ITERATED FUNCTION SYSTEMS. A CRITICAL SURVEY

Transcription

1 ITERATED FUNCTION SYSTEMS. A CRITICAL SURVEY MARIUS IOSIFESCU In the last 30 years or so, the phrase iterated function system has become more and more frequent in mathematical papers and in very many publications of applied people. As in many other instances, the notion of an iterated function system IFS is not a new one. Actually, we are faced with the renaming of an old concept, as shown in the first section of the present paper. However, it should be accepted that the study of this notion has been very much deepened under its new clothes. This survey is thus intended to present the state of the art of the IFS notion, its connections with other concepts, as well as to point out to some open problems. The paper is divided into six sections and two appendices. The first section sketches a historical perspective starting from the simplest case of a finite number of self-mappings. Section 2 introduces the general case of an arbitrary family of self-mappings obeying an i.i.d. mechanism. In Section 3 the existence and uniqueness of a stationary distribution are studied while in Section 4 almost sure convergence properties of the backward process are proved. Section 5 is devoted to a study of the support of the stationary distribution. Section 6 takes up the more general case of an arbitrary family of self-mappings obeying a strictly stationary mechanism instead of an i.i.d. one as in the five previous sections. The appendices collect some classical concepts and results on metrics and distances in metric spaces that we are using in the paper. AMS 2000 Subject Classification: 60G10, 60J05. Key words: iterated function system, metric space, Markov process, stationary probability, weak convergence, almost sure convergence, strictly stationary process. 1. A SIMPLE BASIC CASE The simplest iterated function system IFS p, u i 1 i m is defined by a finite collection of measurable self-mappings u i :, 1 i m, m N + := {1, 2,... }, m 2, of a metric space with metric d and Borel σ-algebra B, and a constant probability vector p = p i 1 i m. This allows to define a sequence ζ n n N of -valued random variables by the MATH. REPORTS 1161, ,

2 182 Marius Iosifescu 2 recursive equation ζ n = u ξn ζ n 1, n N +, where ζ 0 = w 0 arbitrarily given in and ξ n n N+ is a sequence of {1,..., m}-valued random variables with common probability distribution p, on the infinite product probability space [This clearly implies that ξ n n N+ {1,..., m}, P {1,..., m}, p i 1 i m N +. is an i.i.d. sequence.] Then ζ n = u ξn u ξ1 w 0, n N +, and it easy to see that ζ n n N is a -valued Markov process starting at w 0 with transition function P w, A = i A w p i, w, A B, where A w = {1 i m u i w A} = {1 i m w u 1 i A}. So, the transition operator U of our process is defined by m Ufw = P w, dw fw = p i fu i w, w, f B, i=1 the last equation being easily verified starting with indicator functions f = I A, A B. Here, B is the linear space of complex-valued bounded B - measurable functions defined on. A lot of work has been devoted to such iterated function systems p, u i 1 i m in the last three decades. As already mentioned, IFS is not at all a new concept. It only became fashionable in the framework of fractals and chaos but, before that, it appeared as the simplest case of a random system with complete connections and, in particular, as the Bush-Mosteller model for learning with experimenter-controlled-events [see, e.g., Herkenrath, Iosifescu, and Rudolph [23] as well as the review MR b:60078 of Barnsley and Elton [6]; above all see Iosifescu and Grigorescu [29, Chapter 1]. Even if objects now defined as fractals have been known to artists and mathematicians for centuries, the word fractal was coined by Benoit Mandelbrot in the late 1970s to designate a set whose Hausdorff dimension is not an integer. In less formal terms, a fractal object is one that is self-similar and sub-divisible: subsections of it are similar in some sense to the whole object while no matter how small is a subdivision of it, this contains no less details than the whole. Chaos is a subject brought forward by the study of nonlinear dynamics and has connections with fractal geometry. Chaotic systems are characterized by major changes in their behaviour caused by minor changes in the parameters that control them. Often used to illustrate the concept is the butterfly

3 3 Iterated function systems. A critical survey 183 effect : the breeze produced by the beating of a butterfly s wings may eventually generate a hurricane. In Crilly, Earnshow, and Jones Eds. [12] the reader will find a lot of interesting material on fractals, chaos, and their interrelationship, as well as many references. See also the site For historical material see [1]. Sketchily for more details see further on Section 5, using an IFS, a fractal can be constructed as follows. Assume, d is a bounded subset of R 2 in view of computer graphics applications, and the u i, 1 i m, are contraction self-mappings of, i.e., d u i w, u i w r dw, w for any w, w and 1 i m, where r is a positive number strictly less than 1. For any compact subset A of define SA = u i A. 1 i m Then there is a unique compact subset K of such that SK = K. This is called the attractor of the self-mappings u i, 1 i m, or of the deterministic IFS u i 1 i m note that no use was still made of the probability vector p = p i 1 i m, and in many cases has a fractal structure. Equally, for any compact subset A of, the sequence A n n N, where A 0 = A and A n = S A n 1, n N +, converges to K in the Hausdorff metric see A2.2 regardless of the choice of A. In applications, the u i are usually taken to be affine, that is of the form u i w = M i w + b i, 1 i m, for w R 2, where the M i are 2 2 matrices and the b i two-dimensional real vectors. In such a case, K is encoded by 6m real numbers. Traditional fractals as the middle thirds Cantor set and the Sierpinski triangle or arrowhead or gasket can be generated in this way. A sequence w n n N+ of points in is called an orbit of the deterministic IFS u i 1 i m if w n+1 = u in w n, where i n I, n N +. Then K above is an attractor in the sense of dynamical systems, since every orbit does approach K as n. Moreover, for any w K there are i n = i n w {1,..., m}, n N +, such that the sequence u in u i1 w 0 n N+ converges to w as n for any w 0. This clearly connects IFS with chaos as described above. There are several procedures to plot the attractor K on a computer screen. See, e.g., Bressloff and Stark [9]. In this reference a neural network formulation of a deterministic IFS can also be found. Note that a deterministic IFS can only generate a black and white image. Instead, a random IFS u i 1 i m, p as defined before is able to generate both colour and grey images. See again Bressloff and Stark op. cit.. In such a framework, the attractor K of the deterministic IFS u i 1 i m is the support

