |ln b|

m X j d(ξ+ζ) k ≤ #Φ ≤ c6 |ln b| m. (4.11) j=1 INFINITE ITERATED FUNCTION SYSTEMS 19

Now, let b = λn with |ln b| > 1. By combining the inequalities (4.8)–(4.11), we have

k X Rs ≤ k max{Rs : 1 ≤ i ≤ k} ii ii i=1 ≤ (c |ln λn|d(ξ+ζ) m)(c |ln λn|ζ+ξ)s min{Rs : 1 ≤ j ≤ m} 6 5 jj m X ≤ (c |ln λn|d(ξ+ζ))(c |ln λn|ζ+ξ)s Rs 6 5 jj j=1 n d(ξ+ζ) n ζ+ξ s n 2dζ n ξ+2ζ s ≤ (c6 |ln λ | )(c5 |ln λ | ) (c4 |ln λ | )(c3 |ln λ | ) k X × Rs , s ≥ 0. (4.12) ii i=1

Similarly, we have

k X Rs ≤ k max{Rs : 1 ≤ i ≤ k} ii ii i=1 s n d(ξ+ζ) 1 s ≤ (c6 |ln λ | m) min{R : 1 ≤ j ≤ m} n ξ+2ζ jj c3 |ln λ | n d(ξ+ζ) m c6 |ln λ | X ≤ Rs n ξ+2ζ s jj (c3 |ln λ | ) j=1 n 2dζ n d(ξ+ζ) k (c4 |ln λ | )(c6 |ln λ | ) X ≤ Rs , s < 0. (4.13) n ξ+2ζ s n ζ+ξ s ii (c3 |ln λ | ) (c5 |ln λ | ) i=1

From (4.12), (4.13), and Theorem 2.16, we get 1 X s Qλ(s) = sup lim ln inf RΨ n→∞ n Ψ∈P∗ (λn) J∈F J,U ψ∈Ψ 1 X s = sup lim ln sup RΨ . (4.14) J∈F n→∞ n ∗ n Ψ∈PJ,U (λ ) ψ∈Ψ The conclusion now follows from (4.6), (4.7), (4.12), (4.13), (4.14), Proposition 3.8 and Theorem 2.16. This completes this proof.

Using the fact Qλ(0) = −γ ln λ and (4.14), we have ln supΨ∈P∗ (b) #Ψ sup lim J,U = γ. J∈F b→0+ − ln b The following proposition follows from (4.12), (4.13) and Deﬁnition 2.10.

Proposition 4.5. Let X, {Si}i∈I and U satisfy the hypotheses of Deﬁnition 2.9. If (LDP) and (L*DP) hold, then

ln(supΦ∈ (b) #Φ) sup lim PJ,U = γ J∈F b→0+ − ln b 20 CHIH-YUNG CHU# AND SZE-MAN NGAI* if and only if

ln(supΨ∈P∗ (b) #Ψ) sup lim J,U = γ, J∈F b→0+ − ln b where γ is deﬁned as in Deﬁnition 2.10.

5. Growth dimension and proof of Theorem 2.18

We study the growth dimension in this section and prove Theorem 2.18.

Proposition 5.1. Let {Si}i∈I be an FIFS of essential contractions on X that has (LDP), and ξ be deﬁned as in Deﬁnition 2.6(a). Then the following hold.

ln #AI,b (a) limb→0+ − ln b exists. (b) Let αb be the unique nonnegative number satisfying

X αb Rϕ = 1. ϕ∈AI,b

If limb→0+ αb exists, then

−1 ln #AI,b ln #AI,b (1 + ξ) lim ≤ lim αb ≤ lim . (5.1) b→0+ − ln b b→0+ b→0+ − ln b

Proof. Let b1, b2 ∈ (0, 1). For any i ∈ SI,b1b2 , the deﬁnition of SI,b1b2 shows that there is a decomposition i = i1i2 with i1 = i1 ··· ik satisfying i1 ∈ SI,b1 and i2 = i2 ··· in. Notice that i2 could be the empty word. Let Si2 be the identity and thus Ri2 = 1 if that happens. It follows from (3.2) that b1b2 Ri ≥ ξ , (5.2) c1 |ln b1b2| and we have by using (3.2) and (2.6) that if n > k,

b2 < Rik+1···in−1 . (5.3) In fact, inequality (5.3) follows from

b1b2 < Ri1···in−1 ≤ Ri1···ik Rik+1···in−1 ≤ b1Rik+1···in−1 .

