ITERATED FUNCTION SYSTEMS

ITERATED FUNCTION SYSTEMS

ALEXANDRU MIHAIL

We present a generalization of the Hutchinson measure. This is the invariant measure associated with an iterated function system with probabilities, an IFSp for short. We prove the existence and study the invariant measure for a GIFS.

Generalized iterated function systems are a generalization of iterated function systems. The idea of this generalization is to consider contractions fromX^mto X rather then contractions from a metric spaceX to itself.

AMS 2000 Subject Classification: 28A80.

Key words: iterated functions system, generalized iterated function system, Hutchinson measure, Hutchinson distance, attractor.

1. INTRODUCTION

First, we fix the notation and present the main results concerning GIFS and IFSp.

Let (X, d_X) and (Y, d_Y) be two metric spaces. By C(X, Y) we will un- derstand the set of continuous functions from X to Y. On C(X, Y) we will consider the generalized metric d:C(X, Y)×C(X, Y) → R+ =R+∪ {+∞}

defined by

d(f, g) = sup

x∈X

dY(f(x), g(x)).

IfZ ⊂X thend_Z(f, g) = sup

x∈Z

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

For f_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.

Let (Xk, dk) be metric spaces, 1 ≤ k ≤ m. On ×^m

k=1Xk we consider the distances

d^max: _m

×

k=1X_k

× _m

×

k=1X_k

→R+

REV. ROUMAINE MATH. PURES APPL.,54(2009),4, 297–316

defined as

d^max((x₁, x₂, . . . , x_m),(y₁, y₂, . . . , y_m)) =max^m

k=1 d_k(x_k, y_k) and

d^a: m

×

k=1X_k

× m

×

k=1X_k

→R+

defined as

d^a (x₁, x₂, . . . , x_m),(y₁, y₂, . . . , y_m)

=

m

X

k=1

a_kd_k(x_k, y_k), where a= (a1, a2, . . . , am)∈(R^∗+)^m.

Definition 1.1. Let (X, d_X) and (Y, d_Y) be metric spaces. For a function f :X →Y we denote by Lip(f)∈[0,+∞] the Lipschitz constant associated with f, that is,

Lip(f) = sup

x,y∈X;x6=y

d_Y(f(x), f(y)) d_X(x, y) =

= inf{c|d_Y(f(x), f(y))≤cd_X(x, y) for all x, y∈X}.

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}.

Definition 1.2. Let (X, dX) and (Y, dY) be metric spaces. For a function f :X^m→Y the number

Lip(f) = inf{c|d_Y(f(x₁, x₂, . . . , x_n), f(y₁, y₂, . . . , y_n))≤

≤cmax^m

j=1 d_X(x_j, y_j) for allx_j, y_j ∈X}, or, equivalently,

sup

x1,...,xm,y1,...,ym∈X max{dX(x1,y1),...,dX(xm,ym)}>0

d_Y(f(x1, x2, . . . , xm), f(y1, y2, . . . , ym)) max{d_X(x₁, y₁), . . . , d_X(x_m, y_m)} , is denoted byLip(f) and it is called the Lipschitz constant off.

A functionf :X^m→Y is said to be a Lipschitz function ifLip(f)<+∞

and a contraction if Lip(f)<1. Let

Conm(X) ={f :X^m →X |Lip(f)<1} ⊂C(X^m, X), Lip^m_a(X, Y) ={f :X^m →Y |Lip(f)≤a} ⊂C(X^m, Y),

Lip^m_a1,a2,...,am(X, Y) =

= n

f :X^m→Y |dY(f(x1, x2, . . . , xn), f(y1, y2, . . . , yn))≤

m

X

j=1

ajdX(xj, yj) o

, and

Lip^∗,m_a (X, Y) = [

a1+a2+···+am≤a a1>0, a2>0,...,am>0

Lip^m_a1,a2,...,am(X, Y) =

={f :X^m →Y | there existsa1, a2, . . . , am∈(0,∞) such that f ∈Lip^m_a1,a2,...,am(X, Y) and a1+a2+· · ·+am ≤a}, where a, a₁, a₂, . . . , a_m ∈(0,∞).

Then

Lip^m_a1,a2,...,am(X, Y)⊂Lip^∗,m_{a1+a2+···+am}(X, Y)⊂

⊂Lip^m_{a1+a2+···+am}(X, Y)⊂C(X^m, Y).