4 184 Marius Iosifescu 4 of the unique invariant measure of the Markov chain ζ n n N introduced at the beginning of this section. Let us conclude it by mentioning that if is a separable complete metric space, then any transition function Q : B [0, 1], w, A Q w, A can be represented as a measure P supported by some subset f i i Y set of all measurable self-mappings of, to mean that of the Q w, A = P {f f i i Y : fw A} for any w and A B. So, an IFS u i 1 i m, p is a very special case of this general context. See Section 2 for more details. 2. THE GENERAL I.I.D. CASE In this section we will take up a more general case. At the expense of some notational complication, nothing prevents us to consider an arbitrary measurable space instead of the finite set {1,..., m}. Let always be a metric space with metric d and Borel σ-algebra B,, an arbitrary measurable space, u : a B, B - measurable mapping, and p a probability measure on. rite u x w := uw, x, x, and note that for any x we have a B -measurable self-mapping u x :. The pair 2.1 p, u x x is called an iterated function system IFS, as in the case where is a finite set. Similarly to the latter case, on a probability space Ω, K, P p consider the -valued sequence ζ n n N defined by ζ 0 = w 0 arbitrarily given in and 2.2 ζ n = u ξn u ξ1 w 0, n N +, where ξ n n N+ is an i.i.d. -valued sequence with common P p -distribution p. To mark dependence on w 0, we shall occasionally write ζ w 0 n to denote the random variables defined by 2.2. Again, ζ n n N is a Markov process starting at w 0 with transition function P defined by P w, A = pa w, w, A B, where A w := {x u x w A} = {x w u 1 x A}. The transition operator U of our process is now defined by 2.3 Ufw= P w, dw fw = fu x wpdx, w, f B,

5 5 Iterated function systems. A critical survey 185 the last equation being again easily verified starting with indicator functions f = I A, A B. More generally, U n fw = fw P n w, dw = pdx 1 pdx n fu xn u x1 w, w, for any n N + and f B, where P n is the n-step transition function associated with P. The probabilistic meaning of U n fw is that it is the mean value of fζn w under P p for any n N +, f B, and w. e shall also consider the more general case where w 0 is chosen at random according to a given probability distribution. More precisely, on a probability space Ω, K, P λ,p let w 0 be a -valued random variable with probability distribution λ prb [= the collection of all probability measures on B ], that is independent of the ξ i, i N +, which always are i.i.d. with common P λ,p -distribution p. In this case, ζ n n N defined by 2.2 is still a - valued Markov process with initial distribution λ and transition function P. Consequently, its transition operator is always U. Clearly, the probability P λ,p reduces to P p when λ = δ w = probability measure concentrated at some w. Note that U is a bounded linear operator of norm 1 on B, which is a Banach space when endowed with the supremum norm f = sup fw, f B. Under a natural continuity assumption, namely that for p-almost all x the self-mapping u x : is continuous, the same assertion holds for U acting on C [= the linear space of complex-valued bounded continuous functions defined on ], also endowed with the supremum norm. The only thing needing proof is that U is now a Feller operator, to mean that Uf C for any f C. To proceed, fix arbitrarily w and consider any sequence w n n N+ in such that w n w as n. Clearly, according to the assumption made, for any f C and any x not contained in the p-null exceptional set we have lim f u xw n = f u x w. Then, by bounded convergence, lim p dx f u x w n = p dx f u x w, that is, lim Ufw n = Ufw. As w has been arbitrarily chosen, we conclude that Uf C.

6 186 Marius Iosifescu 6 Remark. The assumption of the continuity of u x : for p-almost all x will be tacitly assumed throughout. As a rule, we shall only mention when it is not necessary. Clearly, U maps into itself the collection of B -measurable extended realvalued functions f defined on such that Uf + w, and Uf w, w, are not simultaneously equal to +. An important special case where U is well defined for possibly unbounded functions f is described below. Define lx = l x; d = s u x := sup w w w,w d u x w, u x w d w, w, x. If the metric space is assumed to be separable, then it is easy to see that the mapping x lx of into R is, B R -measurable. Assume that d u x w, u x w 2.4 l := sup w w d w, w p dx < 1. w,w e clearly have l lxp dx. Hence, if the integral in the inequality above is less that 1, then we also have l < 1, but the converse does not hold. Assume also that for some w 0 we have 2.5 d w 0, u x w 0 p dx <. Under assumptions 2.4 and 2.5, the operator U takes Lip 1 into itself. See A1.2. For, 2.5 holds for any w in place of w 0 since dw, u x w dw, w 0 + dw 0, u x w 0 + du x w 0, u x w d ux w 0, u x w + 1 dw, w 0 + dw 0, u x w 0, d w 0, w which yields 2.6 dw, u x wpdx l+1dw 0, w+ Next, for any f Lip 1 we have hence dw 0, u x w 0 pdx<, w. f u x w fw + d w, u x w, x, w, Ufw f u x w p dx <, w,

7 7 Iterated function systems. A critical survey 187 while s Uf 1 is an immediate consequence of 2.4. Consider now another linear operator, closely related to U, defined on prb by V µa = µ dw P w, A, A B, for any µ pr B. Actually, this is a kind of adjoint of U, to mean that 2.7 µ, Uf = V µ, f, µ pr B, f B, where µ, f is defined as the integral fdµ. Equation 2.7 is easily established by using Fubini s theorem. It is also easy to check that V can be expressed by means of an integral over. e namely have V µa = p dx µu 1 x A, A B, for any µ pr B, where µu 1 x A := µ u 1 x A, x, A B. Note that V n µa = µ dw P n w, A, A B, or, alternatively, V n µa = p dx 1 p dx n µ u x1 u xn 1 A, A B, for any n N + and µ pr B. The probabilistic meaning of V n is that V n λa = P λ,p ζ n A for any λ pr B, A B, and n N +. From the equation above we also have that 2.8 V n λa = P λ,p u ξ1 u ξn w 0 A for any n N +, A B, and λ pr B, with P λ,p w 0 A = λa. The result below is well-known in the case where f B, cf Its proof does not differ from that working when f B, namely, Fubini s theorem. Proposition 2.1. If Ufdµ exists for some real-valued B -measurable function f and probability µ prb, then fdv µ also exists and the two integrals are equal. In particular, Proposition 2.1 shows that in the case where U is a Feller operator the pair U, V is a Markov-Feller pair according to Zaharopol [50, p. 3]. The problem raised at the end of the preceding section, namely, the possibility of representing a given transition probability function Q : B [0, 1], w, A Q w, A, as the transition probability function P : B [0, 1], w, A P w, A = p A w, of an IFS u x x, p, where A w = {x u x w A}, w, can be also answered in the present more general case. Assume that, d is a separable