ln #AI,b Now we can proceed as in [2, Proposition 4.1], and conclude that limb→0+ − ln b exists. By (5.2), we have b X αb αb 1 = Rϕ ≥ (#AI,b)( ξ ) . (5.4) c1 |ln b| ϕ∈AI,b It follows from (5.4) and the fact 1 = P Rαb ≤ (# )bαb that for 0 < b < 1, ϕ∈AI,b ϕ AI,b

b ξ − ln b 1 − ln( ξ ) − ln b + ln(c |ln b| ) −(1 + ξ) ln b + ln c ≤ ≤ c1|ln b| = 1 ≤ 1 . ln #(AI,b) αb ln #AI,b ln #AI,b ln #AI,b Letting b → 0 yields (5.1). INFINITE ITERATED FUNCTION SYSTEMS 21

Proposition 5.2. Let {Si}i∈I be an FIFS of essential contractions on X that has (LDP) and satisﬁes (WSC). Assume that K is the associated attractor with dimH K = α. Then Hα(K) > 0.

Proof. Note that for any E ⊂ K, we have [ E ⊆ {S(K): S ∈ AI,|E|,S(K) ∩ E 6= ∅}.

Since {Si}i∈I satisﬁes (WSC), we have from [9, Proposition 3.1(c)] that

#{S(K): S ∈ AI,|E|,S(K) ∩ E 6= ∅} < ∞.

−1 For each S ∈ AI,|E| with S(K) ∩ E 6= ∅, consider the map S : S(K) → K. For any x, y ∈ S(K), let x0, y0 ∈ K such that x = S(x0) and y = S(y0). By (2.4) and (2.7), we get

0 0 0 0 −1 −1 |x − y| = S(x ) − S(y ) ≤ RS x − y ≤ |E| S (x) − S (y) . It follows from [5, Theorem 2] that Hα(K) > 0 .

Proposition 5.3. Let α, K and {Si}i∈I satisfy the hypotheses of Propositions 5.1 and 5.2. α If H (K) < ∞, then there exists a constant c3 > 0 such that for any 0 < b < 1, −α −1 −α b c b ≤ #(AI,b) ≤ c3 . (5.5) 3 |ln b|ξ Consequently, ln #AI,b α = dimH K = lim . (5.6) b→0+ − ln b

Proof. For any 0 < b < 1, by Proposition 4.3, we have K = S ϕ(K). It follows ϕ∈AI,b from (WSC) and [9, Proposition 3.1] that each x ∈ K is covered by at most κ elements of

{ϕ(K): ϕ ∈ AI,b}. Hence X Hα(K) ≤ Hα(ϕ(K)) ≤ κHα(K). (5.7)

ϕ∈AI,b

For each ϕ ∈ AI,b, by using (3.2), we get α α α α b α H (ϕ(K)) ≥ rϕH (K) ≥ ξ H (K) (5.8) c1 |ln b| and α α α α α H (ϕ(k)) ≤ RϕH (K) ≤ b H (K).

It follows by summing the inequalities over ϕ ∈ AI,b and using (5.7) that α b α α ξ H (K)#(AI,b) ≤ κH (K) c1 |ln b| and α α α H (K) ≤ b H (K)#(AI,b). 22 CHIH-YUNG CHU# AND SZE-MAN NGAI*

By the assumption Hα(K) < ∞ and Proposition 5.2, we have 0 < Hα(K) < ∞. Thus (5.5) holds. Taking logarithm on both sides of (5.5), we get b − ln c3 − α ln b ≤ ln #(AI,b) ≤ ln c3 − α ln = ln c3 − α ln b + αξ ln |ln b| . |ln b|ξ The dimension formula (5.6) follows since limb→0+(ln |ln b|)/ ln b = 0.