We will simplify this notation when no confusion can arise.

For a setX, P(X) denotes the set of subsets ofX andP^∗(X) =P(X)− {∅}. For a setA⊂P(X) we put A^∗=A− {∅}.

Let (X, d) be a metric space; K(X) denotes the set of compact subsets of X and B(X) denotes the set of closed bounded subsets of X. It is obvious that K(X)⊂B(X)⊂P(X).

OnP^∗(X) we consider the generalized Hausdorff-Pompeiu semidistance h:P^∗(X)×P^∗(X)→[0,+∞] defined by

h(A, B) = max(d(A, B), d(B, A)), where

d(A, B) = sup

x∈A

d(x, B) = sup

x∈A

y∈Binf d(x, y) .

The Hausdorff-Pompeiu semidistance is a distance onB^∗(X) and in particular on K^∗(X). Both (B^∗(X), h) and (K^∗(X), h) are complete metric spaces if (X, d) is a complete metric space (for a proof see [4] for B^∗(X) and [8], for K^∗(X) for example).

Definition 1.3. Let (X, d) be a complete metric space. A generalized iterated function system on X of order m, a GIFSm for short, consists of a finite family of continuous functions (fk)_k=1,n,fk:X^m→X. It is denoted by S = (X,(fk)_k=1,n). Usually, these functions are contractions.

Definition 1.4. Let f : X^m → Y be a function. The function F_f : P(X)^m→P(Y) defined by

F_f(K₁, K₂, . . . , K_m) =f(K₁×K₂× · · · ×K_m) =

={f(x₁, x₂, . . . , x_m)|x_j ∈K_j forj= 1, m}

is called the set function associated with f.

Let S = (X,(f_k)_k=1,n) be a GIFSm. The function FS : P^∗(X)^m → P^∗(X) defined by

FS(K1, K2, . . . , Km) =

n

[

k=1

Ff_k(K1, K2, . . . , Km) =

n

[

k=1

fk(K1×K2× · · · ×Km) is called the set function associated with the GIFS S = (X,(f_k)_k=1,n). Since the f_k are continuous, we haveFS(K^∗(X)^m)⊂K^∗(X).

Clearly, Lip(FS)≤max

k=1,n

Lip(f_k). It follows that if f_k are contractions, then FS also is a contraction.

We will also consider GIFSs wherefk ∈Lip^m_a1,a2,...,am(X, X), (a1, a2, . . . , am)∈(R^∗+)^mand a1+a2+· · ·+am <1. Such a GIFS is called an (a1, a2, . . . , a_m)-GIFS. In this case,FS ∈Lip^m_a1,a2,...,am(X, X) and we can obtain a better estimation of the speed of convergence.

Using an extension of the Banach contraction principle and related results in [6] sufficient conditions have been obtained for existence and uniqueness of the attractor of a GIFS. These results will be also used to study the genera- lized Hutchinson measure. We state here these results and their applications to GIFS.

Theorem1.1 (generalized Banach contraction principle for Con_m(X)).

For every f ∈ Conm(X) there exists a unique α ∈ X such that f(α, α, . . . , α) = α. For every x₀, x₁, . . . , xm−1 ∈ X, the sequence (x_n)n≥1 defined by x_k+m=f(x_k+m−1, x_k+m−2, . . . , x_k) for all k∈N converges to α.

As for the rate of convergence, we have

d(xn, α)≤ m(Lip(f))^[mn]max{d(x₀, x1), d(x1, x2), . . . , d(xn−1, xn)}

1−Lip(f) ,

for every n∈N.

Theorem 1.2. Let (X, d) be a complete metric space and S = (X, (f_k)_k=1,n) a GIFS of order m with c = max

k=1,n

Lip(f_k) < 1. Then there ex- ists a unique A(S) ∈ K(X) such that FS(A(S), A(S), . . . , A(S)) = A(S).

Moreover, for any H₀, H₁, . . . , Hm−1 ∈ K(X) the sequence (H_n)n≥1 defined by

H_k+m=FS(Hk+m−1, Hk+m−2, . . . , H_k)

converges to A(S). As for the rate of convergence we have h(H_n, A(S)) ≤

mc^[mn]

1−c max{h(H₀, H1), h(H1, H2), . . . , h(Hm−1, Hm)}. In particular, h(H0, A(S))≤ m

1−cmax{h(H₀, H1), . . . , h(Hm−1, Hm)}.