8 188 Marius Iosifescu 8 complete metric space. Then, with = 0, 1 and with Λ the Lebesgue measure restricted to B 0,1, there exists a B B 0,1, B -measurable mapping v : 0, 1 such that Q w, A = Λ s 0, 1 v s w A, w, A B, with v s w := v w, s. In particular, if is R or a Borel subset of it, then one can take v s w = inf {y R Qw,, y] s} for any s 0, 1, w, and A B. Explicit expressions for v do also exist in the general case R. See Kifer [32, Theorem 1.1]. Earlier results can be found in Bergmann and Stoyan [8] and O Brien [43]. See also Athreya and Stenflo [3], where it is shown that the condition on, d to be a separable complete metric space can be replaced by that of being a standard Borel metric space, i.e., Borel measurably isomorphic to a Borel set on the real line. A still unsolved problem is whether nice solutions v s s do exist. For example, can the v s, s, be continuous or Lipschitz mappings? See Dubischar [16] for hints at this matter. 3. THE STATIONARY DISTRIBUTION: EISTENCE AND UNIQUENESS If U is a Feller operator and, d is compact, then the Markov chain ζ n N has invariant probabilities. See, e.g., Krengel [35, p. 178]. Nevertheless, it is important that the latter be approached in some sense by the n-step transition probabilities of the chain as n. It is clear that such a convergence might only hold if extra conditions on the self-mappings u x, x, are imposed. Pursuing such an idea, we shall deal here with the asymptotic behaviour as n of the distribution of ζ n under P λ,p. e shall see that, in our context, compactness of is not necessary. In what follows the reader should refer to the Appendices A1 and A2 at the end. The key result on which our approach is based is Proposition 3.1. Assume that 2.4 and 2.5 hold. Let µ, ν pr B such that ρ H µ, ν <. Then ρ H V µ, V ν l ρ H µ, ν. Proof. e have already seen that under our assumptions the operator U takes Lip 1 into itself. By Proposition 2.1 we then have { } 3.1 ρ H V µ, V ν = sup fd V µ fd V ν f Lip 1 { } = sup Uf dµ Ufdν f Lip 1.

9 9 Iterated function systems. A critical survey 189 Consider the function g = Uf/l. Note that g Lip 1 since for any w, w, w w, by the very definition of l we have gw g w dw, w = 1 f u x w f u x w l d w, w p dx 1 d u x w, u x w l d w, w p dx 1. Then, by 3.1, { ρ H V µ, V ν = l sup { l sup fdµ and the proof is complete. g dµ fdν g dν g = Uf } f Lip 1 Clearly, A1.2 and the result just proved imply } l, f Lip 1 = l ρ H µ, ν, Corollary 3.2. Under the assumptions in Proposition 3.1 we have for any n N +. ρ L V n µ, V n ν l n ρ H µ, ν By only using contraction properties of the operators U and V we can now prove the important result below. Cf. Iosifescu [28]. Theorem 3.3. Let, d be a separable complete metric space. Assume that 2.4 and 2.5 hold. Then the Markov chain ζ n n N has a unique stationary distribution π and 3.2 ρ L P n w,, π ln 1 l d w, u x w p dx for any n N and w. On Ω, K, P π,p the sequence ζ n n N is an ergodic strictly stationary process. Proof. Step 1. Let µ pr B such that ρ H µ, V µ <. For the existence of such a µ, see further Step 2. By Corollary 3.2, for any m, n N + we can write ρ L V n+m µ, V n µ m ρ L V n+k µ, V n+k+1 µ m 1 k=0 k=0 l n+k ρ H µ, V µ ln 1 l ρ H µ, V µ. Since, d is complete, so is prb, ρ L, see A1.2. Hence the sequence V n µ n N is convergent in prb, ρ L to some, say, π pr B.

10 190 Marius Iosifescu 10 Consider another ν pr B such that ρ H µ, ν <. Then, since ρ H ν, V ν ρ H ν, µ + ρ H µ, V µ + ρ H V µ, V ν l + 1 ρ H µ, ν + ρ H µ, V µ, we also have ρ H ν, V ν <. This allows to conclude that V n ν n N converges to the same π as for any n N + we have ρ L V n ν, π ρ L V n µ, π + ρ L V n µ, V n ν ρ L V n µ, π + l n ρ H µ, ν. To sum up, we have proved that if µ pr B satisfies the condition ρ H µ, V µ <, then there exists π = π µ such that 3.4 ρ L V n µ, π ln 1 l ρ H µ, V µ, n N +. [The last inequality follows at once from 3.3.] The same conclusion holds, with the same π, for any other ν pr B for which ρ H µ, ν <. It is easy to prove that π = V π, that is, π is a stationary distribution for ζ n n N. e have ρ L V µ, V ν ρ L µ, ν, µ, ν pr B, by the very definition of the distance ρ L on account of Proposition 2.1. Then ρ L V n+1 µ, V π ρ L V n µ, π 0 as n. Hence both V π and π are equal to the limit in pr B, ρ L of the sequence V n µ n N, that is, π = V π. Step 2. Clearly, δ w probability measure concentrated at w satisfies ρ H δ w, V δ w < for any w since { } ρ H δ w, V δ w = sup f w fd V δ w f Lip 1 = sup {fw Ufw f Lip 1 } by Proposition 2.1 { } = sup fw f u x w p dx f Lip 1 d w, u x w p dx < by 2.5. It follows by Step 1 that the limiting π δ w := π is the same for all w since ρ H δ w, δ w sup { fw f w f Lip 1 } d w, w < for any w, w. Next, any finite linear combination µ = q j δ wj with positive rational coefficients such that q j = 1 satisfies the condition ρ H µ, V µ < since, as is easy to see, ρ H µ, V µ q j ρ H δwj, V δ wj. Moreover, prb, ρ L is separable since, d was assumed to be, see A1.2, and it appears that the class of probability measures µ = q j δ wj just considered is dense in prb, ρ L if we start with a countable dense subset

11 11 Iterated function systems. A critical survey 191 {w j j N + } in. Cf. Hoffmann-Jørgensen [26, p. 83]. Let then λ pr B be arbitrary and for any ε > 0 consider a probability measure µ ε from that class such that ρ L λ, µ ε < ε. Since lim ρ L V n µ ε, π = 0 by Step 1 and ρ L V n λ, π ρ L V n µ ε, π + ρ L V n λ, V n µ ε ρ L V n µ ε, π + ρ L λ, µ ε, n N +, we have lim sup ρ L V n λ, π ε. As ε > 0 is arbitrary, we conclude that the sequence V n λ n N also converges to π in pr B, ρ L. Clearly, 3.2 follows from 3.4 with µ = δ w, w. For an arbitrary λ pr B, a similar upper bound for ρ L V n λ, π holds if we assume that 3.5 λ dw d w, u x w p dx <. Step 3. The uniqueness of π as stationary measure, π = V π, follows now easily. If π pr B satisfies π = V π, then by Step 2 we have lim ρ L V n π, π = 0 and, at the same time, V n π = π, n N +. Hence π = π. Next, the ergodicity of π, that is, ζ n n N is an ergodic strictly stationary sequence on Ω, K, P π,p, follows from the very uniqueness of π. See, e.g., Proposition in Hernández-Lerma and Lasserre [24]. Corollary 3.4. Under the assumptions in Theorem 3.3, for any realvalued bounded Lipschitz function f on we have U n fw fdπ ln dw, u x w p dx max oscf, sf, 1 l n N +, w, with oscf = sup fw inf fw. w w Proof. Clearly, if f is constant there is nothing to prove. If f const. then it is enough to note that for the function g := f inf w fw max oscf, sf Lip 1 we have 0 g 1, and to recall the definition of ρ L V n δ w, π.

