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].

2010Mathematics 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

For FIFSs of contractive similitudes satisfying (WSC), Zerner proved in [12] that if the attractorK is not contained in a hyperplane, then the Hausdorff dimension ofK 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.

Definition 2.1. LetXbe a non-empty compact subset ofR^d, equipped with the Euclidean metric, and let I be a countable set with at least two elements, S_i :X → X, i ∈I, be a countable family of mappings. If there exists a metricρonXthat is topologically equivalent to the Euclidean metric and a constantη ∈(0,1) such that

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

Let

I⁰: ={∅}, I^k:={(i₁, . . . , i_k) :i₁, . . . , i_k∈I} fork≥1, I^∗: =

∞

[

k=0

I^k, I^∞:={(i₁, i2, . . .) :ij ∈I for all j≥1}.

We call I^∗ the space of finite words with alphabets i ∈ I, and I^∞ the coding space. For i= (i1, . . . , i_k)∈I^k, we denote by |i|=kthelength ofi and writeS_i:=Si1 ◦ · · · ◦Si_k(S_∅ is defined to be the identity). We often denote i= (i1, . . . , i_k) simply by i =i1. . . i_k and let i^−m :=i1. . . ik−m be the word obtained fromiby deleting its lastm(1≤m≤k) alphabets.

Ifi∈I^∗S

I^∞ andn≥1 does not exceed the length ofi, we denote byi|_n the wordi1· · ·in

consisting of the firstn alphabets ofi.

Since ρ is topologically equivalent to the Euclidean metric, (X, ρ) is a compact metric space in R^d and {S_i}_i∈I is a family of contraction mapping on (X, ρ). For E ⊂ R^d, we denote the diameter of E by diam(E) = sup_x,y∈Eρ(x, y). For each i =i₁i₂· · · ∈ I^∞, the compact sets S_i|n(X), n≥1, are decreasing and their diameters converge to zero. In fact, by (2.1),

diam(S_i|n(X))≤ηⁿdiam(X). (2.2)

This implies that the set

π(i) =

∞

\

n=1

Si|_n(X)

is a singleton and therefore this formula defines a map π:I^∞→X, called thecoding 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{S_i}_i∈I be an IFS described as above. Define thelimit set as K := [

i∈I^∞

∞

\

n=1

Si|_n(X).

IfK is closed, it is called the attractor (orfixed point) of the IFS.

The limit setKis 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 satisfiesK =S

i∈ISi(K). Since Sj(π(i)) =π(ji) for any j∈I andi∈I^∞, we have K =π(I^∞) =π

[

j∈I

[

i∈I^∞

ji

= [

j∈I

[

i∈I^∞

π(ji) = [

j∈I

[

i∈I^∞

S_j(π(i))

=[

j∈I

S_j [

i∈I^∞

π(i)

= [

j∈I

S_j(π(I^∞)) = [

j∈I

S_j(K).

Thus the limit setK satisfies

K=[

i∈I

S_i(K).

Notice that ifI is finite, thenK is compact. Unlike FIFSs,K is not necessarily compact if I is countably infinite. Also, ifKis not compact, it is not necessarily the unique non-empty bounded subset of R^d satisfying K =S

i∈ISi(K) (see, e.g., [11, Example 2.2]). It follows from Banach’s fixed point theorem and uniform contractivity that {S_i}_i∈I has a unique fixed point K, and if U ⊂ X is a nonempty invariant set (i.e., S

i∈IS_i(U) ⊂ U), then K ⊂K⊂U. It is well known that ifI is finite, then

K =

∞

\

n=1

[

i∈Iⁿ

S_i(X), (2.3)

and thus K is a Borel set. When I is countably infinite, Mauldin and Urba´nski in [10]

showed that (2.3) also holds under the assumption that the system{S_i :i∈I}ispointwise finite (meaning that each element ofXbelongs to at most finitely many elements ofSi(X)).

