# Attractors of Compactly Generated Semigroups of Regular Polynomial Mappings

Authors: Azza Alghamdi; Maciej Klimek; Marta Kosek
Journal: Complexity
 (2018)

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## Abstract

We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.


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## Body

## 1. Introduction

In the recent paper [1] it was shown, as a part of the investigation of the space of pluriregular sets, that it is possible to approximate composite Julia sets generated by finite families of proper polynomial mappings in ℂN in a probabilistic manner. This can be done in the spirit of the theory of iterated function systems (IFSs) and the so-called chaos game. The aim of this paper is to prove similar results in the case of infinite compact families of polynomial mappings. Inevitably, the topological and probabilistic aspects get more complicated than those in the finite case. The main motivation for this study is the wish to gain a better understanding of the metric space R of compact, pluriregular, and polynomially convex subsets of ℂN. This, however, requires a very careful analysis of different types of Julia-like sets arising naturally in this context. This variety of Julia sets is easier to grasp, if one looks at them as corresponding to the topological semigroup generated by infinite compact families of proper polynomial mappings. This is consistent with the point of view adopted by a number of researchers in one complex variable (see [2] and, e.g., the work of Stankewitz and Sumi [3–5]).As a visual hint of the additional complexity that infinite families bring about, we can consider what happens in the complex plane when instead of inspecting the filled-in Julia set of a single polynomial, in this casepcz=z2+c with c=c0, we examine the filled-in Julia set generated by the compact infinite family of polynomials pc:c∈K, where K is a closed-square centered at c0. In the following pictures, c0=0.3+0.5i and K=c0+−0.1,0.1+i−0.1,0.1. Figure 1 shows the autonomous Julia sets of the polynomials pc, one with c=c0 and the other eleven with c selected at random from K according to the uniform probability distribution.Figure 1Bearing in mind that this is just a tiny selection of Julia sets of the simplest (autonomous) type, one can appreciate the infinite variety of Julia sets (autonomous or not) that can be obtained by using just this family of simple quadratic polynomials. The union of all these sets would constitute the composite nonautonomous Julia set corresponding to all combinations ofc∈K. An approximate outline of this set is depicted in Figure 2, to the right of the filled-in Julia set for pc0 included for comparison. All these sets were plotted with the help of measuring the escape time of the orbits of the points under the iteration process. The shades of grey mark how quickly the considered orbits go beyond the radius of escape. Obviously, the situation gets more involved with more complicated polynomial mappings and in higher dimensions.Figure 2The paper is divided into seven sections including the introduction.In Section2, we take a closer look at the nature of convergence in the compact-open topology in the space of polynomial mappings in ℂN and, in particular, at the link to the coefficients of such mappings and their compositions. We also recall the definition of regular polynomial mappings and the concept of radius of escape and its basic properties. Moreover, we propose to regard the topological semigroups generated by compact families of regular mappings, with the composition of mapping as the semigroup operation, as the principal objects that give rise to the composite Julia sets that we want to study.In Section3, we recall the definition of the pluricomplex Green function of a nonempty compact subset of ℂN and the concept of pluriregularity. We also review the definition of the metric space R of all polynomially convex pluriregular compact sets in ℂN. The Julia sets we are studying in this paper reside precisely in that space. The space R is known to be complete (see [6]), and separable (see [1]) but not proper, in the sense that bounded closed sets do not have to be compact (see [1]). The topology of R still holds many unanswered questions. We discuss in some detail the intricacies of the closure operation in R. Namely, given a subset G of a subset of R, we compare the closure of the union in ℂN of the sets which are elements of G with the union in ℂN of the sets which are elements of the closure of G in R. It turns out that equality between these sets requires additional assumptions.In Section4, we prove that if P:ℂN→ℂN is a regular polynomial mapping, then the contraction AP:R∋E↦P−1E∈R is a similitude, which is also continuous when regarded as a function of two variables P,E↦APE. We also show that if F is a compact family of regular polynomial mappings of a fixed degree and K⊂R is compact, then AFK=⋃P∈FAPK is also compact. Furthermore, AF:κR→κR is a contraction whose fixed point SF can be described as the atlas of the Julia sets generated by sequences from the topological semigroup generated by F. We could also describe the set SF as the attractor of the infinite iterated function system AP:P∈F.Section5 begins with restating the definitions of autonomous filled-in Julia set JP generated by a single regular polynomial mapping P and a nonautonomous filled-in Julia set JPnn=1∞ generated by a sequence Pnn=1∞ of regular mappings. If the sequence comes from a compact family F of regular mappings with a fixed degree, then we show that JPm∘…∘P1 converges to JPnn=1∞ as m→∞. Moreover, the speed of the convergence can be estimated in terms of the natural metric on R. We also furnish the code space Fℕ with a metric like the one used in the classical case when the family F is finite. We close this section by linking the attractor to other types of Julia sets. Namely, SF consists of all sets JPnn=1∞ with Pnn=1∞∈Fℕ and the union of all sets constituting SF is the partly filled-in composite Julia set JtrF generated by F, whereas the polynomially convex hull of JtrF is the filled-in composite Julia set JF generated by F. We also include some comments to justify the use of semigroup terminology in this context.The last two sections contain a counterpart of Theorem2 in [1] in the case of a compact infinite family F of regular polynomial mappings of the same degree. Section 6 presents an extension of Theorems 2(a) and 2(b) from [1]. Essentially, we show how much the attractor SF pulls iterations of sets from the surrounding space towards itself if the polynomial mappings used in the iteration process come from F. In Section 7 we extend Theorem 2(c) from [1] to the case of compact infinite families F and we prove that the chaos game approximation of the partly filled-in composite Julia sets remains also valid in this case. We describe first the deterministic version based on disjunctive sequences and then the more familiar probabilistic version. Finally, we close the article with a few comments linking the mathematical context we have investigated to the study of invariant measures associated with general probabilistic approach to iteration function systems described in [7].A few words about the notation used in this paper are in order. For any nonempty setsA and B, the symbol BA will denote the set of all functions from A to B. If F is a collection of nonempty subsets of a set G, the symbol ⋃F will always denote ⋃F∈FF⊂G. Let X,d be a metric space. The symbol κX will denote the set of all nonempty compact subsets of X; Bda,r will denote the open ball with center a and radius r whereas distd, diamd, and χd will denote the distance of a point from a set, the diameter of a set, and the Hausdorff distance between two compact sets, respectively. The set x∈X:distdx,E≤ε, where ε>0, will be referred to as the ε-dilation of the set E⊂X. In the case of the Euclidean metric in ℂN, we will drop the subscript d. A norm symbol with a subscript will always denote the supremum norm. We will use the convention that ℤ+ stands for nonnegative integers and ℕ for natural numbers (excluding zero). Other notational conventions will be described later as the need for them arises.