Let {Si}i∈I be an FIFS of essential contractions on X and let λ ∈ (0, 1). Following [2], we deﬁne ∗ 1 X s Qλ(s) := lim ln Rϕ , s ∈ R, n→∞ n ϕ∈AI,λn

∗ 1 X s Qλ(s) := lim ln Rϕ , s ∈ R, n→∞ n ϕ∈AI,λn ∗ ∗ ∗ and denote by Qλ(s) the common value if Qλ(s) = Qλ(s). ∗ ∗ Proposition 5.4. Let {Si}i∈I be an FIFS of essential contractions on X, and Qλ, Qλ and ∗ s Qλ be deﬁned as above. Let K be associate attractor with H (K) < ∞, where s = dimH K. Assume that {Si}i∈I has (LDP) and satisﬁes (WSC). Then for any λ ∈ (0, 1), dimH K = α, ∗ ∗ ∗ ∗ ∗ where α is unique zero of Qλ(s) or Qλ(s). In particular, Qλ(α) = Qλ(α) = Qλ(α) = 0.

Proof. Similar to the proof of [3, Proposition 14], we have that [2, Proposition 2.3] holds for FIFS of essential contractions.

For any positive integer n, we can see by combining (3.1), the deﬁnition of AI,λn , and n Proposition 4.3 that {ϕ(K): ϕ ∈ AI,λn } is a δn-cover of K with δn = λ |K| → 0 as n → ∞. Let r be deﬁned as in (2.5). Then r > 0 here. Combining the above discussion and the fact n −1 that λ < Rj−1 ≤ r Rj for j ∈ SI,λn , we have s X ns s X −s s s Hδn (K) ≤ λ |K| ≤ r Rϕ |K| . ϕ∈AI,λn ϕ∈AI,λn Furthermore, we have from (5.7) and (5.8) that X 1 α X Hα(K) ≥ κ−1 Hα(ϕ(K)) ≥ κ−1 Hα(K) Rα n ξ ϕ c1 |ln λ | ϕ∈AI,λn ϕ∈AI,λn Next, using Proposition 5.2 and the assumption Hs(K) < ∞, and a similar derivation as that in [2, Theorem 3.3], we can see that the assertions hold.

Lemma 5.5. Let {Si}i∈I be a IIFS of essential contractions on X and 0 < b0 < 1 . Assume that (LDP) and (WSC) hold, and K is both a WSC-set and a quasi s-set. Then for all b ∈ (0, b0), ∗ ∗ X αJ ξαJ αJ X ∗ ξα α Rϕ ≤ |ln b| c1 κ for all J ∈ F and sup Rϕ ≤ |ln b| c1 κ, (5.9) J∈F ϕ∈AJ,b ϕ∈AJ,b J ∗ where αJ = dimH K , α = sup{αJ : J ∈ F}, and F, c1, and κ are deﬁned as in (2.9), (3.2), and (2.11) respectively. INFINITE ITERATED FUNCTION SYSTEMS 23

Proof. Let J ∈ F. By the assumption, each x ∈ K ⊇ KJ is covered by no more than κ sets of the form ϕ(K) ⊇ ϕ(KJ ), where ϕ ∈ AJ,b. It follows from the assumption that K is a WSC-set and (3.2) that for all b ∈ (0, b0),

αJ X αJ X αJ αJ κH (KJ ) ≥ H (ϕ(KJ )) ≥ rϕ H (KJ ) ϕ∈AJ,b ϕ∈AJ,b b αJ X αJ ≥ ξ H (KJ ) c1 |ln b| ϕ∈AJ,b 1 αJ X αJ αJ ≥ ξ Rϕ H (KJ ). c1 |ln b| ϕ∈AJ,b Thus (5.9) follows from Proposition 5.2 and the assumption that K is a quasi s-set.

Note that for each b ∈ (0, b0), {ϕ(K): ϕ ∈ AI,b} is a δ := b |K|-cover for K. Similar to the proof of [2, Theorem 1.2], the following Lemma 5.6 follows from Proposition 5.1 and Proposition 5.3.

Lemma 5.6. Let {Si}i∈I be an FIFS of essential contractions on X that has (LDP). Then

(a) dimH K ≤ dG.