In this case,K is anFσδ and is thus a Borel set. As pointed out in [10],K may even have a

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 r_i:= inf

x6=y∈X

|S_i(x)−S_i(y)|

|x−y| , R_i := sup

x6=y∈X

|S_i(x)−S_i(y)|

|x−y| , i∈I^∗, (2.4) r := inf

i∈Iri, R := sup

i∈I

Ri, (2.5)

We also write rϕ :=ri and Rϕ:=Ri ifϕ=Si for somei∈I^∗. Since for any i,j∈I^∗,

|S_ij(x)−S_ij(y)|

|x−y| = |S_ij(x)−S_ij(y)|

|S_j(x)−Sj(y)| ·|S_j(x)−S_j(y)|

|x−y| , the following inequalities hold (see [2, 3, 9]):

R_ij≤R_iR_j, r_ij≤R_ir_j, r_ij≤r_iR_j,

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

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

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

For anyE ⊂R^d, we use dimH(E),dimP(E),dimB(E),H^s(E),L^d,|E|, andE^◦ 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 #Adenote 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 setU ⊂X and let 0< b <1. Define SI,b :={i= (i1, . . . , in)∈I^∗ :Ri ≤b < R_i⁻},

S_I,b^∗ (U) :={i= (i1, . . . , in)∈I^∗:L^d(Si(U))≤b^dL^d(U)<L^d(S_i⁻(U)}, AI,b :={S_i:i∈SI,b},

AI,b^∗ (U) :={S_i :i∈SI,b^∗ (U)}.

(2.7)

Remark 2.4. SinceRcan be greater than 1, for (i₁, i₂, . . .)∈I^∞, there could be more than one prefix i = (i₁, . . . , i_n) ∈ I^∗ such that i ∈ SI,b (resp. S_I,b^∗ (U)). However, by Remark 2.3, the number of such prefixes must be finite.

Remark 2.5. It is possible that S_i = S_i⁰ for distinct i,i⁰ ∈ I^∗; we identify such S_i and S_i⁰. For IFSs of contractive similitudes, SI,b =S_I,b^∗ (U) and so AI,b =A_I,b^∗ (U). In general, however, they need not be the same. For an IIFS, both AI,b and A_I,b^∗ (U) are countably infinite sets.

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

(a) We say that {S_i}i∈I has the logarithmic distortion property (LDP) if there is a constantξ >0 such that

b→0lim⁺ sup

i∈SI,b

b

ri|lnb|^ξ = 0.

(b) LetU ⊂X be an invariant set. We say that{S_i}_i∈I has thelogarithmic* distortion property (L*DP) if there is a constant ζ >0 such that

lim

b→0⁺ sup

i∈S_I,b^∗ (U)

b

r_i|lnb|^ζ = 0 and lim

b→0⁺ sup

i∈S_I,b^∗ (U)

R_i

r_i|lnb|^ζ = 0. (2.8) Mauldin and Urba´nski [10] defined thebounded distortion property (BDP) of an IFS of injectiveC¹ conformal contractions. We extend this definition directly to IFSs of essential contractions as follows.

Definition 2.7. An IFS{S_i}_i∈I of essential contractions as in Definition 2.1 is said to have thebounded distortion property (BDP) if there exists a constantc >0 such that

R_i

r_i ≤c for all i∈I^∗.

Remark 2.8. In the above definitions, we do not assume that the IFS maps are differen- tiable.

Definition 2.9. Let X and {S_i}_i∈I satisfy the hypotheses of Definition 2.6, U ⊂X be a bounded invariant set that is open in the relative topology ofX withL^d(U)>0, and Φ be a subset of {S_i :i ∈I^∗}. We call a subcollection{ϕ_j}_j∈J ⊂Φ a packing family of Φ with respect toU if it satisfies the following two conditions:

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

(ii) for anyϕ∈Φ, ϕ(U) intersects at least oneϕ_j(U).

We denote by PI,U(b) the class of all packing families of AI,b with respect to U, and denote byP_I,U^∗ (b) the class of all packing families ofA_I,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 : Jis finite}. (2.9) LetJ ∈ F and{S_i}_i∈J be the associated FIFS. For any 0< b <1, letPJ,U(b) andP_J,U^∗ (b) be the corresponding packing families as in Definition 2.9.

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

γ := sup

J∈F b→0+lim

ln(sup_Φ∈PJ,U(b)#Φ)

−lnb <∞. (2.10)

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

Definition 2.12. Let X,{S_i}_i∈I and U satisfy the hypotheses of Definition 2.9 and fix λ∈(0,1). Define

Q_λ(s) := sup

J∈F

lim

n→∞

1 nln

Φ∈PinfJ,U(λⁿ)

X

ϕ∈Φ

R^s_ϕ

, s∈R, Q_λ(s) := sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,U(λⁿ)

X

ϕ∈Φ

R^s_ϕ

, s∈R.

