THE SHIFT SPACE OF A RECURRENT ITERATED FUNCTION SYSTEM

FUNCTION SYSTEM

ALEXANDRU MIHAIL

Recurrent iterated function systems are a generalization of iterated function sys- tems. They were dened in [4] and in a more general settings in [5]. Instead of considering contractions from a metric spaceXto itself, we consider contractions fromX×X toX. Some of the properties of recurrent iterated function systems are similar to those of iterated function systems. In this paper we dene and study the shift or code space associated with a recurrent iterated function system.

AMS 2000 Subject Classication: 28A80.

Key words: iterated function system, recurrent iterated function system, attrac- tor, shift space (or code space).

1. INTRODUCTION

The paper is divided in to four sections. The rst section is the intro- duction. In the second section we shortly describe the shift spaces of an IFS and prove the continuity of the projection map. The next section contains the main results concerning the shift space of a RIFS. The last section contains an example.

We start by a short presentation of iterated function systems, IFS for short, and recurrent iterated function systems, RIFS for short. We will also x the notation.

For a set X, P(X) denotes the sets of subsets of X. Let (X, dX) and (Y, d_Y) be two metric spaces. By C(X, Y) we denote the set of continuous function from X to Y. On C(X, Y) we will consider the generalized metric d:C(X, Y)×C(X, Y)→R+=R+∪ {∞} dened by

d(f, g) = sup

x∈X

d_Y(f(x), g(x)).

IfZ⊂X thend_Z(f, g) = sup

x∈Z

d_Y(f(x), g(x)). Forf_n, f ∈C(X, Y),f_n→^s f denotes pointwise convergence, f_n ^u.c→ f uniform convergence on compact sets and fn u

→ f uniform convergence, that is, convergence in the generalized metric d.

REV. ROUMAINE MATH. PURES APPL., 53 (2008), 4, 339355

Let (X, dX) and (Y, dY) be two metric spaces. On X×Y we consider the distance d=dX×Y : (X×Y)×(X×Y)→R+ dened by dX×Y((x₁, y₁), (x2, y2)) = max{d_X(x1, x2), dY(y1, y2)} (if not otherwise stated).

Let (X, d) be a metric space. B(X) denotes the set of closed bounded nonvoid subsets of X and K(X) the set of compact nonvoid subsets of X. We have K(X) ⊂ B(X) ⊂ P(X). B(X) and K(X) are metric spaces under the Hausdor-Pompeiu distance h :B(X)×B(X) → [0,∞) (or h : K(X)× K(X)→[0,∞)) dened by

h(A, B) = max(d(A, B), d(B, A)) = min{r/A⊂B(B, r) and B ⊂B(A, r)}, where

d(A, B) = sup

x∈A

d(x, B) = sup

x∈A

y∈Binfd(x, y)

and

B(A, r) ={x|there existsy∈Asuch that d(x, y)< r}= S

x∈A

B(x, r).

As for the spaces(B(X), h)and(K(X), h)we have the results below (see [6] and [7]).

Theorem 1.1. (B(X), h) is a complete metric space if (X, d) is a com- plete metric space; (K(X), h) is a complete metric space if (X, d) is a com- plete metric space, compact if (X, d) is compact, and separable if (X, d) is separable.

For a set A from the metric space (X, d), d(A) represents the diameter of A dened asd(A) = sup

x,y∈A

d(x, y).

Let (X, d) be a metric space. For a function f : X → X we de- note by Lip(f) ∈ R+ the Lipschitz constant associated with f : Lip(f) =

sup

x,y∈X;x6=y

d(f(x),f(y))

d(x,y) . f is said to be a Lipschitz function if Lip(f)<+∞ and a contraction if Lip(f)<1. Let Con(X) ={f :X →X|Lip(f)<1}.

Letm∈N^∗. For a functionf :X^m= ^m×

k=1

X→X, the number

inf{c|d(f(x₁, x₂, . . . , x_m), f(y₁, y₂, . . . , y_m))≤cmax{d(x₁, y₁), . . . , d(x_m, y_m)}

for all {x₁, x₂, . . . , x_m, y₁, y₂, . . . , y_m ∈X},that coincides with sup

x1,x2,...,xm,y1,y2,...,ym∈X max{d(x1,y1),...,d(xm,ym)}>0

d(f(x1, x2, . . . , xm), f(y1, y2, . . . , ym)) max{d(x₁, y₁), . . . , d(x_m, y_m)}

is denoted by Lip(f) and it is called the Lipschitz constant of f. Clearly, Lip(f) ∈ R+. A function f : X^m → X is called a Lipschitz function if

Lip(f)<+∞and a contraction if Lip(f)<1. Let Conm(X) ={f :X^m →X|

Lip(f)<1}.