We also have

h(H0, A(S))≤ 1

1−ch(H0, FS(H0, H0, . . . , H0)).

Theorem1.3. If f, g∈Conm(X) have the fixed points α and β, then d(α, β)≤min

1

1−Lip(f)d(f(β, β, . . . , β), β), 1

1−Lip(g)d(α, g(α, α, . . . , α))

≤ 1

1−min{Lip(f),Lip(g)}d(f, g).

Theorem 1.4. If S = (X,(fk)_k=1,n) and S⁰ = (X,(gk)_k=1,n) are two GIFSs of order m such that

c= max{Lip(f₁), . . . ,Lip(fn)}<1andc⁰= max{Lip(g₁), . . . , Lip(gn)}<1, then

h(A(S), A(S⁰))≤ 1

1−min(c, c⁰)max{d(f₁, g₁), . . . , d(f_n, g_n)}.

Theorem 1.5. Let f_n, f ∈ Con_m(X) with fixed points α_n and α such that sup

n≥1

Lip(fn)<1 and fn s

→f on a dense set in X^m. Then αn→α.

Theorem1.6 (generalized Banach contraction theorem forLip^m_a1,...,am(X, X)). Let (X, d) be a complete metric space. Suppose there exists a mapping f ∈Lip^m_a1,a2,...,am(X, X)witha1+a2+· · ·+a_m<1. Then there exists a unique α ∈X such that f(α, α, . . . , α) =α. Moreover, for any x0, x1, . . . , xm−1 ∈X the sequence (x_n)n≥0 defined by x_k+m=f(xk+m−1, xk+m−2, . . . , x_k)converges to α. As for the rate of convergence we have

d(xn, α)≤X

k≥n

z_k≤ m(a1+a2+· · ·+am)^[mn] 1−(a₁+a₂+· · ·+a_m)

maxm

j=1 d(xj−1, xj),

where the sequence (z_n)n≥1 is defined by z_n+m =a₁zn+m−1+a₂zn+m−2+· · ·+

amzn and z0=d(x0, x1),z1=d(x1, x2), . . . , zm−1 =d(xm−1, xm).

Proof. If f ∈ Lip^m_a1,a2,...,am(X, X) then f ∈ Lip^m_{a1+a2+···+am}(X, X). It follows that the assumptions of Theorem 1.3 are fulfilled. This means that there exists a unique α ∈ X such that f(α, α, . . . , α) = α and for every x₀, x₁, . . . , xm−1 ∈ X the sequence (x_n)n≥0 defined by x_k+m = f(xk+m−1, xk+m−2, . . . , x_k) converges toα. It remains to prove that

d(xn, α)≤X

k≥n

z_k≤ m(a1+a2+· · ·+am)^[mn] 1−(a₁+a₂+· · ·+a_m)

maxm

j=1 d(xj−1, xj).

We haved(xn+1, xn)≤zn. This can be proved by induction. Forn∈ {0,1, . . . , m−1}this holds by the definition of thez_k, 0≤k≤m−1. By induction we have

d(x_n+m, xn+m−1) =

=d(f(xn+m−1, xn+m−2, . . . , x_n), f(xn+m−2, xn+m−3, . . . , xn−1))≤

≤a₁d(xn+m−1, xn+m−2) +a₂d(xn+m−2, xn+m−3) +· · ·+a_md(x_n, xn−1)≤

≤a1zn+m−1+a2zn+m−2+· · ·+amzn=zn+m. It follows that

d(xn+p, xn)≤d(xn+p, xn+p−1) +d(xn+p−1, xn+p−2) +· · ·+d(xn+1, xn)≤

≤zn+p−1+zn+p−2+· · ·+z_n, and d(xn, α)≤ P

k≥n

zk.

Letm_n= max(z_n, zn−1, . . . , zn−m+1). We havem_n+1 ≤m_n. Indeed, zn+1 =a1zn+a2zn−1+· · ·+amzn−m+1≤mn(a1+a2+· · ·+am)≤mn

and

mn+1= max(zn+1, zn, . . . , zn−m+2)≤max(zn+1, mn)≤mn.