## 2. Semigroups of Regular Polynomial Mappings

Ifd∈ℤ+, then by Pd we denote the vector space of all polynomial mappings P:ℂN→ℂN of degree not greater than d. Since Pd is of finite dimension, all norms defined on it are equivalent. In particular, if E⊂ℂN is compact and determining for polynomials (i.e., E is not contained in the zero set of a nonconstant polynomial), then a natural choice is the supremum norm PE=supPz:z∈E, where P∈Pd, and ℂN is endowed with the Euclidean norm. Another natural choice would be to transfer the norm from the Euclidean space of Taylor’s coefficients using the natural isomorphism:
(1)Td:Pd∋P↦DαP0α!∣α∣≤d∈ℂNNd=ℂN·Nd,where the multi-indices in ℤ+N are ordered according to the graded lexicographic order and
(2)Nd=N+dd.WhenER is the closed polydisc with the center at the origin and radius R>0, then we can use Cauchy’s estimates to establish a quantitative link between these two norms. For any P∈Pd, if Pz=∑∣α∣≤dzαpα, with some pα∈ℂN, then we have the following:
(3)PER≤∑∣α∣≤dpαRα≤NdNPER.Consequently, the topology onPd is the topology of uniform convergence of polynomial mappings on compact sets or, equivalently, the topology of convergence of the coefficients of polynomial mappings. To put it differently, it is the topology induced on Pd from the set P of all polynomial mappings P:ℂN→ℂN furnished with the compact-open topology, that is, the smallest topology containing all the sets of the form NK,U=P∈P:PK⊂U, where K⊂ℂN is compact and U⊂ℂN is open. The following statement will be useful later on.Proposition 1.
The composition mapping(4)P×⋯×P∋P1,…,Pk↦Pk∘…∘P1∈Pis continuous. In fact, if polynomial mappings are identified with their ordered sets of coefficients, then the mapping
(5)Pd1×⋯×Pdk∋P1,…,Pk↦Pk∘…∘P1∈Pd1⋅…⋅dk,is a polynomial mapping between the respective spaces of coefficients.Proof 1.
It suffices to considerk=2.
The first statement can be checked directly on the sets from the neighbourhood subbaseNK,U:K—compact,U—open of the topology of P. If Q∘P∈NK,U, then for some compact set L⊂ℂN, we can have the inclusions PK⊂intL and QL⊂U. This means that NK,intL×NL,U is contained in the inverse image of NK,U under the composition mapping, which completes the proof of continuity.
As for the second statement, in view of (1) and (3) it is enough to observe that the mapping
(6)ℂN⋅Nd1×ℂN⋅Nd2∋Td1P1,Td2P2↦Td1d2P2∘P1∈ℂN⋅Nd1d2is a polynomial.IfP∈Pd, we will denote by P^ the homogeneous component of P of degree d. We say that P∈Pd is regular if P^−10=0. The subset of all regular maps in Pd, denoted by Pd⋆, is an open subset of Pd (see Section 2 of [8]). Regular polynomial mappings are proper (cf. [9], Theorem 5.3.1) and so they are closed. As proper holomorphic mappings, they are also open and hence surjective (see [10], p. 301).Throughout this paper,BR will denote the closed Euclidean ball in ℂN with center at the origin and radius R>0. If P∈Pd⋆, then P−1BR=intP−1BR¯.In what follows, letPn denote the nth iterate of P, that is, the composition of n copies of P. We call R>0 an escape radius for P∈Pd⋆, if for every z∈ℂN\BR, we have
(7)limn→∞Pnz=∞.Note that ifR>0 is an escape radius for P, then all numbers bigger than R are also escape radii for the same mapping.In [11] (Lemma 1), it was proved that there exists a continuous function,
(8)Pd⋆∋P↦rP∈0,∞,such that rP (given by a constructive formula) is an escape radius for P. Another useful observation is that if R≥rP, then P−1BR⊂intBR (cf. [11], Lemma 1).In our investigation, we will consider a nonempty compact subsetF of Pd⋆. It is worth mentioning that such a family is regular in the sense defined in [12]. Indeed, a subset of Pd is regular there if and only if it is relatively compact in Pd⋆. One simple example of such a compact family was already mentioned in the Introduction section.If the composition of mappings is the semigroup operation, then because of Proposition1, any nonempty compact subfamily F of Pd⋆ generates a topological semigroup denoted by F, which in turn can naturally be associated with a Julia-type set. The primary objective of this article is to investigate such Julia sets and, more specifically, the approximation of such sets. The reason for invoking the concept of a semigroup in this context will be explained at the end of Section 5.