An iterated function system onXconsists of a nite family of contractions (f_k)_k=1,n on X and is denoted byS = (X,(f_k)_k=1,n).

A recurrent iterated function systems onX consists of a nite family of contractions (f_k)_k=1,n,f_k:X×X→X and is denoted byS = (X,(f_k)_k=1,n).

We remark that an IFS is a special case of an RIFS.

For an IFS (respectively, an RIFS) FS : K(X) → K(X) (respectively, FS : K(X)×K(X) → K(X)) is the function dened by FS(B) =

n

S

k=1

f_k(B) (respectively, FS(K, H) =

n

S

k=1

f_k(K, H), where for a function f :X×X →X we denef(K, H) =f(K×H) ={f(x, y)|x∈K, y ∈H}).

In both cases the functionFSis a contraction with Lip(FS)≤max

k=1,n

Lip(f_k). By the Banach contraction theorem for an IFS or an RIFS, there exists a unique set A(S) such that FS(A(S)) =A(S), respectively FS(A(S), A(S)) = A(S).We state the results for RIFS. The proof can be found in [4] and some generalization in [5]. More results about IFS can be found in [1], [2], [3] or [7].

Theorem 1.2. Let(X, d)be a complete metric space andS= (X,(f_k)_k=1,n) an RIFS with c= max

k=1,nLip(f_k)<1. Then there exists a uniqueA(S)∈K(X) such that FS(A(S), A(S)) =A(S). Moreover, for any H₀, H₁∈K(X) the se- quence (Hn)n≥1 dened by Hn+1 =FS(Hn, Hn−1) is convergent to A(S). As for the speed of convergence we have

h(H_n, A(S))≤ 2c^[n2]

1−cmax{h(H₀, H₁), h(H₁, H₂)}.

In particular, h(H₀, A(S))≤ _1−c² max{h(H₀, H₁), h(H₂, H₁)}. As for the esti- mation of the distance h(H0, A(S)) we also have

h(H₀, A(S))≤ 1

1−ch(H₀, f(H₀, H₀)).

We remark that A(S) =

n

[

k=1

f_k(A(S), A(S)) =

n

[

k=1

[

x∈A(S)

f_k({x}, A(S))

=

n

[

k=1

[

x∈A(S)

f_k(A(S),{x})

=

n

[

k=1

[

x,y∈A(S)

f_k({x},{y})

. As for the continuity of the attractor of an RIFS we have

Theorem 1.3. Let (X, d) be a complete metric space and S = (X,(f_k)_k=1,n) and S⁰ = (X,(g_k)_k=1,n) two RIFSs with c= max

k=1,n Lip(f_k) <1 and c⁰ = max

k=1,n Lip(g_k)<1. Then

h(A(S), A(S⁰))≤ 1

1−min(c, c⁰) max

k=1,n

d(fk, gk).

A variant of this theorem is

Theorem 1.4. Let (X, d) be a complete metric space, n ∈ N^∗, and let S_m = (X,(f_k^m)_k=1,n), m ∈ N^∗, and S = (X,(f_k)_k=1,n) be recurrent iterated function systems. Let c_m= max

k=1,nLip(f_k^m) and c₀ = max

k=1,nLip(f_k). If sup

n≥1

c_n= c <1 and f_k^m →^s f_k on a dense set in X×X for every k= 1, n, then c₀ ≤c and A(S_m)→A(S).

2. THE SHIFT SPACE OF AN IFS

In this section we present the shift space of an IFS. At the end of it we prove the continuity of the projection map.

We start with some notation: Rdenotes the real numbers,Nthe natural numbers,N^∗=N− {0},N^∗n={1,2, . . . , n}.For two nonvoid setsAandB, B^A denotes the set of functions fromAtoB. ByΛ = Λ(B)we will understand the set B^N∗ and by Λn = Λn(B) the set B^N∗n. The elements of Λ = Λ(B) = B^N∗ will be written as innite words ω = ω1ω2. . . ωmωm+1. . ., where ωm ∈ B while the elements of Λ_n = Λ_n(B) = B^N∗n as words ω = ω₁ω₂. . . ω_n; Λ(B) is the set of innite words with letters from the alphabet B and Λn(B)is the set of words of length n. By Λ^∗ = Λ^∗(B) we will understand the set of all nite words Λ^∗ = Λ^∗(B) = S

n≥1

Λn(B). If ω = ω1ω2. . . ωmωm+1. . . or if ω = ω1ω2. . . ωn and n ≥ m, then [ω]m = ω1ω2. . . ωm. For two words α ∈ Λ_n(B) and β ∈ Λ_m(B) or β ∈ Λ(B) by αβ we will understand the concatenation of the wordsαandβ, namely,αβ=α1α2. . . αnβ1β2. . . βm, and respectively αβ=α₁α₂. . . α_nβ₁β₂. . . β_mβ_m+1. . ..