Also, zn+2 ≤(a1+a2+· · ·+am)mn+1 ≤(a1+a2+· · ·+am)mn, and so on.

It follows that

mn+m= max(zn+m, zn+m−1, . . . , zn)≤(a1+a2+· · ·+am)mn

and

z_n≤(a₁+a₂+· · ·+a_m)^[mn]mm−1. Therefore,

d(xn, α)≤X

k≥n

zk≤ m(a₁+a₂+· · ·+a_m)^[mn] 1−(a1+a2+· · ·+am)

maxm

j=1 d(xj−1, xj).

Remark 1.1. Assume m = 2. Let (z_n)n≥1 be the sequence defined by z_n+2=az_n+1+bz_n,z₀ >0 andz₁ >0. Thenz_n= ^z1_r^−r1z0

1−r2 (r₁)ⁿ+^z1_r^−r2z0

2−r1 (r₂)ⁿ and P

k≥n

z_k = ^z1_r^−r1z0

1−r2

(r1)ⁿ

1−r1 +^z1_r^−r2z0

2−r1

(r2)ⁿ

1−r2, where r1 and r2 are the solutions of the equationr²−ar−b= 0. That is,r1 = ^a+

√a²+4b

2 and r2 = ^a−

√a²+4b 2 . We have |r₂|< r1 <√

a+b. So, lim

n→∞

(r1)ⁿ

(a+b)^[n2] = 0 and lim

n→∞

zn

(a+b)^[n2] = 0.

2. ITERATED FUNCTION SYSTEMS WITH PROBABILITIES We now consider the IFSp.

Definition 2.1. An iterated function system with probabilities, an IFSp for short, consists of a metric space (X, d), a finite family of contractions fk :X → X and a system of probabilities (pk)_k=1,n. That is, pk ∈ (0,1) and p1+p2+· · ·+pn= 1. We denote the IFSp by S= (X,(fk)_k=1,n,(pk)_k=1,n).

Definition 2.2. Let (X, d_X) be a complete metric space. Then 1)B(X) denotes the space of finite signed Borel measures on X;

2)B₊(X) denotes the space of finite positive Borel measures on X;

3)supp(µ) denotes the support of the measureµ∈ B(X),|µ|the total variations of µ and kµk = |µ|(X) is the norm of the measureµ; supp(µ) =

T

H=H;µ(H)=µ(X)

H if µ ∈ B₊(X) while supp(µ) = supp(µ+) ∪ supp(µ−), where µ=µ+−µ− is the Hahn decomposition ofµ∈ B(X);

4) B_c(X) denotes the space of finite signed Borel measures on X with compact support;

5) B_1,+(X) denotes the space of positive Borel measures on X with µ(X) = 1;

6)M(X) denotes the space of finite positive Borel measures onX with compact support such that µ(X) = 1. M(X) =B_1,+(X)∩ B_c(X).

Definition 2.3. M(X) with the distance dH : M(X)× M(X) → R+

given by

dH(µ, ν) = sup

Z

X

fdµ− Z

X

fdν

f ∈Lip₁(X,R)

is a metric space. The distance dH is called the Hutchinson distance.

Remark 2.1. With the above notation we have 1)d_H(µ, ν) = sup{|R

Xfdµ−R

Xfdν|

f ∈Lip_a(X,R)} |afor everya >0.

2) If x0 ∈ X then dH(µ, ν) = sup{|R

Xfdµ−R

Xfdν|

f ∈ Lip₁(X,R) and f(x₀) = 0}.

In fact, the topology of the space (M(X), d_H) is the weak topology on M(X). So, we have (see [2], [3] or [8]) the results below.

Theorem2.1. Let (X, d_X) be a compact metric space. Then

a)on B_1,+(X) =M(X) the topology generated by the metric dH is the weak topology;

b) the metric space (B_1,+(X), dH) is compact.

Definition 2.4. Let (X, dX) be a complete metric space and S = (X, (f_k)_k=1,n, (p_k)_k=1,n) an IFSp. The Markov operator associated with S is the function MS : B(X) → B(X) defined by MS(µ) = p₁µ◦f₁⁻¹+p₂µ◦f₂⁻¹+

· · ·+p_nµ◦f_n⁻¹.