## 3. The SpaceR of Pluriregular Sets

IfE is a nonempty compact subset of ℂN, its pluricomplex Green function will be denoted by VE. For the background, we refer the reader to [9]. Recall that
(9)VE=logΦE,where ΦE is the Siciak extremal function
(10)ΦEz=supppz1/degp,z∈ℂN,with the supremum being taken over all nonconstant complex polynomials p:ℂN→ℂ such that pE≤1. It is easy to check that for any compact set E, the zero set of VE is equal to the polynomially convex hull of E. A compact set E is said to be pluriregular if VE is continuous.LetR be the family of all compact, pluriregular, and polynomially convex subsets of ℂN. Endowed with metric Γ defined by
(11)ΓE,F=maxVEF,VFE=VE−VFℂN,E,F∈ℛ,R turns out to be a complete metric space (see Theorem 1 in [6]). It is worth observing that the above formula defining ΓE,F can also be used for pluriregular sets E and F which are not necessarily polynomially convex. In this case, we obtain a pseudometric on the set of all pluriregular compact subsets of ℂN. Note also that if E,F∈R, and C is a set such that E∪F⊂C, then ΓE,F=VE−VFC.It was shown in Theorem1(a) from [1] that if K is compact in R, then
(12)⋃K=⋃K∈KK⊂ℂNis compact, being bounded and closed. In contrast, according to Theorem 1(d) in [1], a closed and bounded set in R does not need to be compact, since the space is not proper. In connection with these results, we would like to address here two questions, the answers to which can facilitate a better understanding of the topology of space R.The first question concerns the operations of closure inR and in ℂN. Let G⊂R. Is it true that
(13)⋃G¯=⋃G¯?It turns out that the answer depends on these additional assumptions:(i)
It is affirmative, ifG is compact in R. Indeed, ⋃G¯=⋃G=⋃G¯, where the second equality follows from Theorem 1(a) in [1].
(ii)
However, the equality (13) is not true in the general case. To be more precise, we have the following properties:
(1)
IfG is relatively compact in R, the inclusion “⊃” in (13) holds. Namely, ⋃G¯ is closed by Theorem 1(a) in [1] and the inclusion follows from
(14)⋃G¯⊃⋃G.
(2)
The inclusion “⊂” in (13) does not hold in general, even for a relatively compact set G. To see this, consider the following example from Section 3 in [6]. Take Ej=eit:t∈0,2π−j−1 and G=Ej:j∈1,2,…. We have
(15)⋃G¯=z∈ℂ:z≤1,⋃G¯=z∈ℂ:z=1.
(3)
IfG is not relatively compact, the inclusion “⊃” in (13) does not need to hold either. To see this, recall Example 3.6 from [13]. Take G=En:n∈1,2,… with
(16)En≔1,2∪⋃j=0n−1jn,jn+εn,where εn>0 is so small that
(17)capEn≤cap1,2+1/n,with cap· denoting the logarithmic capacity. There exists a∈0,1/2 such that
(18)limn→∞VEna=V1,2a>0.
On the other hand, ifx∈⋃G¯, then there exists K∈G¯ with x∈K, which means that we can find a subsequence Enk such that ΓEnk,K→0 as k→∞. At the same time,
(19)0≤VEnkx≤VEnkK≤ΓEnk,K.
Therefore, in this case,VEnkx→0 as k→∞. Thus, a∉⋃G¯. Hence, ⋃G¯⊅0,2=⋃G¯.The other question concerns the fact that inℂN the compactness of a subset is equivalent to being closed and bounded, but it is not the case in R. It is natural to ask whether compactness is needed in the assumption of Theorem 1(a) in [1] mentioned earlier. Let K be closed and bounded in R. Does ⋃K have to be compact in ℂN? The answer is no, it does not. Take K=G¯ and a∈0,1/2 from point (3) above (we use once again Example 3.6 in [13]). Since a∈⋃G¯, there exists a sequence ann=1∞⊂⋃G with an→a. Since a∉⋃K and an⊂⋃K, the set ⋃K is not closed.