On Λ = Λ(N^∗n) = (N^∗n)^N∗we consider the metric d_s(α, β) =

∞

P

k=1 1−δ_αk^βk

3^k , where

δ^y_x=

1 if x=y, 0 if x6=y.

Denition 2.1. The pair(Λ(N^∗n) = (N^∗n)^N∗, d_s) is a compact metric space called the shift space with nletters.

Let Fk : Λ(N^∗_n) → Λ(N^∗_n) be dened by Fk(ω) = kω for k = 1, n. The functions F_k are continuous and are called the right shift functions. We have

ds(F_k(α), F_k(β)) = ds(α, β)

3 .

The function R : Λ(N^∗_n) → Λ(N^∗_n) dened by R(ω = ω1ω2. . . ωmωm+1. . .) = ω₂ω₃. . . ω_mω_m+1. . . also is continuous and is called the left shift function.

We have

d_s(R(α), R(β)) = 3d_s(α, β)−(1−δ_α^β1₁)≤3d_s(α, β).

Remark 2.1. (1)R◦F_k(ω) =ω and F_k◦R(ω) =kω2ω3. . . ωmωm+1. . .. (2)Λ =

n

S

k=1

F_k(Λ), soΛis the attractor of the IFS S = (Λ,(F_k)_k=1,n). Notation 2.1. Let(X, d_X)be a complete metric space,S= (X,(f_k)_k=1,n) be an IFS on X and A = A(S) the attractor of S. For ω = ω₁ω₂. . . ω_m ∈ Λm(N^∗_n) set fω =fω1◦fω2 ◦. . . .◦fωm andAω=fω(A).

Notation 2.2. Let f :X → X or f :X×X →X be a contraction. We denote by e_f the xed point of f. Iff =f_ω then we denote by e_fω or by e_ω the xed point of f =fω.

Denition 2.2. An IFS S = (X,(f_k)_k=1,n) is said to be disconnected if fi(A(S))∩fj(A(S)) = ∅ for any two dierent indices i, j = 1, n, and totally disconnected if f_ω(A(S))∩f_ω⁰(A(S)) = ∅ for any ω, ω⁰ ∈ Λ_m(N^∗n) and any m∈N^∗ such thatω6=ω⁰.

If an IFSS= (X,(f_k)_k=1,n)is disconnected andf_kare injective functions then it is totally disconnected.

The main results concerning the relation between the attractor of an IFS and the shift space is contained in

Theorem 2.1 ([7]). If A = A(S) be the attractor of the IFS S = (X,(f_k)_k=1,n).

(1) For ω∈Λ,A_[ω]m+1 ⊂A_[ω]m andd(A_[ω]m)≤c^md(A), sod(A_[ω]m)→0 as m→ ∞.

(2) If a_ω is dened by {a_ω} = T

n≥1

A_[ω]m then d(e_[ω]m, a_ω) → 0 when m→ ∞.

(3)A=A(S) = S

ω∈Λ

{a_ω},A_α = S

ω∈Λ

{a_αω}for every α∈Λ^∗. (4) The set {e_[ω]m|ω∈Λ and m∈N^∗} is dense in A.

(5) The function π : Λ → A dened by π(ω) = aω is continuous and surjective.

(6)π◦Fk=fk◦π fork= 1, n.

(7) If the IFS S = (X,(fk)_k=1,n) is disconnected and fk are injective functions then π also is injective.

Denition 2.3. The function π : Λ → A from the Theorem 2.1 is called the canonical projection from the shift space on the attractor of the IFS.

We now prove the continuity of the canonical projection. The following result is obvious.

Lemma 2.1. Let (X, d) be a metric space. If f :X→X then d(f(A))≤ Lip(f)d(A) for every A ⊂ X. In general, if f : ×ⁿ

k=1X → X then for every A1, A2, . . . , An⊂X we have

d(f(A₁, A₂, . . . , A_n))≤Lip(f) max{d(A₁), d(A₂), . . . , d(A_n)}.

Lemma 2.2. Let (X, d) be a complete metric space and m∈N^∗. Let f :

m×

k=1X→Xbe a contraction and Ha closed set such that f(H, H, . . . , H)⊂H. Then ef ∈H, where ef is the xed point of f.

Proof. Leta₀ ∈H and be dene the sequence(a_n)_nby a_n+1 =f(a_n, a_n, . . . , a_n). Thena_n∈H,a_n→e_f and e_f ∈H becauseH is a closed set.