This means that for continuous functions with compact support or for simple functions f :X →Rwe have

Z

X

fd(MS(µ)) =p1

Z

X

f ◦f1dµ+p2

Z

X

f◦f2dµ+· · ·+pn

Z

X

f ◦fndµ.

Clearly,Lip(MS)≤ max

k=1,n

Lip(f_k). It follows that if f_k are contractions, then MS also is a contraction. By the Banach contraction principle, there exists a unique µS such thatMS(µS) =µS. Also, the support of the measure µS is the attractor of the IFS S. The measure µS is called the Hutchinson measure associated withS.

Remark 2.2. On B_1,+(X^m) we have different Hutchinson distances de- pending on the distance we take on X^m as a product space. So, we can consider

d^max_H (µ, ν) = sup

Z

X^m

fdµ− Z

X^m

fdν

f ∈Lip^m₁ (X,R)

, d^a_H^1,a2,...,am(µ, ν) = sup

Z

X^m

fdµ− Z

X^m

fdν

f ∈Lip^m_a1,a2,...,am(X^m,R)

. Remark 2.3. a)Lip^m_a(X,R) =Lip_a((X^m, d^max),R).

b) Lip₁((X^m, d^a1,a2,...,am), X) =Lip^m_a1,a2,...,am(X, X).

3. THE CASE OF AN (a₁, a₂, . . . , a_m)-GIFSp

In this section we define a generalized iterated function system with probabilities and prove that an (a1, a2, . . . , am)-GIFSp witha1+a2+· · ·+a_m<

1 has a unique invariant measure.

Definition 3.1. Let (X, d) be a complete metric space. A generalized iter- ated function system with probabilities onX of orderm, a GIFSpm for short, consists of a finite family of continuous functions (f_k)_k=1,n,f_k:X^m→Xand a system of probabilities (p_k)_k=1,n. It is denoted byS = (X,(f_k)_k=1,n,(p_k)_k=1,n).

Usually, these functions are contractions.

Let (a₁, a₂, . . . , a_m) ∈ (R^∗₊)^m. An (a₁, a₂, . . . , a_m)-generalized iterated function system with probabilities, an (a1, a2, . . . , am)-GIFSp for short, is a GIFSpm such that f_k ∈ Lip^m_a1,a2,...,am(X, X) for k = 1, n and a system of probabilities (p_k)_k=1,n. Letc∈(0,1). A∗c-generalized iterated function system

of order m with probabilities, a ∗c-GIFSpm for short, is a GIFSp of order m such that f_k∈Lip^∗,m_c (X, X) for k= 1, n.

Definition 3.2. Let (X, d_X) be a complete metric space and S = (X, (f_k)_k=1,n, (p_k)_k=1,n) a GIFSpm. The measure operator associated with S is MS :B(X)^m → B(X) defined by MS(µ1, µ2, . . . , µm) =

n

P

k=1

p_{k m}

×

l=1µ_l

◦f_k⁻¹. For m = 2 the function MS : B(X)× B(X) → B(X) is defined by MS(µ, ν) =p1(µ×ν)◦f₁⁻¹+p2(µ×ν)◦f₂⁻¹+· · ·+pn(µ×ν)◦f_n⁻¹.

This means that for continuous functions with compact support and simple functions f :X →Rwe have

Z

X

fd(MS(µ1, µ2, . . . , µm)) =

n

X

k=1

p_k Z

X^m

f ◦f_k(x1, x2, . . . , xm)d _m

×

l=1µ_l(x_l)

. In particular, for m= 2, we have

Z

X

fd(MS(µ, ν)) =

n

X

k=1

pk

Z

X²

f ◦fk(x, y)dµ(x)×ν(y) =

=p₁ Z

X²

f◦f₁(x, y)dµ(x)×ν(y) +· · ·+p_n Z

X²

f◦f_n(x, y)dµ(x)×ν(y).

Definition 3.3. Let (X, d_X) be a complete metric space and S = (X, (fk)_k=1,n, (pk)_k=1,n) a GIFSpm. A measure µis called an invariant measure for the GIFSpmSifµ=MS(µ, µ, . . . , µ), that is,µ=

n

P

k=1

pk

_m

×

l=1µ

◦f_k⁻¹. For m= 2 we then haveµ=p₁(µ×µ)◦f₁⁻¹+p₂(µ×µ)◦f₂⁻¹+· · ·+p_n(µ×µ)◦f_n⁻¹.