## 4. Similitudes of the Space of Pluriregular Sets

Let us recall the transformation formula for regular polynomial mappings from Theorem 5.3.1 in [9]:
(20)VP−1E=1dVE∘P,E⊂ℂN,P∈Pd⋆.Recall also that ifX,d is a metric space and c>0 is a constant, then a mapping f:X→X is referred to as a similitude with the ratioc, if dfa,fb=cda,b for all a,b∈X. As a direct consequence of (20), we can describe a family of similitudes of R.Proposition 2.
IfP∈Pd⋆, then
(21)AP:R∋K↦P−1K∈Ris a contractive similitude with the contraction ratio 1/d.Proof 2.
LetK,L∈R. In view of (20) we have
(22)ΓP−1K,P−1L=VP−1K−VP−1LℂN=1dVK−VLPℂN=1dΓK,L.And this concludes the proof.In particular,AP is a continuous map. Moreover, for each R>0, the mapping
(23)Pd⋆∋P↦P−1BR∈Ris continuous (see Remark 1 in [8]). These observations can be generalized as follows.Proposition 3.
The mapping(24)Pd⋆×R∋P,K↦P−1K∈Ris continuous with respect to the product topology on Pd⋆×R.Proof 3.
FixK∈R and Q∈Pd⋆. In view of the triangle inequality and Proposition 2, if P∈Pd⋆,E∈R, then
(25)ΓP−1E,Q−1K≤ΓP−1E,P−1K+ΓP−1K,Q−1K=1dΓE,K+ΓP−1K,Q−1K.
Hence, it is now enough to prove that ifQn→Q, as n→∞, then
(26)ΓQn−1K,Q−1K⟶0.
Take a sequenceQnn=1∞⊂Pd⋆ which is convergent to Q and consider F≔Qn:n∈ℕ∪Q. This family is compact in Pd⋆ and therefore, ⋃P∈FP−1K is bounded in view of Remark 3.2 in [12], because K∈R. Take ρ>0 such that ⋃P∈FP−1K⊂Bρ. Since ·Bρ is a norm in Pd, there exists m>0 such that QnBρ≤R≔QBρ+1 for n≥m. This means that QnBρ⊂BR for such n. Obviously, QBρ⊂BR, too.
Fixε>0. The Green function VK is continuous; hence, it is uniformly continuous on BR, that is, there exists δ>0 such that if z,w∈BR with z−w<δ, then ∣VKz−VKw∣<ε. Since Qn→Q, there exists k≥m such that Qn−QBρ<δ if n≥k. Therefore,
(27)ΓQn−1K,Q−1K=1d∥VK∘Qn−VK∘Q∥Bρ<εifn≥k.And this concludes the proof.LetF now be a compact subset of Pd⋆. For any subset K of R, put
(28)AℱK≔⋃P∈ℱAPK,where the similitudes AP are as in Proposition 2.Proposition 4.
IfF is a compact subset of Pd⋆ and K is a compact subset of R, then AFK is compact.Proof 4.
Choose a sequenceEn of elements from AFK. Then, there exist sequences Pn⊂F and Kn⊂K such that En=Pn−1Kn. As K is compact, we can assume (passing to a subsequence if needed) that Kn→K in K if n→∞. Since F is compact, so here again (passing to a subsequence if needed), we can assume that Pn→P in F if n→∞. It follows from Proposition 3 that Pn−1Kn→P−1K, if n→∞. Thus, we have shown that every sequence in AFK has a convergent subsequence.Recall thatκX denotes the family of all nonempty compact subsets of the metric space X, furnished with the Hausdorff metric.Corollary 1.
LetF be a nonempty compact subset of Pd⋆. The mapping
(29)Aℱ:κℛ∋K↦⋃P∈ℱAPK∈κℛis well defined and is a contraction with ratio 1/d. In particular, the mapping AF has a unique fixed point SF∈κR.Proof 5.
It is enough to use the inequality(30)Γ⋃j∈JEj,⋃j∈JFj≤supj∈JΓEj,Fj,Ejj∈J,Fjj∈J⊂ℛ,(cf. [12], p. 891, and Corollary 2 in [6]) in combination with Propositions 2 and 4. The second conclusion follows from Banach’s contraction principle.