Theorem 2.2. Let (X, d) be a complete metric space, n ∈ N^∗, and let S_m = (X,(f_k^m)_k=1,n), m ∈ N^∗, and S = (X,(f_k)_k=1,n) be recurrent iterated function systems. Let c_m = max

k=1,nLip(f_k^m)and c₀= max

k=1,nLip(f_k). Let π : Λ→ A(S) and πm : Λ → A(S_m) be the canonical projections. If sup

n≥1

cn = c < 1 and f_k^m→^s

m f_k on a dense set in X for every k= 1, n then c₀≤cand π_n→^u π. Proof. Because the functions of the family (f_k^m)_k=1,n are equally contin- uous and f_k^m →^s

m f_k on a dense set in X for everyk = 1, n, we havef_k^{m u.c}→f_k, f_[ω]^m

l

u.c→f_[ω]l for everyω ∈Λ and l∈N^∗. By Theorem 1.4 we have c0 ≤c and A(S_m) → A(S). Because A(S_m) → A(S) and A(S_m), A(S) ∈ K(X), there exists M > 0 such that d(A(S_m)) < M and h(A(S_m), A(S)) < M for every m∈N^∗. We can also suppose thatd(A(S))< M. We have

(1) h(A(S_m)_[ω]l, A(S)_[ω]l)≤c^lh(A(S_m), A(S)) +h(f_[ω]^m

l(A(S)), f_[ω]l(A(S))) and, because h(A(S_m), A(S))< M,

(2) h(A(S_m)_[ω]l, A(S)_[ω]

l)≤c^lM+h(f_[ω]^m

l(A(S)), f_[ω]

l(A(S))).

Indeed

h(A(S_m)_[ω]l, A(S)_[ω]l) =h(f_[ω]^m

l(A(S_m)), f_[ω]l(A(S)))

≤h(f_[ω]^m

l(A(S_m)), f_[ω]^m

l(A(S))) +h(f_[ω]^m

l(A(S)), f_[ω]l(A(S)))

≤c^lh(A(S_m), A(S)) +h(f_[ω]^m

l(A(S)), f_[ω]l(A(S))).

Becausef_[ω]^m

l

u.c→f_[ω]l, for everyK ∈K(X) we have f_[ω]^m

l(K)→f_[ω]l(K), in particular,f_[ω]^m

l(A(S))→f_[ω]l(A(S)). Using (1) we obtain thatA(S_m)_[ω]l → A(S)_[ω]

l for everyl∈N^∗ and everyω ∈Λ.

We also have

(3) d(πm(ω), π(ω))≤2c^lM +h(A(S_m)_[ω]l, A(S)_[ω]l).

Indeed, becauseA(S_m)_[ω]l andA(S)_[ω]l are compact, there existx∈A(S_m)_[ω]l and y ∈ A(S)_[ω]l such that h(A(S_m)_[ω]l, A(S)_[ω]l) = d(x, y) (see [7]). Then, taking into account thatπm(ω), x∈A(S_m)_[ω]l andπ(ω), y∈A(S)_[ω]l, we have

d(πm(ω), π(ω))≤d(πm(ω), x) +d(x, y) +d(y, π(ω))

≤d(A(S_m)_[ω]l) +h(A(S_m)_[ω]l, A(S)_[ω]

l) +d(A(S)_[ω]

l)

≤c^ld(A(S_m)) +h(A(S_m)_[ω]l, A(S)_[ω]l) +c^ld(A(S))

≤2c^lM+h(A(S_m)_[ω]l, A(S)_[ω]l).

To see that π_n→^s π it is enough to take in (3) an l large enough and then an m large enough. Let ε > 0 be xed and let l be a natural number such that 6c^lM < ε. Because f_α^{m u.c}→f_αfor everyα∈Λ_l and Λ_l is nite, there existsm_ε such that h(f_α^m(A(S)), f_α(A(S))) < ε/2 for every m ≥mεand every α ∈ Λl. For m≥m_εfrom (1) and (3) we have

d(πm(ω), π(ω))≤2c^lM+h(A(S_m)_[ω]l, A(S)_[ω]l)

≤2c^lM+c^lM +h(f_[ω]^m

l(A(S)), f_[ω]

l(A(S)))≤ε/3 +ε/6 +ε/2 =ε.

3. THE SHIFT SPACE FOR AN RIFS

In this section we present the construction of the shift space of an RIFS and establish the main properties concerning the relation ship between it and the attractor of the RIFS.

Through out this section (X, d_X) will be a xed complete metric space and S = (X,(f_k)_k=1,n) a xed RIFS with ncomponents.

We start with some notation: k, l, m, i, j, m⁰, ij denotes natural numbers if not otherwise stated. For a nonvoid set I and a family of functions f_i : X_i → Y_i, i ∈ I, ×

i∈If_i denotes the function ×

i∈If_i : ×

i∈IX_i → ×

i∈IY_i dened by

×

i∈Ifi((xi)i∈I) = (fi(xi))i∈I.