12 192 Marius Iosifescu 12 Remarks. 1. Since lim ρ LV n λ, π = 0 for any λ prb see Step 2 in the proof of Theorem 3.3, by equation 2.8 the backward process ζ w 0 n = u ξ1 u ξn w 0, n N +, converges in distribution under P λ,p as n to π, with λ prb and P λ,p w 0 A = λa, A B, that is, lim P λ,p ζw 0 n A = πa for any A B whose boundary is π-null. e shall show more, namely, that for any fixed w the sequence ζ n w n N converges P p -a.s. at a geometric rate as n to a -valued random variable ζ such that P p ξ A = πa, A B. See further Theorems 4.1 and As for the nature of the stationary distribution π, according to results of Dubins and Freedman [15] on Markov operators, it should be of pure type under appropriate assumptions. For example, if for some probability measure m prb either mu 1 m for any x or, when is countable, x ν m implies νu 1 x m for any x whatever ν prb, then π is either absolutely continuous or purely singular with respect to m. This also applies to similar further results as, e.g., Theorems 3.5 and 3.6 or Corollaries 3.7 and 3.8. The type of π appears to be related to the so-called open set condition OSC. The family u x x is said to satisfy the OSC if there is a non-empty bounded open set V such that u x V V for any x and u x V u x V = for any x, x, x x. See, e.g., Lau and Ngai [36] and the references therein. A more general version of Theorem 3.3 is obtained using the fact that d α is still a metric in for any 0 < α 1. [It is enough to note that if a, b, c 0 and c a + b, then c α a + b α a α + b α.] rite then see A1.2 ρ L,α and Lip α 1 for the items associated with the metric space, d α, which correspond for α = 1 to ρ L and Lip 1, respectively. Remark that B is not altered when replacing d by d α. Clearly, l x; d α = [l x; α] α := l α x, x, and then the conditions corresponding to 2.4 and 2.5 are 3.6 l α := sup w w w,w and, respectively, 3.7 d α u x w, u x w d α w, w p dx < 1 d α w 0, u x w 0 p dx < for some w 0, hence for all w 0.

13 13 Iterated function systems. A critical survey 193 e can now state Theorem 3.5. Let, d be a separable complete metric space. Assume that 3.6 and 3.7 hold. Then the Markov chain ζ n n N has a unique stationary distribution π and 3.8 ρ L P n w,, π ln α d α w, u x w p dx 1 l α for any n N + and w. On Ω, K, P π,p the sequence ζ n n N is an ergodic strictly stationary process. Proof. It follows from Theorem 3.3 that 3.8 holds with ρ L,α in place of ρ L. The validity of 3.8 will follow from the inequality ρ L,α ρ L for any 0 < α < 1. e shall in fact prove that 3.9 {f f Lip 1, 0 f 1} {f f Lip α 1, 0 f 1} for any 0 < α 1, which clearly implies ρ L,α ρ L. To proceed, note that if f Lip 1 = Lip 1 1 and 0 f 1, then for any 0 < α 1 we can write = max max sup w w dw,w 1 sup w w dw,w 1 f w f w sup w w d α w, w = fw f w d α w, w, sup dw,w >1 fw f w dw, w f w f w d α w, w, some quantity not exceeding 1 max sf, 1 1. e used the inequality x α > x which holds for 0 < α, x < 1. Hence f Lip α 1, showing that 3.9 holds. Remarks. 1. It is obvious that the assumptions in Theorem 3.5 are weaker than those in Theorem 3.3, so that the former is a real generalization of the latter. Also, the result corresponding to Corollary 3.4 under assumptions 2.4 and 2.5 also holds. Clearly, both Theorem 3.5 and the corresponding corollary have versions holding when 3.5 is replaced by the condition 3.10 λ dw d α w, u x w p dx <.

14 194 Marius Iosifescu 14 For example, if 3.6, 3.7, and 3.10 hold, then ρ L V n λ, π ln α λ dw d α w, u x w p dx 1 l α for all n N To compare Theorem 3.5 and Theorem 5.1 in Diaconis and Freedman [13, pp ] let us first note see, e.g., Hewitt and Stromberg [25, p. 201] that the condition 3.11 L α := l α x p dx < 1 for some 0 < α 1, which is stronger than 3.6, implies the inequality 3.12 log l x p dx < 0. Conversely, if L β := lβ x p dx < for some β > 0 and 3.12 holds, then there exists α > 0 such that L α < 1. The assumptions in Theorem 5.1 in Diaconis and Freedman op.cit. are 3.12 and a so-called algebraic-tail condition on l and d which amounts to the existence of positive constants a and b such that 3.13 p {x l x > y} < ay b, p {x dw 0, u x w 0 > y} < ay b for y > 0 large enough and some w 0, hence for all w 0. e are going to prove that these assumptions are equivalent to 3.11 in conjunction with 3.7, so that they are stronger than those in Theorem 3.5. First, on account of the equation 3.14 Eη = 0 P η > y dy which holds for any non-negative random variable η, it is clear that 3.11 and 3.7 imply both 3.12 and, via Markov s inequality, Second, if 3.13 holds, then for any α > 0 we have p {x l α x > y} < ay b/α, p {x d α w 0, u x w 0 > y} < ay b/α for y > 0 large enough. Choosing α < min b, 1, it follows from 3.14 that both L α and dα w 0, u x w 0 dx are finite. But L α < in conjunction with 3.12 implies the existence of 0 < α < α such that L α < 1, as has just been mentioned. The proof is complete. 3. The average contractibility condition 3.6 can be weakened to average contractibility after a given number of steps. To introduce it, for any n N + and x n = x 1,..., x n n put u x n = u xn u x1

15 15 Iterated function systems. A critical survey 195 and consider the IFS pn, u x n x n n, where p n denotes the nth product measure of p with itself. Clearly, for any fixed n N + we have a new IFS for which condition 3.6 reads as 3.15 l α,n := d α u sup, u w xn w w n d α w, w p n dx n < 1. w,w It is not difficult to check that l α,n n N+ is a submultiplicative sequence, that is, l α,m+n l α,m l α,n, m, n N +. Hence, if l α,k 1 for some k N +, then l α,nk n N+ is a non-increasing sequence. In particular, it follows that condition 3.15 for some n 2 is weaker than the condition l α,1 < 1, that is, 3.6. It is easy to see that Theorem 3.5 carries over to an IFS satisfying condition 3.15 for some fixed n = n 0 together with the condition 3.16 d α w 0, u x n 0 w 0 p n0 dx n 0 <. n 0 for some w 0, hence for all w 0. The latter corresponds to condition 3.7 and reduces to it when n 0 = 1. More precisely, the following result holds. Theorem 3.6. Let, d be a separable complete metric space. Assume that 3.15 and 3.16 hold for some fixed n 0 N +. Then the Markov chain ζ nn0 n N has a unique stationary distribution π and 3.17 ρ L P nn 0 w, π ln α,n 0 d α w, u 1 l x n 0 w p n0 dx n 0 α,n0 for any n N + and w. On Ω, K, P π,p the sequence ζ nn0 n N is an ergodic stationary process. Note that this is just a transcription of Theorem 3.5 for the IFS p n0, ux n 0 x n 0 n 0. It does not yield a stationary distribution for the whole Markov chain ζ n n N. To ensure that π occurring in the statement above is a stationary distribution for ζ n n N, more assumptions are to be made. e namely first have Corollary 3.7. Let n 0 2. Under the assumptions in Theorem 3.6, if for some 1 r < n 0 we have 3.18 d α w, u x rw p r dx r < r n 0