## 5. Julia-Type Sets

IfP∈Pd⋆, its (autonomous) filled-in Julia set is defined as follows:
(31)JP=z∈ℂN:Pnzn=1∞isbounded.As shown in [6], this set is the unique fixed point of the similitude AP:R∋K↦P−1K∈R. Hence, the standard argument used to prove the Banach contraction principle yields the equality
(32)JP=limn→∞Pn−1E,E∈ℛ.Moreover, ifR>0 is an escape radius of P, then we also have the equality
(33)JP=⋂n≥1Pn−1BR.Before turning our attention to other types of Julia sets, we need to point some useful estimates. IfR>0 is an escape radius for P∈Pd⋆, then (cf. Equation 7 in [1])
(34)ΓP−1BR,BR≤∥P∥∂BRRd.More generally, ifF is a compact family in Pd⋆, then due to the continuity of the mapping in (8), a common escape radius R>0 for all mappings in F can be found. Also,
(35)M≔supP∈FP∂BRis finite because of the compactness of F. Thus, as an immediate consequence of (34) we obtain
(36)ΓP−1BR,BR≤MRd,P∈ℱ.For a sequencePnn=1∞ of mappings from F, we define its filled-in Julia set (nonautonomous if the sequence is not constant) as follows:
(37)JPnn=1∞=z∈ℂN:Pn∘…∘P1zn=1∞isbounded.The estimate (36) allows the use of the enhanced version of Banach’s contraction principle (Lemma 4.5 in [12]) for sequence APnn=1∞. As a consequence, we can see that
(38)JPnn=1∞=limn→∞Pn∘⋯∘P1−1E,E∈ℛ,JPnn=1∞=⋂n≥1Pn∘⋯∘P1−1BR,if R>0 is as in (36). For some background on (a larger family of) nonautonomous Julia sets in the complex plane, see [14, 15].It turns out that nonautonomous filled-in Julia sets can be approximated by autonomous filled-in Julia sets. Before making this statement more precise, let us establish some notations. IfF is a compact family in Pd⋆, the symbol Fℕ will denote the code space overF, defined as the Cartesian product of countably many copies of F with the usual product topology. By Tychonoff’s theorem, Fℕ is compact and it can be furnished with the metric (see, e.g., Theorem 4.2.2 in [16]):
(39)ρPnn=1∞,Qnn=1∞=∑j=1∞Pj−QjBR2j,Pnn=1∞,Qnn=1∞∈ℱℕ.Proposition 5.
LetF be a compact family in Pd⋆. Then, for each Pnn=1∞∈Fℕ and m∈ℕ
(40)ΓJPnn=1∞,Pm∘⋯∘P1−1BR≤MRdmd−1,where M≔supP∈F∥P∥∂BR. In particular,
(41)JPnn=1∞=limm→∞JPm∘⋯∘P1.Proof 6.
To show (40) one can repeat the proof of the enhanced version of Banach’s contraction principle (Lemma 4.5 in [12]). Namely, in view of (36), we have
(42)ΓPn+m∘⋯∘P1−1BR,Pm∘⋯∘P1−1BR≤MRd∑j=1n1dm+j−1=MRdm∑j=1n1dj.Letting n go to infinity gives (40).
As for (41), in view of (40) we can write
(43)ΓJPnn=1∞,JPm∘⋯∘P1≤ΓJPnn=1∞,Pm∘⋯∘P1−1BR+ΓPm∘⋯∘P1−1BR,JPm∘⋯∘P1≤2MRdmd−1.For a finiteF, Proposition 5 was shown in [17].We define thepartly filled-in composite Julia set of the compact family F⊂Pd⋆ as
(44)Jtrℱ=⋂m∈ℕ⋃P1,…,Pm∈ℱPm∘⋯∘P1−1BR.This set is compact (see proof of Theorem 4.6 in [12]), and its polynomially convex hull JF is the unique fixed point of the mapping:
(45)ℛ∋K↦⋃P∈ℱAPK^∈ℛ.JF is called the filled-in composite Julia set of F. Here, the hat marks the operation of taking the polynomially convex hull of the set under the hat. The subscript tr stands for the word truncated.The following theorem describes the connection between the Julia sets from this section and the attractorSF from the end of the previous section (Corollary 1).Theorem 1.
LetFbe a nonempty compact family inPd⋆. Then,
(1)
SF=JPnn=1∞:Pnn=1∞∈Fℕ;
(2)
JtrF=⋃SF.Proof 7.
This fact can be deduced from general theory in [18] but we give here the proof in this special case to make our work consistent (cf. also [19] for the case of a finite family).The familyS=SF is the unique fixed point of AF:κR→κR (cf. Corollary 1). Therefore,
(46)S=AℱS=⋃P1∈ℱAP1S=⋃P1,P2∈ℱAP1AP2S=⋯=⋃P1,P2,…,Pn∈ℱAP1∘AP2∘⋯∘APnS,n∈ℕ.Since by Proposition 2 the function APj is a contraction,
(47)diamΓAP1∘AP2∘⋯∘APnS⟶0,n⟶∞.The sequence AP1∘AP2∘⋯∘APnSn=1∞ is also decreasing with respect to inclusion. Therefore, its limit is a singleton, and by the definition of JPnn=1∞, we have the equality
(48)⋂nAP1∘AP2∘⋯∘APnS=JPnn=1∞.Thus, S=JPnn=1∞:Pnn=1∞∈Fℕ.From the definition ofJtrF, it is obvious that ⋃SF⊂JtrF. Let us fix a common escape radius R>0 for all P∈F.Now, takez∈JtrF. First, we claim that for any n∈ℕ, there exists En contained in the 1/n-dilation of SF and such that z∈En. Indeed, given n∈ℕ, one can choose m∈ℕ so that the inequality
(49)MRdmd−1<1nis satisfied with M defined in Proposition 5. Since z∈JtrF, by the definition of the latter, there exist P1,…,Pm∈F such that z∈Pm∘⋯∘P1−1BR. The inequalities (49) and (40) imply the estimate
(50)ΓPm∘⋯∘P1−1BR,JPm∘⋯∘P1<1n,and this means that En=Pm∘⋯∘P1−1BR fulfills our claim.To finish the proof, we want to show thatz∈⋃SF. Since SF is compact, the sequence En has an accumulation point E∈SF. But then, since convergence of sets in R,Γ means uniform convergence of the corresponding pluricomplex Green functions, we can conclude that VEz=0, which means that z∈E⊂⋃SF.Remark 1.
It is worth emphasizing that all of the types of Julia sets defined in this section correspond one way or another to sequences in the semigroupF. This is the reason why conceptually it is natural to see the set SF not only as the attractor associated with the semigroup F but also as a kind of atlas of all Julia sets associated with that semigroup. Indeed, this is exactly the meaning of Theorem 2 combined with the definition of JtrF.