SetΩk=N^∗

n^2k−1 fork∈N^∗. OnΩkwe consider the metricdk: Ωk×Ω_k→ R+ given bydk(x, y) = 1−δ^yx, whereδx^y =

1 if x=y, 0 if x6=y.

Consider the bijectionφ_kbetweenΩ_k×Ω_k =N^∗_n2k−1×N^∗_n2k−1 andΩ_k+1= N^∗_n2k given byφ_k(a, b) =n^2k−1(a−1) +b.

Letp^k₁ : Ωk×Ωk → Ωkand p^k₂ : Ωk×Ωk →Ωkbe dened by p^k₁(x, y) = x and p^k₂(x, y) = y. Let ψ^k₁ : Ω_k+1 → Ω_kand ψ₂^k : Ω_k+1 → Ω_k be dened by ψ₁^k(x) = p^k₁ ◦(φ_k)⁻¹(x) and ψ₂^k(x) = p^k₂ ◦(φ_k)⁻¹(x). More generally, we let φ_lk: (Ω_l)^2k−l→Ω_k, where

φ_lk=φk−1◦(φk−2×φk−2)◦. . .◦

₂k−l−1

×

i=1 φ_l

for 0< l < k. In particular, φ_(k−1)k=φk−1.

Let also θ_k : Ω_k → Ωk−1×Ωk−1 the inverse of φk−1 and θ_kl : Ω_k → (Ω_l)^2k−l be the inverse of φ_lk. That is θ_k=φ⁻¹_k−1 and

θ_kl=

2^k−l−1

×

i=1 φ⁻¹_l

◦

2^k−l−2

×

i=1 φ⁻¹_l−1

◦. . .◦

φ⁻¹_k−2×φ⁻¹_k−2

◦φ⁻¹_k−1

=

₂k−l−1

×

i=1 θ_l+1

◦

₂k−l−2

×

i=1 θl−1

◦. . .◦(θk−1×θk−1)◦θ_k fork > l >0.

We have θk=θk(k−1).

Set pⁿ_j = p^n,X_j : (X)²ⁿ → X, p^n,l_j (x1, x2, . . . , x2ⁿ) = xj, where X is a nonvoid set. In particular, if X = Ωl set p^n,l_j =p^n,Ω_j ^l. Then p^n,l_j : (Ωl)²ⁿ →Ωl

and p^n,l_j (x1, x2, . . . , x2ⁿ) =xj. Set τ_j^k,l =p^k−l,l_j ◦θkl : Ωk → Ωl for 0 < l < k and j∈ {1,2, . . . ,2^k−l}.

Remark 3.1. With the above notation we have (1)τ_j^k+1,k =ψ^k_j, wherej∈ {1,2};

(2)₂^k−l

×

i=1 θ_lm

◦θ_kl=θ_km, where0< m < l < k; (3)p^m,l_i

2 ◦p^m_i ^0,X

1 =p^m_i ^0+m,l

2+2^m(i1−1), wherem⁰, m, l∈N^∗,i₁ ∈ {1,2, . . . ,2^m0}, i2 ∈ {1,2, . . . ,2^m}and X= (Ωl)^2m;

(4) θ_mk ◦p^u,m_i = p^u,X_i ◦ ₂^u

×

i=1 θ_mk

, where 0 < k < m, u ∈ N^∗ and X = (Ω_k)^2m−k, so that, in particular, if u = l−m then θ_mk ◦p^l−m,m_i = p^l−m,X_i ◦₂^l−m

×

i=1 θmk

, wherek < m < l and X= (Ωk)^2m−k; (5)τ_j^m,k◦τ_i^l,m=τ_j+2^l,k m−k(i−1), wherek < m < l;

(6)τ_j^k,l=ψ_i^l₁◦ψ_i^l+1

2 ◦. . .◦ψ^k−1_i

k−l, where k < l,i1, i2, . . . , ik−l∈ {1,2}and j =i1+ (i2−1)2 +· · ·+ (ik−l−1)2^k−l−1.

Indeed, the assertions above can be justied as follows.

(1)τ_j^k+1,k =p^1,k_j ◦θ_k+1,k =p^1,k_j ◦φ⁻¹_k =ψ^k_j. (2)₂^k−l

×

i=1θ_lm

◦θ_kl = ₂^k−l

×

i=1

h₂^l−m−1

×

i=1 φ⁻¹_m

◦. . .◦

φ⁻¹_l−2×φ⁻¹_l−2

◦φ⁻¹_l−1 i

◦ ₂^k−l−1

×

i=1 φ⁻¹_l

◦. . .◦

φ⁻¹_k−2×φ⁻¹_k−2

◦φ⁻¹_k−1 =

₂^k−m−1

×

i=1 φ⁻¹_m

◦. . .◦

φ⁻¹_k−2×φ⁻¹_k−2

◦ φ⁻¹_k−1 =θkm.