16 196 Marius Iosifescu 16 for any w, then π is the unique stationary distribution of the Markov chain ζ nn0 +r n N and 3.19 l n α,n 0 n 0 ρ L P nn 0 +r w,, π d α w, u x n 0 w p n0 dx n 0 + d α w, u 1 l x rw p r dx r α,n0 r for any n N + and w. Proof. e clearly have 3.20 ρ L P nn 0 +r w,, π ρ L P nn 0 +r w,, P nn 0 w, + ρ L P nn 0 w,, π for any w and n N +. Coming back to the proof of Theorem 3.5 and using Corollary 3.2 we can write ρ L P nn 0 +r w,, P nn 0 w, = ρ L V nn 0 +r δ w, V nn δ w ρ L,α V nn 0 +r δ w, V nn 0 δ w l n α,n0 ρ H,α V r δ w, δ w while 3.22 ρ H,α V r δ w, δ w = sup {U r fw fw f Lip α 1 } sup p dx r fw f u x rw f Lip α 1 r d α w, u x r w p r dx r < r for any w. Now, 3.19 follows from 3.17, 3.20, 3.21, and Let us note that as in the case n 0 = 1, condition 3.16 for just one w 0 in conjunction with 3.15 implies that the former holds for any w 0. In the case of 3.18, assumed to hold for just one w, a similar conclusion would follow when assuming in addition that r d α sup w w ux rw, u x r w d α w, w p r dx r <. Clearly, such a condition is not implied by only 3.15, as simple examples show.

17 17 Iterated function systems. A critical survey 197 Corollary 3.8. Let n 0 2. Under the assumptions in Theorem 3.6 in conjunction with 3.18 for any 1 r < n 0, we have 3.23 ρ L P n w,, π d α w, u l n+1 1 x n 0 w p n0 dx n 0 n 0 α,n 0 n 0 + max 1 l α,n0 1 r<n 0 r d α w, u x rwp r dx r for any n 2n 0 1 and w. The Markov chain ζ n n N has π as unique stationary distribution and is an ergodic strictly stationary process on Ω, K, P π,p. Proof. This follows from Theorem 3.6 and Corollary 3.7 taking into account that both 3.17 and 3.19 hold actually with ρ L,α in place of ρ L, see the proof of Theorem 3.5. Next, we have to note that ρ L,α P n w, = ρ L,α V n δ w, π, w, n N +, and then follow the reasoning from Steps 2 and 3 in the proof of Theorem A natural and interesting question now arises. hat does it happen when condition 3.15 does not hold for any n N +, that is, if l α,n 1 for any n N +? First, there is an interesting special case where l α,n = 1 for any n N +, namely, that of = = R +, u x w = w x, w, x R +, while the probability p on B R+ is such that 0 < E p ξ 1 <. It can be proved that π pr B R+ given by πa = A P p ξ 1 > x dx, A B R+, E p ξ 1 is the only stationary probability distribution for the Markov chain ζ n n N when ξ 1 is not supported by a lattice. This case has been considered in Feller s 1971 classical book. The result above first appears in Knight [33] and Leguesdron [38]. A recent treatment based on the reversed sequence ζ w 0 n n N has been given by Abrams et al. [2]. Very few is known in the lattice case and no rate of convergence if any to π in the non-lattice case is given. Coming back to the case l α,n 1 for all n N +, if l α,n > 1 for at least one n N +, then we should necessarily have l α,1 > 1 by the submultiplicativity of the sequence l α,n n N+. Our guess is that if l α,n > 1 for infinitely many n N +, then there cannot exist a stationary probability π for ζ n n N. 5. It is possible to ensure the existence of the stationary distribution π for ζ n n N without assuming global contraction and drift conditions. Instead, some local contraction conditions and appropriate drift conditions can be considered.

18 198 Marius Iosifescu 18 For example, Jarner and Tweedie [30] considered a separable complete metric space, d with finite diameter, that is, 3.24 sup d w, w <, w,w and assumed that i the maps u x, x, are non-separating on average, to mean that 3.25 E p dζ w 1, ζ1 w d w, w for all w, w ; ii there exist a positive number r < 1 and a set C B such that contraction occurs after reaching C, to mean that 3.26 E p d ζτ w C w τ C w, ζw τ C w τ C w rd w, w for all w, w, where sup w C τ C w = inf {n N + ζ w n C }, w ; iii there exists a measurable function L : [1, such that Lw < and for some positive constants q < 1 and a the inequality 3.27 U Lw = p dx L u x w qlw + ai C w holds for all w. These authors showed that under assumptions i through iii the conclusions of our Theorem 3.5 all hold with a convergence rate O b n L 1 2 w, w, as n, for some positive constant b < 1, with the constant implied in O independent of n N + and w. It is clear that in the special case C = assumptions i ii reduce to the only condition E p d ζ1 w, ζ1 w rdw, w, w, w, for some positive constant r < 1, that is, to condition 2.4 while iii is satisfied with L 1. As 3.24 implies 2.5, the case C = is covered by Theorem 3.5. On the other hand, a condition like ii seems quite difficult to be checked in the case where C. 4. ALMOST SURE CONVERGENCE PROPERTIES e now come back to Remark 1 following Corollary 3.4 concerning the convergence in distribution of the backward process 4.1 ζw 0 n = u ξ1 u ξn w 0, n N +,