## 6. On the Attracting Nature ofSF

Recall that we use the symbolBΓE,r to denote the open ball in R,Γ with center at E∈R and radius r>0.The next theorem is a counterpart of Theorems2(a) and 2(b) in [1] in the case of infinite compact regular families of polynomial mappings.Theorem 2.
LetF be a nonempty compact family in Pd⋆.
(a)
Letπnn=1∞⊂F. If E∈R and U⊃SF is an open subset of R, then almost all elements of the sequence
(51)E=Aπn∘⋯∘Aπ1E:n≥1belong to U. In particular, all accumulation points of this sequence are in SF and so E¯ is compact in R.
(b)
LetE∈SF. For every neighbourhood V of E, there exists an open set U⊃SF and mappings Q1,…,Qm∈F, such that
(52)AQm∘⋯∘AQ1U⊂V.
Moreover,m can be made arbitrarily large.Proof 8.
a Fix a common escape radius R>0 for all P∈F. Let M be like in Proposition 5. Fix P∈F. In view of the proof of (40) (but with E replacing BR) combined with the triangle inequality and (36), we have the following estimates:
(53)distΓπ1∘⋯∘πm−1E,Sℱ≤Γπ1∘⋯∘πm−1E,Jπ1∘⋯∘πm≤supΓE,P−1E:P∈ℱdm−1d−1≤supP∈ℱΓE,BR+ΓBR,P−1BR+ΓP−1BR,P−1Edm−1d−1≤d+1ΓE,BR+M/Rdmd−1⟶0ifm⟶∞,which is what is needed, as distΓR\U,SF>0.
b Take E∈SF and ε>0 such that BΓE,ε⊂V. Fix n∈ℕ. Without loss of generality, we may suppose that d−ndiamΓSF<ε/4. It follows from Theorem 1 that E=JPnn=1∞ for some Pnn=1∞∈Fℕ. Moreover, JPnn=1∞=limm→∞JPm∘⋯∘P1 by (41). Therefore, we can choose m>n such that
(54)ΓE,JPm∘⋯∘P1<ε4.DefineQj=Pm+1−j for j∈1,…,m and let
(55)U=F∈R:distΓF,SF<ε.IfF∈U, then there exists G∈SF such that ΓF,G<ε. Therefore,
(56)ΓQ1∘⋯∘Qm−1F,Q1∘⋯∘Qm−1G≤d−mΓF,G<ε2.Moreover,
(57)ΓJPm∘⋯∘P1,Q1∘⋯∘Qm−1G=ΓPm∘⋯∘P1−1JPm∘⋯∘P1,Pm∘⋯∘P1−1G≤d−mΓJPm∘⋯∘P1,G≤d−mdiamΓSℱ<ε4.Combining (54), (56), (57), and using the triangle inequality, we see that
(58)ΓE,Q1∘⋯∘Qm−1F<ε,as required.