(3) Let(x1, x2, . . . , x₂m0+m) = (y1, y2, . . . , y₂m0)∈(Ωl)^2m

0+m

= ((Ωl)^2m)^2m

0, where y_i = (x₁₊₂^m_(i−1), x₂₊₂^m_(i−1), . . . , x₂^m₊₂^m_(i−1)) ∈ X = (Ω_l)^2m. Then p^m_i ^0+m,l

2+2^m(i1−1)(x₁, x₂, . . . , x₂m0+m) =x_i2+2^m_(i1−1) and

p^m,l_i2 ◦p^n,X_i1 (x1, x2, . . . , x₂m0+m) =p^m,l_i2 ◦p^m_i1^0,X(y1, y2, . . . , y₂m0)

=p^m,l_i

2 (y_i1) =p^m,l_i

2 (x₁₊₂m(i1−1), x₂₊₂m(i1−1), . . . , x₂m+2^m(i1−1)) =x_i2+2m(i1−1). (4) Let (x₁, x₂, . . . , x₂^u) ∈ (Ω_m)^2u. Then ₂^u

×

i=1θ_mk

(x₁, x₂, . . . , x₂^u) = (θ_mk(x₁), θ_mk(x₂), . . . , θ_mk(x₂^u))andp^u,X_i ◦2^u

×

i=1θ_mk

(x₁, x₂, . . . , x₂^u) =θ_mk(x_i). Also p^u,m_i (x₁, x₂, . . . , x₂^u) =x_i and θ_mk◦p^u,m_i (x₁, x₂, . . . , x₂^u) =θ_mk(x_i).

(5)τ_j^m,k◦τ_i^l,m=p^m−k,k_j ◦θ_mk◦p^l−m,m_i ◦θ_lm =p^m−k,k_j ◦p^l−m,X_i ◦₂^l−m

×

i=1θmk

◦ θ_lm =p^l−k,l_j+2m−k(i−1)◦θ_lk=τ_j+2^l,k m−k(i−1), whereX= (Ω_k)^2m−k =θ_mk(Ω_m).

(6) The proof can be made by induction onk. The rst step k=l+ 1is point (1). Let j₁ =i₁+ (i₂−1)2 +· · ·+ (ik−l−1)2^k−l−1 and

j =j1+2^k−l(ik−l+1−1) =i1+2(i2−1)+· · ·+(i_k−l−1)2^k−l−1+(ik−l+1−1)2^k−l. From point (5) and the induction hypothesis we then have

τ_j^k+1,l =τ_j^k,l

1 ◦τ_i^k+1,k

k−l+1 =τ_j^k,l

1 ◦ψ^k_ik−l+1 =ψ_i^l₁◦ψ^l+1_i

2 ◦. . .◦ψ^k−1_i

k−l◦ψ_i^k_k−l+1. Denition 3.1. The space (Ω, dΩ), where Ω = Q

k≥1

Ω_k and dΩ is the dis- tance dΩ : Ω×Ω→R+ given bydΩ(α, β) = P

k≥1

dk(αk,βk)

3^k = P

k≥1 1−δ^βk_αk

3^k , is called the shift space or the code space of an RIFS with ncomponents. An element ω ∈ Ω will be written as an innite word ω = ω1ω2. . . ωmωm+1. . ., where ω_m∈Ω_m. In other words, Ω ={f :N^∗→N^∗|f(k)≤n^2k−1,k≥1}.

We remark that convergence in (Ω, dΩ) is component-wise convergence.

It is easy to prove

Lemma 3.1. (Ω, d_Ω) is a compact metric space.

We now dene the contractions (or the right shift functions) on(Ω, d_Ω). Denition 3.2. LetF_k: Ω×Ω→Ωbe dened by

F_k(α, β) =ω=kφ1(α1, β1)φ2(α2, β2). . . .φm(αm, βm). . . , that is, ω₁=kand ω_m =φm−1(αm−1, βm−1)for k= 1, n.

We also dene the reduction function (or the left shift function) on (Ω, dΩ).

Denition 3.3. (1) LetR₁ : Ω→Ωbe dened by R₁(ω) =ψ₁¹(ω₂)ψ²₁(ω₃) . . . ψ₁^m(ωm+1). . ., that is, R1(ω)n=ψⁿ₁(ωn+1).

(2) LetR₂ : Ω→Ωbe dened byR₂(ω) =ψ¹₂(ω₂)ψ²₂(ω₃). . . ψ^m₂ (ω_m+1). . ., that is, R2(ω)n=ψⁿ₂(ωn+1).