19 19 Iterated function systems. A critical survey 199 to the stationary distribution π under P λ,p, with λ prb and P λ,p w 0 A = λa, A B. e shall namely prove almost sure convergence properties of this process under the assumptions already made. Before proceeding, we shall recall a result of Letac [39] that reads as follows. Proposition 4.1 Letac s lemma. If for p-almost all x the mapping u x : is continuous and if ζ := ζ w 0 n exists P p -a.s. and lim does not depend on w 0, then the probability distribution µ = P p ζ 1 of ζ under P p is the only stationary distribution of ζ w 0 n n N+. The proof of this result is very simple. Let π w 0 n denote the probability distribution of both ζ w 0 n and ζ w 0 n, n N +. e clearly have π w 0 n = V π w 0 n 1, n 2, with the operator V defined as in Section 2, where it has been shown that for any bounded continuous real-valued function g on the function Ug : w Ugw = g u xw p dx is bounded and continuous, too. According to equation 2.7, for any n 2 we have gwπ w 0 n dw = gwv π w 0 n 1 dw = Ugwπ w 0 n 1 dw. Since ζ w 0 n converges P p -a.s., letting n we get gwµ dw = Ug w µ dw, showing that µ is a stationary distribution for ζ w 0 n n N+. If µ is another stationary distribution for ζ w 0 n n N+, then it is the probability distribution of ζ w 0 n, hence of ζ w 0 n, for any n N +. As the latter distribution should converge to µ, we have µ = µ. Remarks. 1. No assumption on the metric space, d is needed in Proposition A weak variant of Proposition 4.1, that implies it under its stronger assumptions, see Athreya and Stenflo [3], is as follows. ith, d and u x x unrestricted, assume that for some w 0 there exists a random variable ζ w 0 to which ζ w 0 n converges in distribution under P p as n. Then i ζ w 0 n also converges in distribution under P p as n to ζ w 0, and ii if U is a Feller operator, the probability distribution µ w 0 = P p ζ w 0 1 of ζ w 0 under P p is a stationary distribution for the Markov chain ζ w 0 n n N+ while if µ w 0 does not depend on w 0, it is the unique stationary distribution. e start with

20 200 Marius Iosifescu 20 Theorem 4.2. For any w, under the assumptions of Theorem 3.5, the backward process ζ w n = u ξ1 u ξn w, n N +, converges P p -a.s. at a geometric rate as n to a -valued random variable ζ. e have π = P p ζ 1, that is, πa = P p ζ A, A B. Proof. By the very definition of l α we have d ux w, u x w α d w, w p dx l α < 1 for any x and w w, w, w. This amounts to E p d α u ξn+1 w, u ξn+1 w l α d α w, w for any n N and w, w. Since ζn+1 w = u ξ n+1 ζn w, for any w and n N +, the above inequality yields E p d α ζn+1, w ζn+1 w ξ 1,..., ξ n l α d α ζn w, ζn w P p -a.s. for any n N and w, w. As [ ] E p E p d α ζn+1, w ζn+1 w ξ 1,..., ξ n we deduce that = E p d α ζn+1, w ζn+1 w, E p d α ζn+1, w ζn+1 w l α E p d α ζn w, ζn w for any n N + and w, w, which implies E p d α ζn w, ζn w l n αd α w, w for any n N + and w, w. Note now that for any n 1 the nth product measure of p with itself is symmetrical, which allows us to also write 4.2 E p d α ζw w n, ζ n l n αd α w, w for any n N + and w, w. w and n N +, we can write E p d α ζw n, ζ n+1 w = E p [E p d α ζw n, Finally, since ζ w n+1 = ζ u ξ n+1 w n ζ u ξ n+1 w n ξn+1 ] l α [ n Ep d α w, u ξn+1 w ξ n+1 ] by 4.2 = l α n d α w, u x w p dx. for any

21 21 Iterated function systems. A critical survey 201 Therefore, 4.3 E p d α ζw n, ζ n+1 w l n α d α w, u x w p dx for any n N + and w, with dα w, u x w p dx < by 3.7. Hence for any w the series n N + P p d α ζw n, ζ n+1 w > l n/2 α is convergent since by Markov s inequality we have P p d α ζw n, ζ n+1 w E p d ζw α n, ζ > l n/2 n+1 w α l n/2 α l n/2 α d α w, u x w p dx for any n N + and w. It follows from the Borel-Cantelli lemma that the inequality d ζw α n, ζ n+1 w l n/2 α holds P p -a.s. for all sufficiently large n. So, for any w, ζ n w n N+ is P p -a.s. a Cauchy sequence in, d α that converges at a geometric rate to a -valued random variable, say, ζ,w. Note also that it follows from 4.3 that E p d α ζw n, ζ,w E p d α ζw 4.4 n, ζ n+j+1 w j N ln α d α w, u x w p dx 1 l α for any n N +. Let us show that, actually, ζ,w does not depend on w. Indeed, fix w. For any w and n N + by 4.2 and 4.4 we have E p d α ζw n, ζ,w E p d α ζw n, ζ n w + E p d α ζw n, ζ,w l n α d α w, w + d α w, u x w p dx / 1 l α, whence P p d α ζw n, ζ,w l n/2 α l n/2 α d α w, w + d α w, u x wpdx/1 l α by Markov s inequality again. e thus conclude as before that and we are done. lim ζ w n = ζ,w P p -a.s.,

22 202 Marius Iosifescu 22 Let us from now on write ζ for the unique P p -a.s. limiting random variable. To prove the equation π = P p ζ 1, we notice that for any f C we have E p f ζn w = E p f ζw n, n N +, w. But, on one hand, by Theorem 3.5, lim E p f ζ w n = fdπ, w, and, on the other hand, by bounded convergence, lim E p f ζw n = f ζ dp p = fd P p ζ 1, w. Therefore, fdπ = fd P p ζ 1 for any f C. Hence π = Pp ζ 1 a well known result see, e.g., Parthasarathy [44, Theorem 5.9]. Remarks. 1. The equation π = P p ζ 1 implies that the support supp π of π, that is, the smallest closed subset of having π-measure 1, defined as supp π = {w : π {v d v, w < ε} > 0 for all ε > 0} = {w P p d ζ, w < ε > 0 for all ε > 0} and the range of ζ, that is, the set ζ Ω, where Ω is the random event {ω Ω : ζ ω exists}, with P p Ω = 1, are strongly related. Indeed, we clearly have 1 = P p Ω = P p ζ 1 ζ Ω = π ζ Ω and whence and 1 = π supp π = P p ζ supp π, π ζ Ω supp π = 0 P p Ω ζ supp π = 0, where stands for symmetric difference of sets. 2. A special case where supp π can be precisely described is further considered in Theorem 5.1. Theorem 4.2 can be slightly generalized by considering the case of a random initial point w 0 with P λ,p w 0 A = λa, A B, for any given λ prb. e namely have Theorem 4.3. Assume that 3.6 and 3.7 hold. Then the backward process ζ w 0 n n N+ converges P λ,p -a.s. at a geometric rate as n to by

23 23 Iterated function systems. A critical survey 203 a -valued random variable ζ. e have π = P λ,p ζ 1, that is, πa = P λ,p ζ A, A B. Clearly, Remark 1 after Theorem 4.2 also has a version in which w is replaced by λ pr B. e now consider assumptions more general than those we assumed before. These new assumptions involve a kind of local contractibility at one point only instead of 3.6. e thus introduce Condition C below. Cf. u and Shao [49] and Herkenrath and Iosifescu [22]. Condition C. i There exist w 0, α 0, 1], r 0, 1 and - measurable functions ϕ n : [0,, n N +, such that both R:= 1 lim sup δ 1/n n strictly exceed r for any w, where and R w := lim sup 1 ϕ 1/n n w δ n := E p ϕ n ζ w 0 1 dα w 0, ζ w 0 1, n N +. ii For any w and n N + one has E p d α ζ w n, ζ w 0 n r n ϕ n wd α w, w 0. It is clear that R and R w, w, are the convergence radii of the power series δ n t n and ϕ n wt n, t R, n N + n N + respectively. Also, i and ii generalize 3.6 and 3.7, respectively, where ϕ n 1, n N +. Theorem 4.4. Assume Condition C holds. j There exists a -valued σ ξ 1, ξ 2,... -measurable random variable ζ to which ζ n w converges P p -a.s. as n for any w and, moreover, E p d α ζw n, ζ r n ϕ n wd α w, w 0 + r m δ m, n N +. m n jj Let π denote the probability distribution of ζ and let w 0, w 0 be -valued random variables independent of ξ n n N+ and with common distribution π. Then E π,p d α ζ w 0 n, ζ w 0 2 r m δ m, n N +. m n n