We callQ_λ (resp. Q_λ) thelower (resp. upper)topological pressure function with scale λ. If Q_λ =Q_λ, we denote byQ_λ the common function and call it atopological pressure function with scale λ. Note that λis fixed and sis the variable of the functions Q_λ(s) and Q_λ(s).

Similar to Definition 2.12, we define Q^∗_λ(s) := sup

J∈F

lim

n→∞

1 nln

Φ∈PinfJ,U^∗ (λⁿ)

X

ϕ∈Φ

[L^d(ϕ(U))]^s/d

, s∈R, Q^∗_λ(s) := sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,U^∗ (λⁿ)

X

ϕ∈Φ

[L^d(ϕ(U))]^s/d

, s∈R.

Definition 2.13. LetX and {S_i}i∈I satisfy the hypotheses of Definition 2.6. We say that {S_i}_i∈I satisfies the weak separation condition (WSC) if there exist an invariant subset D⊂X withD^◦ 6=∅, called aWSC-region, and a constant κ∈N such that

sup

x∈X

#

ϕ∈AI,b :x∈ϕ(D) ≤κ for allb∈(0,1). (2.11) IfE ⊂X is an invariant subset and (2.11) holds, we callE aWSC-set.

For any J ∈ F, we denote by K^J the attractor of the FIFS {S_i}_i∈J and write αJ :=

dimHK^J.

Definition 2.14. LetX and {S_i}_i∈I satisfy the hypotheses of Definition 2.6. We callK a quasi s-set ifH^αJ(K^J)<∞ for any J ∈ F.

Remark 2.15. If{S_i}_i∈Isatisfies (BDP) and (WSC), then it follows from [9, Theorem 3.2]

thatK is a quasis-set.

We now state the first two main results of this paper.

Theorem 2.16. Let X,{S_i}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 {S_in,j}^k_j=1^Jn ∈PJ,U(λⁿ), n∈N, the following hold:

(a) for all s∈R,

Q_λ(s) =Q_λ(s) = sup

J∈F

n→∞lim 1 nln

^k

Jn

X

j=1

R^s_in,j

= (s−dimHK) lnλ;

(b) let γ be defined in Definition 2.11, then

dim_HK = dim_PK= dim_BK =γ = sup

J∈F

n→∞lim lnk_n^J

−nlnλ

.

Theorem 2.17. Let X,{S_i}_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 families {S_in,j}^k_j=1^Jn ∈P_J,U^∗ (λⁿ), n∈N, we have

Q^∗_λ(s) = Q^∗_λ(s) = sup

J∈F

n→∞lim 1 nln

k_n^J

X

j=1

[L^d(S_in,j(U)]^s/d

=Q_λ(s)

= s−dim_HK

lnλ for alls∈R.

LetX⊂R^dbe nonempty compact subset withX^◦6=∅, and{S_i}_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{S_i}_i∈I as

d_G= lim

b→0+

ln #AI,b

−lnb .

For each finite set J ⊂ I, we denote by d^J_G the growth dimension of the associated FIFS {S_i}_i∈J. Following [11], we define the growth dimension of the IIFS{S_i(x)}_i∈I (In this case,

#(AI,b) =∞ for any b∈(0,1)) as

dG= sup{d^J_G :J ⊂Iis finite}. (2.12) Theorem 2.18. Let X,{S_i}_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

dim_HK= dim_PK = dim_BK=γ=d_G, 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,

r_i|E| ≤ |S_i(E)| ≤R_i|E|,

ri(L^d(E))^1/d ≤(L^d(Si(E)))^1/d≤Ri(L^d(E))^1/d. (3.1)

Lemma 3.1. Assume the same hypotheses on X and {S_i}i∈I as in Definition 2.9, and assume that (LDP) holds. Let R_i and r_i 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

c₁|lnb|^ξ ≤r_i≤R_i≤bfor all i∈SI,b and b∈(0, b₀). (3.2) Moreover, c₁(b₀)→0 as b₀ →0.

Proof. By Definition 2.6, for sufficiently smallb0 >0, 0< c₁ := sup

b∈(0,b₀)

sup

i∈SI,b

b

r_i|lnb|^ξ <∞.

This implies that ^b

ri|lnb|^ξ ≤ c₁ for any b ∈ (0, b₀). Thus the conclusion follows from the

definition ofSI,b and the factr_i≤R_i.