(b) If, in addition, (WSC) is satisﬁed, then dimH K = dG

Let {Si}i∈I be an FIFS of essential contractions on X. Using the notation in [2], we deﬁne the topological pressure function P : R → R as 1 X P (s) = lim ln Rs . (5.10) n→∞ n φ n φ∈{SJ :J∈I } The upper and lower topological pressure functions, denoted by P (s) and P (s), respectively, are deﬁned by using lim inf and lim sup, respectively.

Proposition 5.7. Let {Si}i∈I be an FIFS of essential contractions on X and P (s) be described as above. Then P (s) exists and is ﬁnite. Moreover, P (s) strictly decreasing, convex and continuous on [0, ∞), and P (0) ≥ 0.

Proof. Whether (BDP) is satisﬁed or not, we see from the proof of [2, Proposition 2.2] that P (s) exists and is ﬁnite, convex and continuous on [0, ∞), and P (0) ≥ 0. It follows from (2.6) that for any δ > 0, 1 X 1 X ln Rs+δ ≤ ln Rs + δ ln R. n φ n φ n n φ∈{SJ :J∈I } φ∈{SJ :J∈I } Letting n → ∞ and using (5.10), we get P (s + δ) ≤ P (s) + δ ln R < P (s), if s ≥ 0, i.e, P (s) is strictly deceasing on [0, ∞). 24 CHIH-YUNG CHU# AND SZE-MAN NGAI*

We extend Lemma 5.6 to IIFS.

Lemma 5.8. Let {Si}i∈I be an IIFS of essential contractions on X that has (LDP) and

(WSC). If K is both a WSC-set and a quasi s-set, then dimH K = dG.

∗ Proof. Using the notation in [11]. Let t > α and b ∈ (0, b0), where b0 is described as in Lemma 5.5. Then by Lemma 5.5,

X t X t−α∗ α∗ t−α∗ ξα∗ α∗ φb(t) = sup Rτ ≤ sup b Rτ ≤ b |ln b| c1 κ. (5.11) J∈F J∈F τ∈AJ,b τ∈AJ,b ∗ Since limb→0+(ln |ln b|)/ ln b = 0, (5.11) implies Q(t) ≤ Q(t) ≤ α − t < 0, where

1 X t 1 X t Q(t) =: lim ln Rτ and Q(t) =: lim ln Rτ . b→0+ − ln b b→0+ − ln b τ∈AI,b τ∈AI,b Note that according to [2, Lemma 3.4], for ﬁxed real numbers p and q, there is an integer l > 0 (independent of n) such that l ∗ n+q n+p [ k {Sj : j ∈ I , r ≤ Rj ≤ r } ⊆ Si ◦ Sk : Si ∈ AI,rn , k ∈ I . k=0 This result also holds for FIFS of essential contractions. Using Proposition 5.4, Proposition

5.7, and a similar derivation as that in [2, Theorem 1.1], we conclude that αJ is the unique zero of the associated topological pressure function P . Similar to the proof of [11, Theorem ∗ 1.2(a)], we get dimH K = α . The assertions follow by using a proof similar to that of [11, Corollary 5.2(b)].

Proof of Theorem 2.18. Theorem 2.18 follows from Theorem 2.16(b) and Lemma 5.8.

6. Example

In this section we illustrate the applications of our results by some examples.

Example 6.1. Let X = [0, 1], 0 < r ≤ 1/2, and K be the limit set of the following IIFS of similitudes: rk+1x, i = 2k + 1; S (x) = i rk+1x + (1 − rk+1), i = 2k + 2, where k = 0, 1, 2,.... Then (LDP), (L*DP), (BDP) and (CGC) hold. Moreover, dimH K = ln 2 − ln r .