Remark 3.1. With the above notation one can assert that 1) supp(MS(µ₁, µ₂, . . . , µ_m))⊂

n

S

k=1

f_k

supp _m

×

l=1

µ_l

=FS

supp _m

×

l=1

µ_l

=FS

_m

×

l=1

supp(µ_l)

=FS supp(µ₁),supp(µ₂), . . . ,supp(µ_m)

; 2) if µ₁, µ₂, . . . , µ_m ∈ B₊(X) then supp(MS(µ₁, µ₂, . . . , µ_m)) =

Sn

k=1

f_k(supp(µ₁),supp(µ₂), . . . ,supp(µ_m)) =FS(supp(µ₁),supp(µ₂), . . . ,supp(µm));

3) MS(B₊(X)^m)⊂ B₊(X), andMS(B_c(X)^m)⊂ B_c(X);

4) MS(B_1,+(X)^m)⊂ B_1,+(X);

5) MS(M(X)^m)⊂ M(X).

Lemma3.1. 1)If f ∈Lip_c(Y, Z)and g∈Lip^m_a1,...,am(X, Y),then f◦g∈ Lip^m_ca1,...,cam(X, Z).

2) If fi ∈ Lip_ai(X,R), i = 1, n, and h : X^m → R is given by h(x1, x₂, . . . , x_m) =f₁(x₁) +f₂(x₂) +· · ·+f_m(x_m), then h ∈ Lip^m_a1,a2,...,am(X,R).

The function hwill be denoted by ⊕^m

l=1fl.

Notation. For a functionf :X→Y, define the function G^m_l (f) :X^m → Y by G^m_l (f)(x1, x2, . . . , xm) =f(x_l). We haveLip(G^m_l (f)) =Lip(f).

Convention. For two natural numbers m, n such that m < n, ×^m

l=nµ_l and

⊕m

l=nf_l contains no terms. For example, µ× m

×

l=nµ_l

=µ.

Proposition3.1.Let (X, dX)be a complete metric space and µ1, µ2, . . . , µ_m,µ⁰₁, µ⁰₂, . . . , µ⁰_m∈ M(X). Then

1) max{d_H(µ1, µ⁰₁), dH(µ2, µ⁰₂), . . . , dH(µm, µ⁰_m)} ≤d^max_{H m}

×

l=1µl, ×^m

l=1µ⁰_l

; 2)d^max_H

_m

×

l=1µl, ×^m

l=1µ⁰_l

≤

m

P

l=1

dH(µl, µ⁰_l);

3)d^a_H^1,a2,...,am m

×

l=1

µ_l, ×^m

l=1

µ⁰_l

=

m

P

l=1

a_ld_H(µ_l, µ⁰_l).

Proof. 1) It is enough to prove that d^∞_H(µ_l, µ⁰_l) ≤d^∞_{H m}

×

l=1µ_l, ^m×

l=1µ⁰_l

for every l= 1, m. By Fubini’s theorem, we have

d^max_{H m}

×

l=1

µ_l, ^m×

l=1

µ⁰_l

= sup

Z

X^m

hd _m

×

l=1

µ_l

− Z

X^m

hd _m

×

l=1

µ⁰_l

|h∈Lip^m₁ (X,R)

≥sup

Z

X^m

G^m_l (f)d _m

×

l=1µ_l

− Z

X^m

G^m_l (f)d _m

×

l=1µ⁰_l

|f ∈Lip₁(X,R)

= sup

Z

X

fdµ_l− Z

X

fdµ⁰_l

|f ∈Lip₁(X,R)

=d_H(µ_l, µ⁰_l).