## 7. Chaos Game and Approximation of Attractors

We will start with the definition of disjunctive sequences over a finite or countable alphabet.LetA be a nonempty set which is at most countable. A sequence of elements of A, that is, a function τ:ℕ→A is said to be disjunctive, if for any m∈ℕ and any function θ:1,…,m→A there exists n∈ℕ such that θj=τn+j for j∈1,…,m. A simple example of a disjunctive sequence with A=ℕ is given in [20]: the first entry is 1, followed by all 2-letter words over 1,2, then by all 3-letter words over 1,2,3, and so on.IfA is regarded as the alphabet and functions like θ as possible finite words overA, then the sequence τ is disjunctive if it contains all finite words as its finite subsequences. Disjunctive sequences, usually over a finite alphabet, have been used for a long time in study of formal languages, in automata theory and number theory (see [21] for an overview). More recently, disjunctive sequences turned out to be a natural tool for derandomization of the chaos game (see [20]).The next result is a generalization of Theorem2(c) in [1].Theorem 3.
LetF be a nonempty compact subset of Pd⋆ and F0=πn:n∈ℕ a dense countable subset of F. Let τ:ℕ→ℕ be a disjunctive sequence.Then, for anyE∈R,
(59)limm→∞ΓJF,⋃Em¯=0,where
(60)Em=πτ1∘⋯∘πτn−1E:n≥m.Proof 9.
First of all, it should be noted that Theorem2 yields compactness of Em¯. Furthermore, a countable subset F0 of F exists because of the separability of Pd. Recall also that R is separable (see Theorem 1(d) in [1]).
Fix a norm· in Pd.
In view of Theorem2(b) and from Theorem 1(b) in [1], it suffices to prove that
(61)limm→∞χΓSF,Em¯=0,where χΓ denotes the Hausdorff metric corresponding to Γ.Takeε>0. In view of Theorem 2(a), if m is sufficiently large, then the ε-dilation of SF contains Em, and hence also Em¯. In order to prove that for sufficiently large m, the ε-dilation of Em contains SF, it is enough to show that any point from an ε/2-dense finite subset of SF is within ε/2-distance from a point of Em.LetA∈SF be an element of a fixed ε/2-dense finite subset of SF. By Theorem 2(b), there exist ℓ∈ℕ and Q1,…,Qℓ∈F such that for δ∈0,ε/2 the image of the δ-dilation of SF via the mapping
(62)F↦Q1∘⋯∘Qℓ−1Fis a subset of BΓA,ε/4. Using Theorem 2(a) again if necessary, we can increase m so that the δ-dilation of SF contains Em. In particular, if K∈Em, then
(63)Q1∘⋯∘Qℓ−1K∈BΓA,ε4.Proposition3 assures continuity of Pdl⋆×R∋Q,K↦Q−1K∈R. By Proposition 1, the mapping Fl∋q1,…,ql↦q1∘⋯∘qℓ∈Pdl⋆ is continuous, too. Therefore, the mapping
(64)Fl×Em¯∋q1,…,qℓ,K↦q1∘⋯∘qℓ−1K∈Ris uniformly continuous. Thus, there exists η>0 such that if pj,qj∈F with qj−pj<η,j∈1,…,ℓ, and K,L∈Em¯ with ΓK,L<η, then
(65)Γq1∘⋯∘qℓ−1K,p1∘⋯∘pℓ−1L<ε4.SinceF0 is dense in F, there exist P1,…,Pℓ∈F0 such that P1−Q1<η,…,Pℓ−Qℓ<η. Let θ:1,…,ℓ→ℕ be chosen so that Pj=πθj for j∈1,…,ℓ.Sinceτ is disjunctive, for some n≥m, we have θj=τn+j for j∈1,…,ℓ. Consequently, if we put K=πτ1∘⋯∘πτn−1E, we have K∈Em and
(66)πτ1∘⋯∘πτn+ℓ−1E=πθ1∘⋯∘πθℓ−1K=P1∘⋯∘Pℓ−1K.We know that(67)ΓP1∘⋯∘Pℓ−1K,Q1∘⋯∘Qℓ−1K<ε4,because of the choice of η, and so it follows from (63) combined with the triangle inequality that
(68)πτ1∘⋯∘πτn+ℓ−1E∈BΓA,ε2.And this concludes the proof.The next statement is a probabilistic version of the above theorem.Corollary 2.