∗ Proof. (LDP), (L*DP) and (BDP) hold since all Si are contractive similitudes. Let J = n {1, 2}. Take U = (0, 1) and b = r with n ≥ 1. It is easy to see that sup n #Φ = Φ∈PJ∗,U (r ) n ∗ 2 . For any J ∈ F with J 6= J , since S1[(0, 1)] and S2[(0, 1)] are disjoint, S2k+1(x) = INFINITE ITERATED FUNCTION SYSTEMS 25

n n S11 ··· 1(x), and S2k+2(x) = S22 ··· 2(x) for k ≥ 0, we have #Φ ≤ 2 for any Φ ∈ PJ,U (r ). | {z } | {z } k+1 k+1 Thus

ln(sup n #Φ) Φ∈PJ,U (r ) n ln 2 ln 2 sup lim n = lim = , J n→∞ − ln r n→∞ −n ln r − ln r and thus (CGC) holds. Moreover, it follows from Theorem 2.16 that dimH K = ln 2/(− ln r).

The following example shows that (LDP) is strictly weaker than (BDP).

k Example 6.2. Let A be a d × d real matrix, and let Sj(x) = ρjA j (x + dj), j = 1, 2,..., be an IIFS with kj ∈ {0} ∪ N, and 0 < |ρj| < 1. Assume that all eigenvalues of A have moduli 1 and supj≥1{kj} < ∞. Then

(a) (LDP) and (L*DP) are satisﬁed;

(b) (BDP) holds if and only if there is a real invertible matrix B and a real orthogonal matrix Q such that A = BQB−1. In this case, the attractor is similar to a self-similar set generated by the IFS with A replaced by Q.

Proof. The proof is similar to that of [3, Example 23]; we omit the details. Example 6.3. Any FIFS of essential contractions satisﬁes (CGC).

Proof. Assume that {Si}i∈I is an FIFS of essential contractions on X. Then r > 0. For any J ∈ F, it follows from (3.14) that

0 #Φ ≤ (#I)nk0 logr λ−k0 logr C ,

n 0 where Φ ∈ PJ,U (λ ) and C is described as in the proof of Proposition 3.6. Since 0 (nk0 logr λ − k0 logr C ) ln(#I) lim = −k0 log (#I), n→∞ −n ln λ r we get

ln(sup n #Φ) Φ∈PJ,U (λ ) lim ≤ −k0 log (#I) < ∞. n→∞ − ln λn r Thus (CGC) holds.

Example 6.4. Let Sk : [0, 1] → [0, 1], k = 1, 2,..., be an IIFS of contractions as follows: ρ2k−1(x + ak), x ≤ 1/2; Sk(x) = ρ2k(x − 1/2) + ρ2k−1(1/2 + ak), x > 1/2. where 0 < ρk ≤ 1/2. Assume the following conditions hold:

(a) a2 = 0, (ρ1 + ρ2)/2 + ρ1a1 = 1, ρ1 + ρ2 = ρ3 + ρ4, and max{ρ , ρ } c := sup 2k−1 2k : k = 1, 2,... < ∞; min{ρ2k−1, ρ2k} 26 CHIH-YUNG CHU# AND SZE-MAN NGAI*

(b) for each k, either Sk(0, 1) ⊂ (0, 1/2) or Sk(0, 1) ⊂ (1/2, 1); (c) for each k ≥ 2, Sk(1) < S1(0), Sk(0) ≤ Sk+1(0); (d) for each k ≥ 1, S2k([0, 1]) ∩ S2k+1([0, 1]) 6= ∅ with S2kS1 = S2k+1S2, and no other Sj([0, 1]) intersects the union S2k([0, 1]) ∪ S2k+1([0, 1]).

Then (LDP), (L*DP), (BDP) and (CGC) hold, and K is both a WSC-set and a quasi s-set.

∗ Proof. By Remark 3.3, we need only prove that Ri/ri ≤ c for all i ∈ I , i.e., (BDP) holds, where c is described as in condition (a). We use induction on the length of i. Since ρk > 0, it is easy to see that Rk = max{ρ2k−1, ρ2k} and rk = min{ρ2k−1, ρ2k} for k = 1, 2,... , and thus the assertion is true when |i| = 1. Assume it is true for the case |i| ≤ n. To check the case

|i| = n + 1, let i = i1 ··· in+1. If Si2 [0, 1] ⊆ [0, 1/2], then Si(x) = ρ2i1−1(Si2...in+1 (x) + ai1 ) for all x ∈ [0, 1]. Hence Ri = ρ2i1−1Ri2...in+1 and ri = ρ2i1−1ri2...in+1 . If Si2 [0, 1] ⊆ [1/2, 1], using the same argument, we have Ri = ρ2i1 Ri2...in+1 and ri = ρ2i1 ri2...in+1 . In both cases, it follows by induction hypothesis that Ri/ri ≤ c. Thus (LDP) and (L*DP) hold.

