Attractors of compactly generated semigroups of regular polynomial mappings

HindawiComplexit y

Volume2018,ArticleID5698021,11pageshttps://doi .org/10.1155/2018/5698021

Research Article

AttractorsofCompactlyGeneratedSemigroupsofRegular PolynomialMappings

AzzaAlghamdi ,

Klimek ,

andMartaKosek

1Department of Mathematics, Faculty of Science, Albaha University, Al Baha, Saudi Arabia

2Department of Mathematics, Uppsala University, P.O. Box 480, 751-06 Uppsala, Sweden

3Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University,Łojasiewicza 6,30-348 Kraków, Poland

CorrespondenceshouldbeaddressedtoMartaKosek;marta.kosek@im.uj.edu.plReceive d9April2018;Accepted9July2018;Published11November2018Academic Editor:

DimitriVolchenkov

Copyright©2018AzzaAlghamdietal.ThisisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlyc i t e d .

Weinvestigatethemetricspaceofpluriregularsetsaswellasthecontractionsonthatspaceinducedbyinfinitecompactfamiliesof 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 chaosgame.

1. Introduction

In the recent paper[1]it was shown, as a part of the investi- gation of the space of pluriregular sets, that it is possible to approximatecompositeJuliasetsgeneratedbyfinitefamilies of proper polynomial mappings inℂ^Nin 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 thecaseofinfinitecompactfamiliesofpolynomialmappings.

Inevitably,thetopologicalandprobabilisticaspectsgetmore

complicated than those in the finite case. The main motiva- tion for this study is the wish to gain a better understanding of the metric spaceRof compact, pluriregular, and polyno- mially convex subsets ofℂ^N. This, however, requires avery careful analysis of di erent types of Julia-like sets arising naturally in this context. This variety offf Julia sets is easier to grasp, if one looks at them as corresponding to the topo- logical 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 thatinfinite 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 casep_cz=z²+cw i t h

c=c₀, we examine the filled-in Julia set generated by the compact infinite family of polynomialsp_cc∈K, whereKisaclosed-

squarecenteredatc₀.Inthefollowingpictures,c_{0= 0}3+05iandK=c

0+−01,01+i−01,01.Figure1shows the autonomous Julia sets of the polynomialsp_c, one withc=c₀and the other eleven withcselected atr a n d o m

fromKaccording to the uniform probability distribution.

Bearing 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 simplequadratic polynomials.Theunionofallthesesetswouldconstitutethe

composite nonautonomous Julia set corresponding toall combinations ofc∈K.An approximate outline of this set isd e p i c t e d i n F i g u r e 2 , t o t h e r i g h t o f t h e fi l l e d - i n J u l i a

set forp_c

0included 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 consideredo r b i t s

2 Complexity

Figure1

Figure2 go beyond the radius of escape. Obviously, the situation gets more involved with more complicated polynomial mappings and in higherdimensions.

The paper is divided into seven sections including thei n t r o d u c t i o n .

In Section2,we take a closer look at the nature of convergence in the compact-open topology in the space of polynomial mappings inℂ^Nand, in particular, at the link to the coe cients of such mappings and their compositions.ffi Wealsorecallthedefinitionofregularpolynomialmappings

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 theprincipalobjectsthatgiverisetothecompositeJuliasets that we want tostudy.

In Section3,we recall the definition of thepluricomplex Green function of a nonempty compact subset ofℂ^Nand the concept of pluriregularity. We also review the definition of the metric spaceRof all polynomially convexpluriregular

n=1 n=1

n=1 n=1

DαP 0 α

N compact sets inℂ^N. The Julia sets we are studying in this

paper reside precisely in that space. The spaceRis 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 ofRstill holds many unanswered questions. We discuss in some detail the intricacies of the closure operation inR. Namely, given a subsetGof a subset ofR, we compare the closure of the union inℂ^Nof the sets which are elements ofGwith the union inℂ^Nof the sets which are elements of the closure ofGinR. It turns out that equality between these sets requires additionalassumptions.

In Section4,we prove that ifPℂ^N→ℂ^Nis a regular

polynomial mapping, then the

contractionA_PR∋E↦P⁻¹E∈Ris a similitude, which is also continuous when regarded as a function of two variablesP,E↦APE. We also show that ifFis a compact family of regular polynomial mappings of a fixed degree andK⊂Ri s

compact, thenA_FK=⋃_P∈FA_PKis also compact.