24 204 Marius Iosifescu 24 Proof. Note first that ζ w 0 n+1 = ζ u ξ n+1 w 0 n, n N +. By Condition Cii we then have E p d α ζw 0 n, ζ w 0 n+1 = E p [E p d α ζw 0 n, ζ u ξ n+1 w 0 ] n ξn+1 r n E p ϕn uξn+1 w 0 d α w 0, u ξn+1 w 0 = r n E p ϕ n u ξ1 w 0 d α w 0, u ξ1 w 0 = r n E p ϕ n ζ w 0 1 dα w 0, ζ w 0 1 = rn δ n for any n N +. Fix r 0 0, 1 with r R < r 0 < 1. Then, by Markov s inequality, P p d α ζw 0 n, ζ w 0 r n n+1 r0 n δ n, n N +. As by Condition Ci the series with general term r/r 0 n δ n is convergent, the Borel-Cantelli lemma implies that P p d α ζw 0 n, ζ w 0 n+1 r n 0 infinitely often = 0, hence ζ w 0 n ζ, say, P p -a.s. as n by the completeness of and, obviously, ζ is σ ξ 1, ξ 2,... -measurable. Next, by the triangle inequality, 4.5 E p d α ζw 0 n, ζ E p j N E p d α ζw 0 n+j, ζ w 0 n+j+1 r 0 j N d α ζw 0 for any n N +. Finally, by Condition Cii, E p d α ζw n, ζ E p d α ζw n, ζ w 0 n n+j, ζ w 0 n+j+1 δ n := r m δ m m n r n ϕ n wd α w, w 0 + δ n + E p d α ζw 0 n, ζ for any w, whence we conclude that ζ w n ζ P p -a.s. as n, by invoking the argument already used to show that ζ w 0 n ζ P p -a.s. as n. e should now choose r 0 0, 1 with r/ min R, R w < r 0 < 1 and note that both series ϕ n w r/r 0 n δ n and r/r 0 m δ m = n N + n N + m n n r/r 0 n δ n are convergent by Condition Ci. The proof of j is thus n N + complete. jj For any n, m N + and w consider the random variable ζ w n,m = u ξn u ξn+m 1 w,

25 25 Iterated function systems. A critical survey 205 so that ζ n w = ζ 1,n w. It follows from j that the limit ζ n, := ν n = lim ζ w m n,m, n N +, exists P p -a.s., does not depend on w, and has P p -distribution π. Clearly, ζ ν n+1 n = ζ P p -a.s. for any n N +. Also, ν n+1 is independent of ξ i 1 i n for any n N +. Then the triangle inequality and the above facts allow us to write E π,p d α ζ w 0 n, ζ w 0 n E π,p d α ζ w 0 n, ζ w 0 n + E π,p d α ζ w 0 n, ζ w 0 n = 2E π,p d ζw α 0 n, ζ w 0 n = 2E p d α n+1 ζν n, ζ w 0 n = 2E p d α ζ, ζ w 0 n for n N +, and the claim follows from 4.5. Note that π occurring in E π,p in the relations above refers to w 0 and w 0 while w 0 is the fixed point from Condition C. Let us remind that by the Letac lemma, the distribution π of ζ under P p is the unique stationary distribution of the Markov chain ζ n n N+ if for p-almost all x the mapping u x : is continuous, an assumption we agreed about, see Section 2. ithout such an assumption, the uniqueness of the stationary distribution cannot be asserted. Theorem 4.4 allows one to estimate the rate of convergence of the n- step transition probability function Q n of the Markov chain ζ n n N+ to its stationary probability distribution π cf. Theorem 3.5. e namely have Corollary 4.5. Assume Condition C holds. Then for any w and n N + we have ρ L,α Q n w,, π r n ϕ n wd α w, w 0 + m n r m δ m. Proof. By the very definition of ρ L,α, { ρ L,α Q n w,, π = sup fv Q n w, dv π dv f v f v d α v, v }, v, v. 0 f 1, Since for any f C as in the definition above we have fv Q n w, dv π dv = E p f ζ w n f ζ = Ep f ζw n f ζ and Ep f ζw n f ζ Ep d α ζw n, ζ,

26 206 Marius Iosifescu 26 the proof is complete by Theorem 4.4 j. Remark. It is interesting to compare the upper bound of ρ L,α Q n w,, π in Theorem 3.5 and Corollary 4.5, under the assumptions of the former, when ϕ n 1, n N +, and the part of r in Condition Cii is played by l. Then the two bounds are l n d α w, u x w p dx 1 l and l n d α w 0, u x w 0 p dx l n d α w, w 0 +, 1 l respectively, for any w and n N +. As d α w, u x w p dx l + 1 d α w, w 0 + d α w 0, u x w 0 p dx by 2.6, the second bound always exceeds the first one for any w and n N + even if both of them are O l n as n. Theorem 4.4 allows to define -valued random variables ζ i, i 0, in such a way that the equation ζ i = u ξi ζ i 1 that holds for i N + still holds for i 0. Let ξ i i Z be a doubly infinite sequence of i.i.d. -valued random variables with common distribution p. Similarly to the context in Theorem 4.4j, the limit lim m u ξ i u ξi 1 ξ i m w := ζ w i, w, exists P p -a.s. for any i 0, is a σ..., ξ i 1, ξ i -measurable function that does not actually depend on w, has probability distribution π, and satisfy P p -a.s. the equation ζ i = u ξi ζ i 1 for any i 0 if, as noted before in a different context, for p-almost all x the mapping u x : is continuous. Notice that with ζ i, i 0, so defined, the doubly infinite sequence ζ i i Z constructed by using the latter recurrence equation for any i Z := {..., 1, 0, 1,... } is a strictly stationary process, as well as a Markov process, on a suitable probability space, to be still denoted Ω, K, P π,p. The idea of extending a strictly stationary process into the past allowing to consider it to have been going on forever, of which we have just seen an instance, is an old one. The possibility of always extending a real-valued strictly stationary process into the past is proved in Doob [14, pp ]. See also Elton [17]. Recall that a -valued process = n n N or = n n Z on a probability space Ω, K, P is said to be strictly stationary ergodic iff the left shift on N or Z is measure-preserving ergodic for the measure P 1. Proposition 4.6 Doob s lemma. Assume is a separable complete metric space. Given a -valued strictly stationary process = n n N,