Lemma 3.2. Assume the same hypotheses onX, {S_i}_i∈I andU as in Definition 2.9, and assume that (L*DP) holds. LetR_iandr_ibe as in (2.4), andζ >0be defined as in Definition 2.6(b). Then for allb₀ >0 sufficiently small, there is a constant c₂ =c₂(b₀)>0 such that

b

c₂|lnb|^ζ ≤r_i ≤R_i ≤c₂|lnb|^ζb for alli∈SI,b^∗ (U) and b∈(0, b₀). (3.3) Moreover, c₂(b₀)→0 asb₀ →0.

Proof. By Definition 2.6(b), there exists a constantc3 > 0 such that for sufficiently small b0 >0,

0< c3 = sup

b∈(0,b0)

sup

i∈SI,b^∗ (U)

b

r_i|lnb|^ζ <∞ and a constantc₄>0 such that

0< c₄= sup

b∈(0,b0)

sup

i∈S_I,b^∗ (U)

R_i

r_i|lnb|^ζ <∞.

This implies that for anyb∈(0, b0), b

r_i|lnb|^ζ ≤c3 and R_i

r_i|lnb|^ζ ≤c4. (3.4) By the definition ofS_I,b^∗ (U) and (3.1),i∈S_I,b^∗ (U) implies that

ri≤

L^d(Si(U)) L^d(U)

1/d

≤b. (3.5)

By lettingc₂ := max{c₃, c₄}and combining (3.4), (3.5), and the fact thatr_i≤R_i, we obtain

(3.3).

Remark 3.3. If{S_i}_i∈I satisfies (BDP), then there is a constantc >0 such thatRi/ri ≤c for alli∈SI,b∪S_I,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.

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

Proposition 3.4. Letr <1andSI,Ribe defined as in (2.5)and (2.7)respectively. Assume thatr >0. Then there exists an integer k0>0 such that for anyi∈I^∗ with |i|> k0, there is a decomposition i=i1i2 with |i₂| ≤k0 such thati1 ∈SI,R_i.

Proof. As mentioned in Remark 2.3, Ri → 0 uniformly as |i| → ∞. Thus there exists an integerk0 >0 such thatRi≤rfor alli∈I^∗with|i| ≥k0. Leti∈I^∗with|i|=n > k0. First we check R_i⁻¹. IfRi< R_i⁻¹, then we takei1 =i andi2 =∅, and thus the conclusion holds.

If not, we check R_i⁻². If R_i < R_i⁻², then R_i⁻¹ ≤R_i< R_i⁻². Thus the conclusion holds by lettingi₁ =i⁻¹ andi₂ =σⁿ⁻¹(i). Continue. Note that there exists some 1≤k≤n−1 such thatR₂−k+1 ≤R_i< R^−k_i , for otherwise, one would getr ≤R_iⁿ⁻¹ ≤R_i < r, a contradiction.

We further claim that there exists somek∈Nwith 1≤k≤k₀suchR_i^−k+1 ≤R_i< R_i^−k. To see this we suppose on the contrary that there exists some`∈Nsatisfyingk<sub>0</sub>+1≤`≤n−1 such thatR_i−(`−1) ≤R<sub>i</sub>< R<sub>i</sub><sup>−`</sup>. Then one would get

R_i^{−(`−1)</sup> ≤Ri≤R<sub>i</sub><sup>−(`−1)}R_σ^{n−(`−1)</sup><sub>(i)</sub>≤R<sub>i</sub><sup>−(`−1)}r < R_i^−(`−1),

where the third inequality is because |σ<sup>n−(`−1)</sup>(i)|=`−1≥k0. This contradiction proofs the claim. The asserted result now follows by letting i₁ =i^−k and i₂ =σ^n−k(i).

Lemma 3.5. Assume the same hypotheses on X, {S_i}_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. Letk₀ be as in Proposition 3.4. For t∈ I^∗ with |t|=n≥k₀, let n=lk₀+t with 0≤t < k0. Then rⁿ≤rt≤Rt≤r^l. Taking logarithm, we have

nlnr≤lnrt≤lnRt≤llnr≤ nlnr

k₀ , |t| ≥k0. This implies that

lnR_t lnr_t ≥ 1

k₀ >0. (3.6)

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 constantc₁ >0 such that

b

c₁|lnb|^ξ ≤r_i≤R_i≤b for all i∈SI,b andb∈(0, b₀). (3.7) To prove (2.8), we assume, without loss of generality, that|j|> k0 for anyj∈S_I,b^∗ (U) and Rj < b0, since we need only consider sufficiently small b >0. By the definition ofS_I,b^∗ (U), we have

L^d(Sj(U)) L^d(U)