Furthermore,A_FκR→κRis a contraction whosefixed pointS Fcan be described as the atlas of the

Julia sets generated by sequences from the topological semigroup generated byF. We could also describe the setS Fas the attractor of the infinite iterated function systemAPP∈F.

Section5begins with restating the definitions ofautono- mous filled-in Julia setJ Pgenerated by a single regular polynomial mappingPand a nonautonomous filled-in Julia setJ P_n^∞generated by a sequenceP_n^∞of regular

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 sets A and B, the symbol B^Awill denotethesetofallfunctionsfromAtoB.IfFisacollection

of nonempty subsets of a setG, the symbol⋃Fwill always denote⋃_F∈FF⊂G. LetX, d be a metric space. The symbolκXwill denote the set of all nonempty compact subsets

ofX;B_da,rwill denote the open ball with centera

and radiusrwhereas dist_d, diam_d, andχ_dwill denote the distance of a point from a set, the diameter of a set, and the Hausdor distance between two compactff s e t s , respectively. The setx∈Xdist_dx,E≤ε, whereε> 0, will be referred to as theε-dilation of the setE⊂X.In thec a s e o f t h e E u c l i d e a n m e t r i c i nℂ^N,w e w i l l d r o p t h e

subscript d. A norm symbol with a subscript will always denote the supremum norm. We will use the convention thatℤ₊stands for nonnegative integers andℕforn a t u r a l numbers (excluding zero). Other notational conventionswill be described later as the need for themarises.

2. SemigroupsofRegularPolynomialMappings

Ifd∈ℤ₊,thenbyP_dwedenotethevectorspaceofallpoly- nomial mappingsPℂ^N→ℂ^Nof degree not greater thand.SinceP_dis of finite dimension, all norms defined on it are equivalent.Inparticular,ifE⊂ℂ^Niscompactanddetermin- ingforpolynomials(i.e.,Eisnotcontainedinthezerosetofa

nonconstant polynomial), then a natural choice is the supre- mum normP_E= supP z:z∈E, whereP∈Pd, and

mappings. If the sequence comes from a compact family Fof regular mappings with a fixed degree, then we show thatJ Pm∘ … ∘P₁converges toJ Pn∞

asm→ ∞. ℂ^Nis endowed with the Euclidean norm. Another natural choice would be to transfer the norm from the Euclidean Moreover, the speed of theconvergence can be estimated inⁿ⁼¹ space of Taylor’s coe cients using the natural isomorphism:ffi terms of the natural metric onR. We also furnish the code

spaceF^ℕwithametricliketheoneusedintheclassicalcase when the familyFis finite. We close this section byl i n k i n g

T_dP_d∋P↦ _∣α∣≤

d

∈ℂ^{N Nd}=ℂ^N·Nd, 1 the attractor to other types of Julia sets. Namely,S Fc o n - where the multi-indices inℤ^Nare ordered according to the sists of all setsJ Pn∞

withPn∞ ∈F^ℕand theunion graded lexicographicorder and⁺ ofallsetsconstitutingSFi s thepartlyfilled-incomposite Julia

setJ_trFgenerated byF, whereas the polynomially convex hull ofJ_trFis the filled-in composite Juliaset

J Fgenerated byF. We also include some comments to justify the use of semigroup terminology in this context.

ThelasttwosectionscontainacounterpartofTheorem2in[1]in the case of a compact infinite familyFof regular polynomial mappings of the same degree. Section6presents anextensionofTheorems2(a)and2(b)from[1].Essentially, we show how much the attractorS Fpulls iterations ofsets

N_d= N+d 2

d

WhenE Ris the closed polydisc with the center at the origin and radiusR> 0, then we can use Cauchy’s estimates to establish a quantitative link between these two norms.

ForanyP∈P_d,ifPz= ∑_∣α∣≤dz^αp_α,withsomep_α∈ℂ^N,then wehavethefollowing:

from the surrounding space towards itself if the polynomial mappingsusedintheiterationprocesscomefromF.InSec- tion7we extend Theorem2(c)from[1]to the case ofcom-

P_{E R}≤〠

∣α∣≤d

p_αR^α≤N_d P_{E R} 3

pact infinite familiesFand 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 deter- ministic version based on disjunctive sequences and then the more familiar probabilistic version. Finally, we closethe article with a few comments linking the mathematical con- text we have investigated to the study of invariantm e a s u r e s

Consequently, the topology onPdis the topology of uniform convergence of polynomial mappings on compact setsor,equivalently,thetopologyofconvergenceofthecoef-

ficientsofpolynomialmappings.Toputitdi erently,ff itisthe topology induced onP_dfrom the setPof allp o l y n o m i a l

mappingsPℂ^N→ℂ^Nfurnished with the compact-open topology, that is, the smallest topology containing all the sets

d

d

d

d

lim

d

int PB^{1 R}

of the formNK,U=P∈PP K⊂U, whereK⊂ℂ^Nis compactandU⊂ℂ^Nisopen.Thefollowingstatementwillbe useful lateron.