2)

d^max_{H m}

×

l=1µ_l, ^m×

l=1µ⁰_l

=

= sup

Z

X^m

hd _m

×

l=1 µl

− Z

X^m

hd _m

×

l=1 µ⁰_j

|h∈Lip₁(X^m,R)

= sup

h∈Lip₁(X^m,R)

m

X

l=1

Z

X^m

hd l

×

j=1µ_j

× m

×

j=l+1

µ⁰_j

− Z

X^m

hdl−1

×

j=1µ_j

×m

×

j=l

µ⁰_j

≤ sup

h∈Lip₁(X^m,R)

^m X

l=1

Z

X^m

hd _l

×

j=1µj

× _m

×

j=l+1µ⁰_j

− Z

X^m

hd l−1

×

j=1µj

×_m

×

j=lµ⁰_j

≤

m

X

l=1

sup

h∈Lip₁(X^m,R)

Z

X^m

hd _l

×

j=1µj

× _m

×

j=l+1µ⁰_j

− Z

X^m

hd _l−1

×

j=1µj

×_m

×

j=lµ⁰_j

≤

m

X

l=1

sup Z

X^m−1

Z

X

h(x₁, . . . , x_m)dµ_l(x_l)− Z

X

h(x₁, . . . , x_m)dµ⁰_l(x_l) ·

·d l−1

×

j=1µl(xj)

× _m

×

j=l+1µ⁰_l(xj)

|h∈Lip₁(X^m,R)

≤

m

X

l=1

sup

Z

X^m−1

dH(µl, µ⁰_l)d l−1

×

j=1µl(xj)

×

× _m

×

j=l+1

µ⁰_l(x_j)

|h∈Lip₁(X^m,R)

=

m

X

l=1

d_H(µ_l, µ⁰_l).

3) We first prove thatd^a_H^1,a2,...,am _m

×

l=1µl, ^m×

l=1µ⁰_l

≤

m

P

l=1

aldH(µl, µ⁰_l).

As in the proof of point 2), by Fubini’s theorem, we have d^a_H^1,a2,...,am _m

×

l=1

µ_l, ×^m

l=1

µ⁰_l

=

= sup

Z

X^m

hd m

×

l=1

µ_l

− Z

X^m

hd m

×

l=1

µ⁰_l

h∈Lip^m_a1,a2,...,am(X,R)

=

= sup

h∈Lip^m_a

1,a2,...,am(X,R)

m

X

l=1

Z

X^m

hd _l

×

j=1µj

× _m

×

j=l+1µ⁰_j

−

− Z

X^m

hdl−1

×

j=1µ_j

× m

×

j=l

µ⁰_j

≤

≤ sup

h∈Lip^m_a

1,a2,...,am(X,R)

m

X

l=1

Z

X^m

hd _l

×

j=1

µ_j

× _m

×

j=l+1

µ⁰_j

−

− Z

X^m

hdl−1

×

j=1µ_j

× m

×

j=l

µ⁰_j

≤

≤

m

X

l=1

sup

h∈Lip^m_a

1,a2,...,am(X,R)

Z

X^m

hd _l

×

j=1µ_j

× _m

×

j=l+1

µ⁰_j

−

− Z

X^m

hd _l−1

×

j=1µj

× _m

×

j=lµ⁰_j

≤

≤

m

X

l=1

sup Z

X^m−1

Z

X

h(x₁, . . . , x_m)dµ_l(x_l)− Z

X

h(x₁, . . . , x_m)dµ⁰_l(x_l) ·

·dl−1

×

j=1µ_l(x_j)

× m

×

j=l+1

µ⁰_l(x_j)

|h∈Lip^m_a1,a2,...,am(X,R)

.

Since the functionstx1,...,xl−1,xl+1,...,xm ∈Lip_al(X,R), where t_{x1,...,xl−1,xl+1,...,xm}(x_l) =h(x₁, x₂, . . . , x_m), we have

Z

X

h(x₁, x₂, . . . , x_m)dµ_l(x_l)− Z

X

h(x₁, x₂, . . . , x_m)dµ⁰_l(x_l)

≤a_ld_H(µ_l, µ⁰_l).

Then

d^a_H^1,a2,...,am _m

×

l=1µl, ×^m

l=1µ⁰_l

≤

≤

m

X

l=1

sup

h∈Lip^m_a

1,a2,...,am(X,R)

Z

X^m−1

a_ld_H(µ_l, µ⁰_l)d l−1

×

j=1µ_l(xj)

× _m

×

j=l+1µ⁰_l(xj)

=

=

m

X

l=1

a_ld_H(µ_l, µ⁰_l).