1/d

≤b <

L^d(S_j⁻¹(U)) L^d(U)

1/d

, and thus r_j ≤b < R_j⁻¹. AsR_j ≥R_j⁻¹r, we get

rj ≤b < r⁻¹Rj. (3.8)

Combining (3.6) and (3.8), we see that exists some constant ˜c≥1 such that

˜

c⁻¹≤ lnRj

lnb ≤˜c. (3.9)

As|j|> k₀, Proposition 3.4 implies that there exists a decompositionj=j₁j₂ with|j₂| ≤k₀ such that j₁ ∈ SI,Rj. As R_j < b₀, substituting b and i in (3.7) by R_j and j₁ respectively yields

R_j

r_j1|lnR_j|^ξ ≤c1.

Using (2.6) and the fact thatj₁∈SRj and |j₂| ≤k₀, we have

r^k0rj1 ≤rj1rj2 ≤rj≤rj1R^|j2|. (3.10) Combining (3.8)–(3.10), we get

b

r_j|lnb|^ξ ≤ r⁻¹R_jc˜^ξ

r^k0r_j1|lnR_j|^ξ ≤ c₁˜c^ξ

r^k0+1 =:c₃<∞ (3.11) and

R_j

b|lnb|^ξ ≤ R_j

r_j|lnb|^ξ ≤ R_jc˜^ξ

r^k0r_j1|lnR_j|^ξ ≤ c1˜c^ξ

r^k0 ≤ c1˜c^ξ

r^k0+1 =c3 <∞. (3.12) 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,{S_i}_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 Q_λ(s) and Q_λ(s) are real-valued, strictly decreasing, and continuous functions on R that tend to

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

Proof. Let k₀ be described as in Proposition 3.4. Write C := sup{R_i : |i| < k₀}. Then C < ∞. For any J ∈ F, let ϕ = S_i1···i_k ∈ Φ ∈ PJ,U(λⁿ). Let k−1 = lk₀ +m with 0≤m < k₀, wherel∈N∪ {0}. Then we haveR_i1···ik−1 ≤Cr^l≤C⁰r^k/k0 for some constant C⁰. Hence

r^k≤rϕ≤Rϕ≤λⁿ< Ri1···i_k−1 ≤C⁰r^k/k0. (3.13) It follows that

k < nk0log_rλ−k0log_rC⁰, (3.14) and thus

Rϕ ≥r^{nk0logrλ−k0logrC0}. (3.15) It follows that from (3.13) and (3.15) that whens≥0,

r^{(nk0logrλ−k0logrC0)s}≤ X

ϕ∈Φ

R_ϕ^s ≤#Φλ^ns,

and when s <0,

λ^ns≤ X

ϕ∈Φ

R^s_ϕ≤#Φr^{(nk0logrλ−k0logrC0)s}. Thus ifs≥0,

sk₀lnλ−sk₀lnC⁰

n ≤ ln(infΦ∈PJ,U(λⁿ)

P

ϕ∈ΦR_ϕ^s) n

≤ ln(sup_Φ∈PJ,U(λn)

P

ϕ∈ΦR^s_ϕ) n

≤ ln(sup_Φ∈PJ,U(λn)#Φ)

n +slnλ

≤ ln(sup_Φ∈PJ,U(λⁿ₎#Φ)

−lnλⁿ (−lnλ) +slnλ;

ifs <0,

slnλ ≤ ln(infΦ∈PJ,U(λⁿ)

P

ϕ∈ΦR^s_ϕ) n

≤ ln(sup_Φ∈PJ,U(λn)

P

ϕ∈ΦR^s_ϕ) n

≤ ln(sup_Φ∈PJ,U(λn)#Φ)

n +sk0log_rλlnr−sk0log_rC⁰lnr n

= ln(sup_Φ∈PJ,U(λⁿ₎#Φ)

−lnλⁿ (−lnλ) +sk0lnλ−sk0lnC⁰

n .

Lettingn→ ∞ and using (2.10) and the definitions ofQ_λ(s) and Q_λ(s), we get sk0lnλ≤ Q_λ(s)≤ Q_λ(s)≤(s−γ) lnλ, ifs≥0;

slnλ≤ Q_λ(s)≤ Q_λ(s)≤(sk₀−γ) lnλ, ifs <0.