Proposition 1.The composition mapping

P×⋯×P∋P₁,…,P_k↦P_k∘ … ∘P₁∈P 4

In our investigation, we will consider a nonemptycompact subsetFofP^⋆. It is worth mentioning that such a family is regular in the sense defined in[12].Indeed, asubset ofPdis regular there if and only if it is relatively compact inP^⋆. One simple example of such a compact family was already mentioned in the Introduction section.If the composition of mappings is the semigroup opera-tion, then because of Proposition1,any nonempty compact iscontinuous.Infact,ifpolynomialmappingsareidentified subfamilyFofP^⋆generates a topological semigroup with their ordered sets of coefficients, then themapping denoted byF, which in turn can naturally be associated^d

P_d1×⋯×Pd_k∋P₁,…,Pk↦Pk∘…∘P₁∈Pd₁⋅…⋅d_k, 5 is a polynomial mapping between the respective

spacesofcoefficients.

Proof 1.It su ces to considerffi k= 2.

The first statement can be checked directly on the sets from the neighbourhood subbaseN K,U:K—compact,

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 Section5.

3. TheSpaceRofPluriregularSets

IfEis a nonempty compact subset ofℂ^N, its pluricomplex U—open of the topology ofP. IfQ∘P∈N K,U, then

for some compact setL⊂ℂ^N, we can have the inclu- sionsP K⊂intL andQ L⊂U.This meanst h a t

Green function will be denoted byVE

we refer the reader to[9].Recall that . For the background, N K, intL×N L,Uis contained in the inverse image

ofN K,Uunder the composition mapping, which com- pletes the proof of continuity.

As for the second statement, in view of(1)and(3)it is

V_E=logΦ_E, 9

whereΦ_Eis theSiciak extremal function enough to observe that the mapping

ℂ^N⋅Nd1×ℂ^N⋅Nd2∋TdP¹,T^dP1²↦T^ddP²2∘P¹ 12 6

Φ_Ez= sup

p p z^1/degp,

z∈ℂ^N,

10

∈ℂ^N⋅Nd1d2 is a polynomial.

with the supremum being taken over all nonconstant complex polynomialspℂ^N→ℂsuch thatp_E≤1. It is easy to check that for any compact setE, the zero set ofVEi se q u a l t o t h e p o l y n o m i a l l y c o n v e x h u l l o fE .Ac o m p a c t

IfP∈P , we will denote byP̂ the homogeneous setEis said to bepluriregularifV_Eis continuous.

component

̂̂⁻¹

d

ofPof degreed. We say thatP∈P_d isregular LetRbe the family of all compact, pluriregular, and polynomially convex subsets ofℂ^N. Endowed with metricΓ ifP 0 = 0 . The subset of all regular maps inPd,denoted defined by

byP^⋆,isanopensubsetofP_d(seeSection2of[8]).Regular polynomialmappingsareproper(cf.[9],Theorem5.3.1)and

so they are closed. As proper holomorphic mappings, they are also open and hence surjective (see[10],p. 301).

Throughout this paper,B_Rwill denote the closed Euclidean ball inℂ^Nwith center at the origin and radius R> 0. IfP∈P^⋆,thenP⁻¹B

R= .

ΓE,F=max V_EF,V_FE

=V_E−V_FℂN, E,F∈ℛ, 11 Rturns out to be a complete metric space (see Theorem1in[6]).It is worth observing that the above formula defining

ΓE,Fcan also be used for pluriregular setsEandFwhich Inwhat follows, letP^{d n}denote thenth iterate ofP, that is, are not necessarily polynomially convex. In this case, we the composition ofncopies ofP. We callR>0

anescaperadiusforP∈P^⋆, if for everyz∈ℂ^N\B_R, we have obtain a pseudometric on the set of all pluriregular compact subsets ofℂ^N. Note also that ifE,F∈R, andCis

n→∞ Pⁿz =∞ 7

a set such thatE∪F⊂C, thenΓE,F=V_E−V_FC.