(3) LetR: Ω→Ω×Ω be dened byR(α) = (R₁(α), R₂(α)). Remark 3.2. R is a continuous function. Indeed

dΩ×dΩ(R(α), R(β)) = max(dΩ(R1(α), R1(β)), dΩ(R2(α), R2(β)))

= max X

k≥1

1−δ^R_R^1(α)k

1(β)_k

3^k ,X

k≥1

1−δ^R_R^2(α)k

2(β)_k

3^k

!

= max X

k≥1

1−δ^ψ

k 1(αk+1) ψ₁^k(βk+1)

3^k ,X

k≥1

1−δ^ψ

k 2(αk+1) ψ^k₂(βk+1)

3^k

!

≤X

k≥1

max

1−δ^ψ1k(αk+1)

ψ^k₁(βk+1),1−δ^ψk2(αk+1)

ψ₂^k(βk+1)

3^k

≤X

k≥1

1−δ_β^αk+1

k+1

3^k = 3X

k≥2

1−δ_β^αk

k

3^k ≤3X

k≥1

1−δ_β^αk

k

3^k = 3dΩ(α, β).

We also have

Lemma 3.2. The functions F_k: Ω×Ω→Ωdened above are contractions with Lipschitz constant less then 2/3, and Ω =

n

S

k=1

F_k(Ω,Ω). In other words, Ω is the attractor of the RIFS (Ω,(F_k)_k=1,n).

Proof. Obviously, Sⁿ

k=1

F_k(Ω,Ω)⊂Ω. Ifω∈Ωthenω=F_ω1(R₁(ω), R₂(ω)) and so Ω =

n

S

k=1

Fk(Ω,Ω). We also have d_Ω(F_k(α, α⁰), F_k(β, β⁰))≤1

3(d_Ω(α, β)+d_Ω(α⁰, β⁰))≤2

3max(d_Ω(α, β), d_Ω(α⁰, β⁰)).

Indeed,

dΩ(Fk(α, α⁰), Fk(β, β⁰)) =X

k≥1

1−δ_(F^{(Fk(β,β0))k}

k(α,α⁰))_k

3^k

= 1−δ_k^k

3 +X

k≥2

1−δ_(F^{(Fk(β,β0))k}

k(α,α⁰))k

3^k =X

k≥2

1−δ^{φk−1(βk−1,β}

0 k−1) φk−1(αk−1,α⁰_k−1)

3^k

≤ 1 3

X

k≥1

1−δ^β_α^k_k 3^k +X

k≥1

1−δ^β

0 k

α⁰_k

3^k

= 1

3(dΩ(α, β) +dΩ(α⁰, β⁰)).

Lemma 3.3. Consider the functions F_k: Ω×Ω→Ωand R: Ω→Ω×Ω as above. Then R◦Fk: Ω×Ω→Ω×Ωis the identity map and Fk◦R(ω) = kω₂ω₃. . . ω_mω_m+1. . ..

Ωis the set of innite words ω=ω1ω2. . . ωmωm+1. . ., where ωm ∈Ωm. We describe now the sets[Ω]_mof nite words of lengthm. Let[Ω]_m =

Qm

k=1

Ω_k= {f : N^∗_m→N^∗| f(k) ≤ n^2k−1,1 ≤ k ≤ m}. The elements of [Ω]m are words ω =ω₁ω₂. . . ω_m of lengthm, whereω_k∈Ω_k.

Denition 3.4. LetΩ^∗ = S

m≥1

[Ω]_m be the set of nite words. Forω∈Ω^∗,

|ω| denotes the length of ω. If ω ∈ Ω then |ω| = +∞. As above, if ω = ω₁ω₂. . . ω_mω_m+1. . . then [ω]_m = ω₁ω₂. . . ω_m and [ω]_m ∈ [Ω]_m. For α ∈ Ω^∗ and β ∈Ω^∗ orβ ∈Ωwe say thatα≺β if |α| ≤ |β|and [β]|α|=α.

The functions F_kfor k = 1, n, R, R₁, R₂ as above can be dened in a similar manner on nite words, on which they have similar properties. We can take F_k : [Ω]_m ×[Ω]_m → [Ω]_m+1, m ∈ N^∗, or F_k : Ω^∗ ×Ω^∗ → Ω^∗, R₁, R₂ : [Ω]_m → [Ω]m−1, m ∈ N^∗, m ≥ 2, or R₁, R₂ : Ω^∗−[Ω]₁ → Ω^∗ and R : [Ω]m→[Ω]m−1×[Ω]m−1 or R: Ω^∗−[Ω]1→Ω^∗×Ω^∗.

Remark 3.3. With the above notation we have

(1)[Fk(α, β)]m =Fk([α]m−1,[β]m−1), whereα, β ∈Ω∪Ω^∗ with|α| ≥m and |β| ≥m.