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

Next, since Q_λ(s) and Q_λ(s) are real-valued, for any δ >0, we have Q_λ(s+δ) ≤ sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R^s_ϕλ^nδ

= Q_λ(s) +δlnλ <Q_λ(s), (3.16) and by (3.15),

Q_λ(s+δ) ≥ sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R_ϕ^sr^{δ(nk0logrλ−k0logrC0)}

= Q_λ(s) +δk₀lnλ, that is,

Q_λ(s) +δk0lnλ≤ Q_λ(s+δ)≤ Q_λ(s) +δlnλ <Q_λ(s). (3.17)

Moreover, for anyδ >0, we also have Q_λ(s−δ) ≤ sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R_ϕ^sr^{−δ(nk0logrλ−k0logrC0)}

≤ Q_λ(s))−δk0lnλ, Q_λ(s−δ) ≥ sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R^s_ϕλ^−nδ

≥ Q_λ(s)−δlnλ >Q_λ(s), that is,

Q_λ(s)<Q_λ(s)−δlnλ≤ Q_λ(s−δ)≤ Q_λ(s))−δk0lnλ. (3.18) From (3.17) and (3.18), we see that Q_λ(s) is strictly decreasing and continuous on R. We can get analogous results for Q_λ(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 lnx is convex,

Q_λ(ts₁+ (1−t)s₂) = sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R^ts_ϕ¹R^(1−t)s_ϕ ²

= sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

(R_ϕ^s1)^t(R^s_ϕ²)^1−t

≤ tsup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R^s1

+ (1−t) sup

J∈F

n→∞lim 1 nln

sup

Φ∈PJ,Uλⁿ

X

ϕ∈Φ

R^s2

= tQ_λ(s₁) + (1−t)Q_λ(s₂). (3.19) ThusQ_λ(s) is convexity on R. This completes the proof.

We define the effective domains ofQ_λ as

domQ_λ :={s∈R:−∞<Q_λ(s)<∞}.

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

Proposition 3.7. Let X,{S_i}_i∈I and U satisfy the hypotheses of Definition 2.9. Assume thatλ∈(0,1)and (CGC) holds. ThenQ_λ(s)(resp. Q_λ(s)) is strictly decreasing ondomQ_λ (resp. domQ_λ) that tends to −∞ as s tends to∞. Moreover,Q_λ(s) is convex ondomQ_λ, and hence continuous on(domQ_λ)^◦, 0≤ Q_λ(0)≤ Q_λ(0) =−γlnλ <∞, whereγ is defined as in (2.10).

Proof. It follows from (CGC) holds and the definitions ofQ_λ(s) andQ_λ(s) that 0 ≤ Q_λ(0)≤ Q_λ(0) = sup

J∈F

n→∞lim

ln(sup_Φ∈PJ,U(λn)#Φ) n

= sup

J∈F

n→∞lim

ln(sup_Φ∈PJ,U(λn)#Φ)

−lnλⁿ (−lnλ)

=−γlnλ <∞.

Next, for any s ∈ domQ_λ and δ > 0 with s+δ ∈ domQ_λ, we can see from the proof of Proposition 3.6 that (3.16) holds forQ_λ, and exactly the same inequality as (3.16) holds for Q_λ. Moreover, we have from the proof of Proposition 3.6 that (3.19) holds forQ_λ. Hence

the rest of conclusions hold.

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

Lemma 3.8. Let X and {S_i}_i∈I satisfy the hypotheses of Theorem 2.16. Then for any nonempty invariant open set U ⊂ X with L^d(U) > 0 and sequence of packing families {S_in,j}^k_j^Jn of AJ,λn, we have

Q_λ(s) = sup

J∈F

lim

n→∞

1 nln

k_n^J

X

j=1

R^s_in,j

= sup

J∈F

lim

n→∞

1 nln

k^J_n

X

j=1

r^s_in,j

, s∈R

Q_λ(s) = sup

J∈F

n→∞lim 1 nln

k^J_n

X

j=1

R^s_i

n,j

= sup

J∈F

n→∞lim 1 nln

k^J_n

X

j=1

r_i^s

n,j

, s∈R.

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 setU and the packing families ofAJ,λn.

4. Proof of Theorem 2.16 and Theorem 2.17

LetX,{S_i}i∈I andρ be described as in Definition 2.1. We assume that there exists a set p={p_i :i∈I} with 0< p_i <1 and P

i∈Ip_i = 1. Let S ={S_i}_i∈I,Mbe the set of Borel regular probability measures having bounded support onX. Let

BC(X) ={f :X→R:fis continuous and bounded on bounded subsets}.