It was shown in Theorem1(a)from[1]that ifKis compact inR, then

Note that ifR>0is an escape radius forP, then allnum- bersbiggerthanRarealsoescaperadiiforthesamemapping.

In[11](Lemma 1), it was proved that there exists a continuous function,

P^⋆∋P↦rP∈0,∞, 8

such thatr P(given by a constructive formula) is an escape radius forP. Another useful observation is that ifR≥r P,

thenP⁻¹BR⊂intB_R(cf.[11],Lemma 1).

⋃

⋃

K∈K

iscompact,beingboundedandclosed.Incontrast,according to Theorem1(d)in[1],a closed and bounded set inRdoes 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 spaceR .

n=1

1

d

n→∞

d

k

d

The first question concerns the operations of closure in Rand inℂ^N. LetG⊂R. Is it true that

⋃

⋃

(i) It is a rmative, ifffi Gis compact inR.Indeed,

⋃G=⋃G=⋃G,where the second equality follows from Theorem1(a)in[ 1 ] .

(ii) However,theequality(13)isnottrueinthegeneralcase.To bemoreprecise,wehavethefollowingproperties:

(1) IfGisrelativelycompactinR,theinclusion“⊃”

bounded,butitisnotthecaseinR.Itisnaturaltoaskwhether compactnessisneededintheassumptionofTheorem1(a)in

[1]mentioned earlier. LetKbe closed and bounded inR.

Does⋃Khave to be compact inℂ^N? The answer is no, it does not. TakeK=Ganda∈0, 1/2 from point (3) above (weuseonceagainExample3.6in[13]).Sincea∈⋃G,there exists a sequencea_n^∞⊂ ⋃Gwitha_n→a. Sincea∉ ⋃Kanda_n⊂⋃K,the set⋃Kis notclosed.

4. Similitudes of the Space of PluriregularSets

Let us recall the transformation formula for regular poly- nomial mappings from Theorem 5.3.1 in[9]:

VP⁻¹E= VE∘P, E⊂ℂ^N,

P∈P^⋆ 20

in(13)holds. Namely,⋃Gisclosed byTheorem d ^d

1(a)in[1]and the inclusion follows from

⋃

⋃

(1) The inclusion“⊂”in(13)does not hold in general, even for a relatively compact setG.Tosee this, consider the following

example from

Section3in[6].TakeE_j=e^itt∈0,2π−j⁻¹

Recall also that ifX, d is a metric space andc>0 is a constant, then a mappingf X→Xis referred to as asimilitude with the ratio c, if df a,f b=cda,bfor alla,b∈X. As a direct consequence of(20),we can describe a family of similitudes ofR.

Proposition 2.If P∈P^⋆, then

andG=Ejj∈1,2,… . Wehave

⋃

⋃

APR∋K↦P⁻¹K∈ R 21

15

isacontractivesimilitudewiththecontractionratio1/d.Proof2.LetK,L

∈R.Inviewof(20)wehave (2) IfGis not relatively compact, thei n c l u s i o n

“⊃”in(13)does not need to hold either.

To see this, recall Example 3.6 from[13]. ΓP⁻¹ K,P⁻¹ L =VP⁻¹K−V_P−1L

ℂ^N

TakeG=E_nn∈1,2,… with 1

dV_K−V_L

1 ^Pℂ

N 22

n−1j j 16 ₌

dΓK,L

E_n≔1, 2∪

⋃

j=0 n, n+εn,

And this concludes the proof.

whereε_n>0 is so small that

capEn≤cap 1, 2+1/n, 17 with cap · denoting the logarithmic capacity.

There existsa∈0, 1/2 such that

limVEa =V^1,2a >0 18 On the other hand, ifx∈ ⋃G, then there exists

In particular,A_Pis a continuous map. Moreover, for each R> 0, the mapping

P^⋆∋P↦P⁻¹B_R∈R 23

is continuous (see Remark1in [8]).These observations can be generalized asf o l l o w s .

Proposition 3.The mapping

K∈Gwithx∈K,whichmeansthatwecanfind ⋆ −1 24

a subsequenceEn_ksuch thatΓE_n,K→0 as P_d×R∋P,K↦P K∈R

k→ ∞. At the sametime,

iscontinuouswithrespecttotheproducttopologyonP^⋆× R.