From conditions (c) and (d), for i ∈ I∗, it is easy to see by induction on the length of the composition that all overlapping iterates can be expressed in the form SiS2k[0, 1] ∩ SiS2k+1[0, 1], with no other iterates Sj[0, 1] with |j| = |i|+1 intersecting the union SiS2k[0, 1]∪

SiS2k+1[0, 1]. Thus it follows by induction on the length of i with Ri ≤ b < Ri−1 that #{τ ∈ SI,b : x ∈ τ(X)} ≤ 2 for any b ∈ (0, 1), that is, (WSC) holds and X is a WSC- region.

Taking U = (0, 1). For any b ∈ (0, 1) and J ∈ F, we have

ln(supΦ∈P (b) #Φ) ln(#A ) J,U ≤ J,b (6.1) − ln b − ln b

Let K be the limit set of {Si(x)}i∈N. Then K ⊂ X, and is a WSC-set since X is a WSC- region. Moreover, by Remark 2.15, K is a quasi s-set. It follows from (2.12), (6.1) and Lemma 5.8 that ln(sup #Φ) Φ∈PJ,U (b) sup lim ≤ dG = dimH K, J∈F b→0+ − ln b where dG is the growth dimension of {Si}i∈N. Thus (CGC) holds. This completes the proof.

The following is a class of special IIFSs from Example 6.4; see Figure 1(a). √ Example 6.5. Let X = [0, 1], 0 < r + r2 < (5 − 17)/4 ≈ 0.21922..., and (r + r2)(2 − r − r2)/(1 − r − r2) < t < 1/2. Let 2rx + (1 − r − r2), x ≤ 1/2; S (x) = 1 2r2(x − 1/2) + 1 − r2, x > 1/2, INFINITE ITERATED FUNCTION SYSTEMS 27

X=[0,1] X=[0,1]

n=1 n=1

n=2 n=2

K K

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

(a) (b)

Figure 1. (a) First two iterations of [0, 1] of the IIFS in Example 6.5, with r = 1/10 and t = 2/5. The limit set K is also shown. (b) Example 6.6, drawn with r = 0.27 and t = 2/3. and 2 k 2 k−1 S2k(x) = (r + r ) x + t(1 − (r + r ) ),

2 k 2 k−1 2 k 2 S2k+1(x) = (r + r ) x + t(1 − (r + r ) ) + (r + r ) (1 − r − r ) for any k ≥ 1. Then (LDP), (L*DP), (BDP), and (CGC) hold, and K is both a WSC-set and a quasi s-set.

The following example is diﬀerent from Example 6.4 in that condition (b) need not be satisﬁed; see Figure 1(b). √ Example 6.6. Let X = [0, 1], 0 < r < (2− 2)/2 ≈ 0.292893..., r(2−r)/(1−r) < t < 1−r.

Let S1(x) = rx + (1 − r), and let rkx + t(1 − rk−1), i = 2k; S (x) = i rkx + t(1 − rk−1) + rk(1 − r), i = 2k + 1, for any k ≥ 1. Then (LDP), (L*DP), (BDP), and (CGC) hold. Moreover, K is both a √ WSC-set and is a quasi s-set. Finally, dimH K = ln((2 − 2)/2)/ ln r.

Proof. Using a proof similar to that of Example 6.4, we conclude that (CGC) holds. The other assertions follow from [11, Example 6.1].

Acknowledgment. This work was completed when the ﬁrst author was visiting the De- partment of Mathematical Sciences of Georgia Southern University. He thanks the Depart- ment for its hospitality and support during his visit. 28 CHIH-YUNG CHU# AND SZE-MAN NGAI*

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School of science, Northwest A&F University, Yangling, Shannxi, 712100, P. R. China

E-mail address: [email protected]

Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China, Department of Mathematical Sci- ences, Georgia Southern University, Statesboro, GA 30460-8093, USA.

E-mail address: [email protected]