Forµ∈ M, φ∈ BC(X), define µ(φ) =R

φdµ. For ν∈ M, we define S:M → Mby S(ν)(A) =X

i∈I

piν(S_i⁻¹(A)), for any Borel setA⊂X.

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, {S_i}_i∈I and ρ be described as in Definition 2.1. Then there exists a unique Borel regular probability measure µsuch that µ=P

i∈Ipiµ◦S_i⁻¹.

Let X and {S_i}i∈I be described as in Definition 2.1. We say that {S_i}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 S_i(O)∩S_j(O) =∅for all i6=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 {S_i}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

j∈Ir^s_j >1 withr_j <1, then dim_HK≥s.

Proposition 4.3. LetK be the limit set of the IFS{S_i}_i∈I, andAI,b be defined as in (2.7).

Then K=S

ϕ∈AI,bϕ(K).

Proof. For any i ∈ I, we have S_i(K) ⊂ K since K = S

i∈IS_i(K). This implies that for any ϕ = S_i1···in ∈ AI,b, ϕ(K) ⊂ K. Thus S

ϕ∈AI,bϕ(K) ⊂ K. Notice that the following consequences:

K = [

i∈I

Si(K) =[

i∈I

Si([

j∈I

Sj(K)) = [

i,j∈I

Sij(K) =· · ·

= [

i1,...,ip∈I

Si1···ip(K).

Similarly,

Si1···ip(K) = Si1···ip

[

ip+1∈I

Sip+1(K)

= [

ip+1∈I

Si1···ipip+1(K).

Thus if x ∈ K, then there exists i = i₁i₂· · · ∈ I^∞ such that K ⊃ S_i1(K) ⊃ S_i1i2(K) ⊃

· · · ⊃ {x}. Since there exists n such that i|_n ∈ SI,b, we have x ∈ S_i|n(K), and thus K ⊂S

ϕ∈AI,bϕ(K). This completes this proof.

Proposition 4.4. Let K be the limit set of the IFS {S_i}_i∈I and F be defined as in (2.9).

Then K=S

J∈FK^J.

Proof. Since K^J ⊂ K, we haveS

J∈FK^J ⊂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=

∞

\

p=1

Si1···i_p(X).

We write {x} = x_i1i2···. Let i\₁· · ·i_p denote the infinite sequence i₁· · ·i_pi₁· · ·i_p· · · ∈ I^∞ with recurrent blocki₁· · ·i_p. Since

S_j1···jq(x_i1i2···)∈S_j1···jq(

∞

\

p=1

S_i1···ip(X)) =

∞

\

p=1

S_j1···jqi1···ip(X) =x_j1···jqi1···ip···,

we have S_i1···i_p(x_i\

1···i_p) = x_i\

1···i_p. It follows that x_i\

1···i_p is the unique fixed point s_i1···i_p of S_i1···ip. Thus boths_i1···ipandx_i1···ip···belong toS_i1···ip(X). Since limp→∞diam(S_i1···ip(X)) = 0, we have limp→∞s_i1···ip =x_i1···ip···. For anyp≥1, we chooseJ ∈ F such that (i₁, . . . , i_p)∈ J^p ={(i₁, . . . , i_p) : i_j ∈J, 1≤j ≤p}. Then s_i1···ip ∈ K^J, and thus x∈S

J∈FK^J. Notice that S

J∈FK^J is a close set, and for any y ∈K, there exists a sequence {xⁿ} of points in K such that limn→∞xⁿ=y. We see that K ⊂S

J∈FK^J. This completes the proof.

Proof of Theorem 2.16. By Lemma 3.8, we first requireK⊂U. By Proposition 3.7,Q_λ(0) and Q_λ(0) are real numbers. We first prove that domQ= domQ=R, and for any s∈R,

Q_λ(s) =Q_λ(0) +slnλ= (s−α) lnλ,

Q_λ(s) =Q_λ(0) +slnλ= (s−β) lnλ, (4.1) whereα, β are the unique zeroes ofQ_λ(s) andQ_λ(s), respectively. Substitutingb=λⁿ and i=in,j into (3.2) yields

λⁿ

c1(n|lnλ|)^ξ ≤r_in,j ≤R_in,j ≤λⁿ, n= 1,2, . . . , j= 1, . . . , k_n^J. Hence

k^J_nλ^ns≥Pk^J_n j=1R^s_i

n,j ≥k_n^J _c ¹

1(n|lnλ|)^ξ

s

λ^ns, ifs≥0;

k^J_nλ^ns≤Pk^J_n j=1R^s_i

n,j ≤k_n^J _c ¹

1(n|lnλ|)^ξ

s

λ^ns, ifs <0.