0≤V

=

d

x≤V ≤ΓE ,K Proof 3.FixK∈RandQ∈P^⋆. In view of the triangle

E_nk E_nk

K n_k

inequality and Proposition2,if ^d∈P^⋆,E∈R, then

Therefore, in this case,VE_n

x→0 ask→ ∞.

k ΓP⁻¹ E,Q⁻¹ K ≤ΓP⁻¹

P E,P⁻¹

d

K +ΓP⁻¹ K,Q⁻¹K Thus,a∉ ⋃G. Hence,⋃G⊅0, 2 =⋃G

The other question concerns the fact that inℂ^Nthe compactness of a subset is equivalent to being closed and

=1

ΓE,K+ΓP⁻¹K,Q⁻¹K

25

n=1 d

d

ρ ρ

n=1

n→∞

d

d

d

n

n

d R

d

n=1

n=1

ρ

ρ

Hence, it is now enough to prove that ifQ_n→Q, as

n→ ∞, then Proof 5.It is enough to use the inequality

ΓQ⁻¹K,Q⁻¹K⟶0 26 Γ

⋃

⋃

,F_jj∈J⊂ℛ,

n

Take a sequenceQn∞ ⊂P^⋆which is convergent to

j∈J j∈

J

j∈J

30 Qand considerF≔Q_nn∈ℕ∪Q. This family is compact

inP^⋆and therefore,⋃_P∈FP⁻¹Kis bounded in view of

Remark 3.2 in[12],becauseK∈R. Takeρ (cf.[12],p. 891, and Corollary2in [6])in combination with Propositions2and4.The second conclusion follows from Banach’s contraction principle.

>0 such that⋃_P∈FP⁻¹K⊂B_ρ. Since ·_Bis a norm in

P_d,there existsm>0s u c h that Q_nB≤R≔Q_B+1

5. Julia-TypeSets

forn≥m. This means thatQ_nB_ρ⊂B_Rfor suchn. Obvi- IfP∈P^⋆, its(autonomous)filled-in Julia setis defined ously,QB_ρ⊂B_R, too.

Fixε>0. The Green functionV_K is continuous; hence,it

as follows:d

isuniformlycontinuousonB_R,thatis,thereexistsδ>0such

thatifz,w∈B_Rwithz −w< δ,then∣V^Kz −V^Kw ∣<ε. J P=z∈ℂ^NPⁿz^∞is

bounded 31

SinceQ_n→Q, there existsk≥msuch thatQ_n−Q_B<δ ifn≥k. Therefore,

ΓQ⁻¹K,Q⁻¹K =1

∥V∘Q−V∘Q∥ <εifn≥k

As shown in[6],this set is the unique fixed point of the similitudeAPR∋K↦P⁻¹K∈R. Hence, the standard argument used to prove the Banach contraction principle

n d ^{K n K}

Bρ

yields the equality

27 J P=lim Pⁿ⁻¹E

, E∈ℛ 32

And this concludes the proof.

LetFnow be a compact subset ofP^⋆. For any subset KofR, put

A_ℱK≔

⋃

P∈ℱ

where the similitudesAPare as in Proposition2.

Proposition 4.IfFis a compact subset ofP^⋆andKis acompact subset ofR, thenA_FKis compact.

Moreover, ifR>0is an escape radius ofP, then we also have theequality

J P=

⋂

R 33

n≥1

Before turning our attention to other types of Julia sets, we need to point some useful estimates. IfR>0 is an escape radius forP∈P^⋆, then (cf. Equation 7 in [1])

Proof 4.Choose a sequenceE_nof elements fromA_FK. Then, there exist sequencesP_n⊂FandK_n⊂Ksuch thatE_n=P⁻¹K_n.

AsKis compact, we can assume (passing ΓP⁻¹B_R ,B_R ∥P∥_∂B

R 34

to a subsequence if needed) thatKn→KinKifn→ ∞. More generally, ifFis a compact family inPRd ^⋆, then due SinceFis compact, so here again (passing to a subsequence

if needed), we can assume thatP_n→PinFifn→∞. It fol- to the continuity of the mapping in(8),a common escape^d lows from Proposition3thatP⁻¹Kn→P⁻¹K, ifn→ ∞. radiusR>0 for all mappings inFcan be found. Also,

Thus, we have shown that every sequence inA_FKhas aconvergent subsequence.