LetF be a nonempty compact subset of Pd⋆ and F0=πn:n∈ℕa dense countable subset of F. Let τ:ℕ→ℕ be generated according to probabilities p1,p2,…>0 such that ∑n=1∞pn=1, that is, the values τj of τ are chosen at random, independent from each other, so that ℙτj=i=pi for i,j∈ℕ.
Then, for anyE∈R, with probability 1,
(69)limm→∞ΓJF,⋃Em¯=0,where Em=πτ1∘⋯∘πτn−1E:n≥m.Proof 10.
Because of the strong law of large numbers applied to Bernoulli processes, we can conclude that, given a finite word over the alphabetℕ, the sequence τ contains this word with probability 1. Hence, we can use the same reasoning as in the theorem above.We would like to finish the article with a general observation.Let us assume that we have a probability measureW on some σ-algebra of subsets of F, where as in Theorem 3, F is a compact subset of Pd⋆. We will follow the general set-up from [7]. We will be concerned with a Markov chain ZnK, with initial state K∈R and
(70)ZkKPnn=1∞=Kifk=0,APk∘⋯∘AP1Kifk≥1,where Pk are independently and identically distributed (IID) random elements in F with probability distribution W. Let W also denote the induced probability measure on the code space Fℕ.In a more general setting, the initial state can be given by a random elementX0 in R, independent of Pnn=1∞, and with the probability distribution ν. Then, it is natural to define the random elements
(71)ZkνPnn=1∞=ZkX0Pnn=1∞,k∈ℤ+.So in particular,ν is the probability distribution of Z0ν. If we also define Fν as the probability distribution of Z1ν, then the probability distribution of Zkν is Fkν.The reverse order chain is defined to be as follows:(72)Z^kKPnn=1∞=Kifk=0,AP1∘⋯∘APkKifk≥1.Because of the IID property, bothZkν and Z^kν have the same probability distribution Fkν.Note that all of the above definitions make sense because Proposition3 is guaranteeing appropriate measurability of the sets.Letδa denote the Dirac measure concentrated at a, that is, δaE=1Ea. Below, we use the mapping
(73)Π:Fℕ∋Pnn=1∞↦JPnn=1∞∈SF.It is continuous because of the estimate (40) combined with the definition (39) of the metric ρ on Fℕ. Indeed, given ε>0 and Q=Qnn=1∞∈Fℕ, choose m so that M/Rdmd−1<ε/4. If P=Pnn=1∞∈Fℕ is such that ρP,Q<ε/2m+1, then Pm−QmBR<ε/2 and thus ΓΠP,ΠQ<ε in view of (40) combined with the triangle inequality.Proposition 6.
LetF be a nonempty compact subset of Pd⋆, let W be a probability measure on some σ-algebra of subsets of F and let W also denote the induced probability measure on Fℕ.
Letμ be the pushforward measure on R obtained from the measure W on the code space Fℕ via the mapping Π. Then:
(a)
Ifν is a Borel probability measure, then Fnν→μ weakly. In particular, Fμ=μ and μ is the unique probability measure invariant with respect to F.
(b)
For allK∈R and for a.e. Pnn=1∞⊂F
(74)1n∑k=1nδZkKPnn=1∞⟶μweakly.
(c)
The support ofμ is SF; hence, this is the unique fixed point of the iterated function system AP:P∈F. In particular, the support of μ is compact.Proof 11.
(a) and (b) are straightforward consequences of [7] (Theorem 8).
(c) By Theorem 8 (15) from [7], there exists n0, which may depend on Pnn=1∞ and ε, such that
(75)Pn∘⋯∘P1−1BR∈suppμε,ifn≥n0.
Therefore,SF⊂suppμ. On the other hand, ΠFℕ=SF and hence suppμ⊂SF.It should be noted that the novel element in the above observation is the compactness of the support of the measure in the case of infinite family and its invariance under the IFS in this case. Theorem 8 in [7] gives this property, but only in the case of finite iterated function systems.



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