Since

n→∞lim 1 nln

1 c₁(n|lnλ|)

s

= 0,

by using Proposition 3.8 and the fact that Q_λ(0) andQ_λ(0) are real numbers, we have Q_λ(s) = sup

J∈F

lim

n→∞

1 nln

^k

Jn

X

j=1

R_i^s_n,j

= sup

J∈F

lim

n→∞

1

nln(k_n^Jλ^ns)

= Q_λ(0) +slnλ Q_λ(s) = sup

J∈F

n→∞lim 1 nln

k^J_n

X

j=1

R_i^s_n,j

= sup

J∈F

n→∞lim 1

nln(k_n^Jλ^ns)

= Q_λ(0) +slnλ. (4.2)

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

Next, we prove

β≤dim_HK.

Suppose, on the contrary, β > dimHK. Assume dimHK < s < β. By (4.1), Q_λ(s) = (s−β) lnλ >0. This implies, by Proposition 3.8, that there exists J ∈ F such that

n→∞lim 1 nln

k^J_n

X

j=1

r^s_in,j ≥ Q_λ(s) 2 >0,

Thus there exists an integern >0 depending onJ, such that 1

nln

k^J_n

X

j=1

r^s_in,j ≥ Q_λ(s) 2 >0.

Let K_n^J denote the attractor of the new FIFS {S_in,j}^k_j=1^Jn . Then this FIFS satisfies (OSC) with U being an OSC-set. Since U ⊃ K_n^J, by applying Proposition 4.2 to the new FIFS {S_in,j}^k_j=1^Jn , we get dimHK_n^J ≥s. Notice that K_n^J ⊂K^J and the fact

K^J = [

i∈J^∞

∞

\

m=1

Si|_m(X)⊂ [

i∈I^∞

∞

\

n=1

Si|_n(X) =K,

we get dimHK≥dimHK^J ≥dimHK_n^J ≥s, a contradiction. Thus dimHK ≥β.

Now, we prove

Q_λ(s) =Q_λ(s) = (s−dimHK) lnλ. (4.3) To this end we first prove α ≥dim_HK. Let s > α. Then by (4.1), Q_λ(s) <0, and thus s > 0 by Proposition 3.7. By assumption, for each integer n > 0, {S_in,j}^k_j=1^nJ is a packing family of AJ,λⁿ with respect to U, where J ∈ F. This implies from Definition 2.9 that for eachϕ∈AJ,λⁿ, there is at least onejsuch thatϕ(U)∩S_in,j(U)6=∅. Choosex_n,j ∈S_in,j(U).

By (3.1), both |ϕ(U)| and

S_in,j(U)

are less than λⁿ|U|. Thus ϕ(U) is contained in the ballB_2λⁿ_|U|(x_n,j) with radius 2λⁿ|U|. It follows from Proposition 4.3 and the factK^J ⊂U that

K^J = [

ϕ∈AJ,λn

ϕ(K^J)⊂ [

ϕ∈AJ,λn

ϕ(U)⊂

k^J_n

[

j=1

B_2λⁿ|U|(xn,j).

This implies that{B_2λn|U|(xn,j)}^k_j=1^Jn is a 4λⁿ|U|cover ofK^J. Thus {B_2λn|U|(xn,j) :j= 1,2, . . . , k_n^J, J ∈ F } is a 4λⁿ|U|cover ofS

J∈FK^J. Notice thatK =S

J∈FK^J (Proposition 4.4). This implies that

{B_3λn|U|(xn,j) :j= 1,2, . . . , k_n^J, J ∈ F }

is a 6λⁿ|U|cover ofK. By the compactness ofK, there existJ₁, . . . , J_p such that K⊂

p

[

i=1 k^Jin

[

j=1

B_3λⁿ_|U|(x_n,j). (4.4)

It follows from Proposition 3.8 and (4.2) that sup

J∈F

lim

n→∞

1

nln[k^J_n(6λⁿ|U|)^s]

=Q_λ(0) +slnλ=Q_λ(s)<0.

There exists infinitely many integersnsuch thatk^J_n(6λⁿ|U|)^s<1. Notice thatK⊂K and there existsJ^∗∈ F such that

k^J_n¹ +· · ·+k^Jn^p ≤k^J_n^∗. (4.5)