We now prove thatd^a_H^1,a2,...,am m

×

l=1

µ_l, ×^m

l=1

µ⁰_l

≥P^m

l=1

a_ld_H(µ_l, µ⁰_l). As in the proof of point 1), withθ:X →R the zero function, we have

d^a_H^1,a2,...,am _m

×

l=1µl, ×^m

l=1µ⁰_l

=

= sup

Z

X^m

hd _m

×

l=1µl

− Z

X^m

hd _m

×

l=1µ⁰_l

|h∈Lip^m_a1,a2,...,am(X,R)

≥

≥sup

Z

X^m

⊕m l=1f_ld

_m

×

l=1µ_l

− Z

X^m

⊕m l=1f_ld

_m

×

l=1µ⁰_l

|f_l∈Lip_al(X,R), l= 1, m

= sup

m

X

l=1

Z

X^m

l−1

⊕

j=1θ

⊕fl⊕ _m

⊕

j=l+1θ

d _m

×

l=1µl

−

− Z

X^m

l−1

⊕

j=1θ

⊕f_l⊕ _m

⊕

j=l+1θ

d _m

×

l=1µ⁰_l

|f_l∈Lip_al(X,R)

=

= sup

m

X

l=1

Z

X

f_ldµ_l− Z

X

f_ldµ⁰_l

|f_l∈Lip_al(X,R), l= 1, m

,→1).

Sincef_l ∈Lip_al(X,R) if and only if −f_l∈Lip_al(X,R),we have sup

m

X

l=1

Z

X

fldµl− Z

X

fldµ⁰_l

|fl∈Lip_al(X,R), l= 1, m

=

= sup

m

X

l=1

±Z

X

f_ldµ_l− Z

X

f_ldµ⁰_l

|f_l∈Lip_al(X,R), l= 1, m

=

= sup m

X

l=1

Z

X

f_ldµ_l− Z

X

f_ldµ⁰_l

|f_l∈Lip_al(X,R), l= 1, m

=

=

m

X

l=1

sup

Z

X

fldµl− Z

X

fldµ⁰_l

|fl∈Lip_al(X,R), l= 1, m

=

=

m

X

l=1

a_ld_H(µ_l, µ⁰_l).

We have shown thatd^a_H^1,...,am _m

×

l=1

µ_l, ^m×

l=1

µ⁰_l

≥

m

P

l=1

a_ld_H(µ_l, µ⁰_l). This com- pletes the proof since we have proved earlier that d^a_H^1,...,am _m

×

l=1

µ_l, ×^m

l=1

µ⁰_l

≤

m

P

l=1

a_ld_H(µ_l, µ⁰_l).

Proposition3.2. Let (X, dX) be a complete metric space and S = (X, (f_k)_k=1,n, (p_k)_k=1,n) a GIFSp with f_k ∈ Lip^m_a1,a2,...,am(X, X), that is, an (a1, a2, . . .,am)-GIFSp. Then MS ∈Lip^m_a1,a2,...,am(M(X), M(X)).

Proof. Let f : X → R be a Lipschitz function such that Lip(f) ≤ 1 and µ₁, µ₂, . . . , µ_m, µ⁰₁, µ⁰₂, . . . , µ⁰_m ∈ M(X). Then, by Lemma 3.1, f ◦f_k ∈ Lip^m_a1,a2,...,am(X,R) and

Z

X^m

f◦f_k(x1, x2, . . . , xm) d _m

×

l=1µl(xl)

− Z

X^m

f◦f_k(x1, x2, . . . , xm)d _m

×

l=1µ⁰_l(xl)

≤d^a_H^1,a2,...,am _m

×

l=1µ_l, ×^m

l=1µ⁰_l

. Also,

Z

X

fd(MS(µ₁, µ₂, . . . , µ_m))− Z

X

fd(MS(µ⁰₁, µ⁰₂, . . . , µ⁰_m))

≤

≤d^a_H^1,a2,...,am _m

×

l=1µl, ×^m

l=1µ⁰_l

. Indeed,

Z

X

fd(MS(µ₁, µ₂, . . . , µ_m))− Z

X

fd(MS(µ⁰₁, µ⁰₂, . . . , µ⁰_m))

=

=

n

X

k=1

p_k Z

X^m

f◦f_k(x₁, . . . , x_m)d _m

×

l=1

µ_l(x_l)

−

−

n

X

k=1

p_k Z

X^m

f◦f_k(x1, . . . , xm)d _m

×

l=1µ⁰_l(x_l)

≤