(2)[R1(α)]m=R1([α]m+1), whereα∈Ω∪Ω^∗ and|α| ≥m+ 1.

(3)[R2(α)]m=R2([α]m+1), whereα∈Ω∪Ω^∗ and|α| ≥m+ 1.

Letω∈[Ω]_m. We are going to dene the functionf_ω : ⁿ

2m−1

×

k=1

X→Xand its extension fω : ⁿ

2m−1

×

k=1 P(X)→P(X). We will use the same notation for fω

and for its extension. Becausef_ω is a continuous function,f_ω( ⁿ

2m−1

×

k=1

K(X))⊂

K(X), and so we can consider fω : ⁿ

2m−1

×

k=1 K(X) → K(X). The construc- tion will be made by induction on the length of the word ω. For ω ∈ Ω we have f_[ω]1 = fω1 where fω1 : X ×X → X is the ω1-function from the RIFS S = (X,(f_k)_k=1,n); f_[ω]2 = f_ω1ω2 is the function f_[ω]2 : X×X×X× X → X given by f_[ω]2(x1, x2, x3, x4) = f_[ω]1(f_ψ¹

1(ω2)(x1, x2), f_ψ¹

2(ω2)(x3, x4))

=f_[ω]1(f_τ^2,1

1 (ω2)(x1, x2), f_τ^2,1

2 (ω2)(x3, x4));f_[ω]3 =fω1ω2ω3 is the function f_[ω]3 : X⁸ → X given by f_[ω]3(x1, x2, . . . , x8) = f_[ω]1(f_R1([ω]3)(x1, x2, x3, x4), f_R2([ω]3)(x5, x6, x7, x8)),or explicitly,fω1ω2ω3(x1, x2, . . . , x8) =fω1(f_R1([ω]3)(x1, x2, x3, x4), f_R2([ω]3)(x5, x6, x7, x8))=fω1(f_ψ¹

1(ω2)ψ₁²(ω3)(x1, x2, x3, x4),f_ψ¹

2(ω2)ψ₂²(ω3)

(x5, x6, x7, x8)) = fω1(f_ψ¹

1(ω2)[f_ψ¹

1(ψ²₁(ω3))(x1, x2), f_ψ¹

2(ψ₁²(ω3))(x3, x4)], f_ψ¹

2(ω2)

[f_ψ¹

1(ψ₂²(ω3))(x5, x6), f_ψ¹

2(ψ²₂(ω3))(x7, x8)]) = fω1(f_ψ¹

1(ω2)[f_τ^3,1

1 (ω3)(x1, x2), f_τ^3,1

2 (ω3)

(x₃, x₄)], f_ψ¹

2(ω2)[f_τ^3,1

3 (ω3)(x₅, x₆), f_τ^3,1

4 (ω3)(x₇, x₈)]) = f_ω1ω2((f_τ^3,1

1 (ω3)(x₁, x₂), f_τ^3,1

2 (ω3)(x3, x4), f_τ^3,1

3 (ω3)(x5, x6), f_τ^3,1

4 (ω3)(x7, x8)). In general, f_[ω]m(x1, x2, . . . , x₂^m) =f_[ω]1(f_R1([ω]m)(x₁, x₂, . . . , x₂^m−1), f_R2([ω]m)(x₂^m−1₊₁, x₂^m−1₊₂, . . . , x₂^m)).

Let(X, d_X) be a complete metric space and S = (X,(f_k)_k=1,n) a RIFS.

Let H, H_k⊂X be sets. Then

f_[ω]m(H1, H2, . . . , H2^m) =f_[ω]m(H1×H2× · · · ×H2^m)

={f_[ω]m(x₁, x₂, . . . , x₂^m)|x_k∈H_k} and H_[ω]m =f_[ω]m(H, H, . . . , H).

Remark 3.4. With the above notation, if c= max

k=1,n Lip(fk), ω ∈ Ω and m∈N^∗, then Lip(f_[ω]m)≤c^m, and sod(H_[ω]m)≤c^md(H).

Lemma 3.4. With the above notation we have f_[ω]m(x1, x2, . . . , x2^m) = f_[ω]m−1(f_τ^m,1

1 (ωm)(x1, x2), f_τ^m,1

2 (ωm)(x3, x4), . . . , f_τ^m,1

2m−1(ωm)(x2^m−1, x2^m)).

Proof. The result can be proved by induction onm. The rst step follows from the very denition. For the induction step we havef_[ω]m(x₁, x₂, . . . , x₂^m)

= f_[ω]1(f_R1([ω]m)(x₁, x₂, . . . , x₂^m−1), f_R2([ω]m)(x₂^m−1₊₁, x₂^m−1₊₂, . . . , x₂^m)) =