27 27 Iterated function systems. A critical survey 207 there exists a strictly stationary process = n n Z whose finite-dimensional distributions are identical with those of. Also, is ergodic iff is ergodic. 5. THE SUPPORT OF THE STATIONARY DISTRIBUTION e have already noticed that there is a purely deterministic concept of an iterated function system. This is intensively investigated by people working in geometry of sets and measures see, e.g., Mauldin and Urbański [41]. Let, d be a complete metric space, I a countable set with at least two elements, and u i i I a collection of contracting self-mappings of, that is, such that 5.1 s u i := s i < 1, i I. For any n N + and i n := i 1,..., i n I n put v i n = u i1 u in and note that v i u i, i I. Define a scaling operator S : P P by SE = i I u i E, E, so that S n E = v i ne, n N +, E. i n I n A set E is said to be subinvariant invariant under scaling iff SE E SE = E. The main concern is the limit set associated with u i i I, which is an invariant set under scaling, which we are going to define. Assume first that I is finite, hence s := max s i < 1. Then there exists a unique compact subset K of that is invariant under scaling. To show this, consider see A2.2 the collection bcl of non-empty bounded closed subsets of, which under the Hausdorff metric d H is a complete metric space. Consider also the collection c bcl of the compact subsets of. Note that c is a closed subset of bcl. Clearly, S takes bcl and c into themselves. For any A, B bcl or c we have d H SA, SB = d H u i A, u i B i I i I max d H u i A, u i B max s i d H A, B = sd H A, B. i I i I That is, S is a contraction map on both bcl and c in the Hausdorff metric d H. As bcl, d H is a complete metric space, the existence of some K bcl satisfying K = SK follows from the contraction mapping principle. Actually, we have lim Sn A = K in bcl, d H for any A bcl, hence any A c. i I

28 208 Marius Iosifescu 28 As c is closed in bcl, d H and S takes c into itself, we should have K c, as asserted. Remark. It follows in particular that v i := lim v i nw exists and is an element of K for any w and i = i 1, i 2,... I N +. Clearly, v i does not depend on w as d v i nw, v i n w s v i n d w, w s n dw, w 0 as n for any w, w. It is for this reason that K is also called the attractor of u i i I. Theorem 5.1 Hutchinson [27]. The set K is the intersection of the sets S n, n N +, that is, K = n N + i n I n v i n even if, possibly, / bcl. It also is the closure of the set n N + i n I n θ i n : v i n θ i n = θ i n of fixed points of all v i n, i n I n, n N +. For any i 1, i 2,... I N + the limit lim θ i n := k i exists and clearly belongs to K while the mapping θ : I N + K defined by θ i = k i, i = i 1, i 2,... I N +, is continuous onto K. The compact set K supports probability measures in a natural way. Let p i / 0, 1, i I, with i I p i = 1. It then follows from Theorem 5.1 and Remark 1 following Theorem 4.2 that in the case where is the finite set I and s u i < 1 for any i I, the support of the invariant probability measure π of the IFS u i i I, p i I is precisely K. It should be stressed that K is only and uniquely determined by the mappings u i, i I. Assigning them probabilities yields just a probability distribution over K while two different assignments p = p i i I and p = p i i I of probabilities yield in general different probability distributions over the same K. Remark. Hata [21] proved the existence of a unique nonempty compact set K which is invariant under scaling K = S K by only assuming that the u i, i I, are weakly contractive. This means that α i t := sup d u i y, u i z < t, i I, dy,z t

29 29 Iterated function systems. A critical survey 209 for all t > 0. Next, for any given i = i 1,..., i n, i n+1, = i n, i n+1,, the Lebesgue measure of the set v i nk converges to 0 as n. Also, K = v i nk n N + i n I n compare with Theorem 5.1. Assuming that is a compact subset of R m for some m N + with d the Euclidean distance in R m, Lau and Ye [37] showed that these properties still hold if the u i, i I, are continuous and at least one of them is weakly contractive. Here, unicity means that of a smallest such nonempty compact K. hen I is infinite, it is usual to assume that the metric space, d is compact and, instead of 5.1, that 5.2 lim sup diam v i n = 0, i n I n a condition which clearly holds if sup s i < 1, when i I n diam v i n sup s i diam, i n I n, n N +. i I Since for any given i = i k k N+ I N + the closed sets v i n, n N +, are decreasing and by 5.2 their diameters converge to zero uniformly with respect to i n I n as n, the set Π i = v i n n N + is a singleton and the equation above defines a map Π : I N + which is continuous when I N + is given the product topology induced from the discrete topology on each factor thus making it a compact set if I is finite. Then the limit set associated with u i i I is defined as 5.3 J = v i n. n N + i I N + It is also called the fractal set determined by u i i I. As clearly u i Π i k k N+ = Π i, i1, i 2,... for any i I and i k k N+ I N +, we see that J = i I u i J, that is, J is invariant under scaling. Now, assume that any element of belongs to at most finitely many u i, i I, which clearly implies that, whatever n N +, any element of

30 210 Marius Iosifescu 30 belongs to at most finitely many v i n, i n I n. It is immediate that we can then write 5.4 J = v i n, n N + i n I n a conclusion which has also been reached in the case where I is finite. The difference is that if I is finite, then J is compact and coincides with K as described in Theorem 5.1, when max s i < 1. In the present instance, for an i I infinite I, all what can be asserted about J given by 5.4 is that it is a F σδ set that is, an intersection of countable unions of closed sets. Finally, when the assumption leading to 5.4 does not hold, the limit set 5.3 may have a much more complicated descriptive set-theoretic structure. See, e.g., Mauldin and Urbański [40, Section 5]. Example 5.2. Let = [0, 1] with the Euclidean distance, I = {0, 1}, and u 0 w = aw, u 1 w = aw + 1 a, w, with 0 < a < 1. In this case the composition u i1 u in = v i n can be expressed in closed terms as 5.5 v i nw = a 1 1 n i k a k + a n w for any i 1,..., i n I, n N +, and w. This can be easily seen by induction: 5.5 clearly holds for n = 1 and, assuming that k=1 u i2 u in w = a 1 1 n i k a k 1 + a n 1 w for n 2, we have a v i nw = u i1 u in w = u 1 i1 1 n i k a k 1 + a n 1 w k=2 k=2 a a 1 1 n i k a k 1 + a n w if i 1 = 0 k=2 = a a 1 1 n i k a k 1 + a n w + 1 a if i 1 = 1 k=2 = a 1 1 n i k a k + a n w. k=1 Hence, for any i = i 1, i 2,... I N +, the so-called a-expansion, namely, lim a 1 1 n i k a k = a 1 1 k=1 k N + i k a k