Recall thatκXdenotes the family of all nonempty

M≔su p

P∈F

P_∂B

R 35

compact subsets of the metric spaceX, furnished with

theHausdor metric.ff is finite because of the compactness ofF. Thus, as an imme- diate consequence of(34)we obtain

Corollary 1.LetFbe a nonempty compact subset ofP^⋆.

ΓP⁻¹B_R,B_R ≤M ,

P∈ℱ 36

The mapping

A_ℱκℛ∋K↦

⋃

P∈ℱ

For a sequenceP_n^∞of mappings fromF, we define itsfilled-in Julia set(nonautonomousif the sequence is not constant) asfollows:

is well defined and is a contraction with ratio1/d.

Inparticular, the mappingA_Fhas a uniquefixed point SF∈κR.

n=1∞ =z∈ℂ^N P_n∘…∘P₁z^∞ is bounded 37

≤

J P_n

nn=1

n→∞

d

d n=

1

n=

1

n=

1

∞

M

n

P^j − Q^j 2^{j BR}

Rdm d − 1 Theestimate(36)allowstheuseoftheenhancedversion of

Banach’s contraction principle (Lemma 4.5 in[12])f o r We define thepartlyfilled-in composite Julia setof the compact familyF⊂P^⋆as

sequenceA _P^∞. As a consequence, we can see that ^d

nn=1∞ = lim Pn∘ ⋯ ∘ P

1

−1E

, E∈ℛ,

38

J_trℱ=

⋂

m∈ℕ

⋃

P1,…,Pm∈ℱ

P_m∘⋯∘P₁⁻¹B_R 44

∞

n=1 =

⋂

n≥1

This set is compact (see proof of Theorem 4.6 in[12]),and its polynomially convex hullJ Fis the unique fixed

ifR>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

point of the mapping:

ℛ∋K↦

⋃ ^̂

P∈ℱ

∈ℛ 45

be approximated by autonomous filled-in Julia sets. Before making this statement more precise, let us establish some notations. IfFis a compact family inP^⋆, the symbolF^ℕwill denote thecode space overF, defined as the Cartesian product of countably many copies ofFwith the usual product topology. By Tychonoff’s theorem,F^ℕis compact and it can be furnished with the metric (see, e.g., Theorem4.2.2 in[16]):

JFiscalledthefilled-incompositeJuliasetofF.Here,the hat marks the operation of taking the polynomially convex hull of the set under the hat. The subscript tr stands for the wordt r u n c a t e d .

Thefollowingtheoremdescribestheconnectionbetween the Julia sets from this section and the attractorS Ffrom the end of the previous section (Corollary1).

Theorem 1.LetFbe a nonempty compact family in

∞ ∞ ∞ ∞ ∞ ℕ P^⋆ . Then,

ρP_nn=1,Q_nn=1=〠

j=1

, P_nn=1,Q_nn=1∈ℱ 39

d

(1) S F=JP _{nn =1}^∞ :P_nn=1 ∈F^ℕ; Proposition 5.LetFbe a compact family inP^⋆.Then,

foreach Pn∞∈F^ℕand m∈ℕ (2) J_trF=⋃S F.

Proof 7.This fact can be deduced from general theory in [18]

ΓJ P ∞

nn =1 ,Pm ∘ ⋯ ∘ P

1

−1B

R≤ ,

40

but we give here the proof in this special case to make ourwork consistent (cf. also[19]for the case of a finite family).

The familyS=S F is the unique fixed pointo f whereM≔sup_P∈F∥P∥^∂BR.Inparticular, A_FκR→κR(cf. Corollary1).Therefore,

J Pn∞

= limJ P_m∘ ⋯ ∘ P¹ 41 S=A_ℱS=⋃_P∈ℱAP S=⋃_P,P∈ℱA_P A_PS

n=1 m→∞ 1 1 12 1 2

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

ΓP_n+m∘⋯∘P₁⁻¹B_R,P_m∘⋯∘P₁⁻¹B_R

=⋯=⋃_P

1,P₂,…,P_n∈ℱA^P1∘AP₂

∘ ⋯ ∘AP_n

S, n∈ℕ

46 Since by Proposition2the functionA_Pjis a contraction,

diam A∘A∘ ⋯ ∘ A S⟶ 0, ∞

Mⁿ 1 Mⁿ1

42 Γ P₁ P₂ P_n n⟶

≤Rd〠 d^m+j−1=

Rd^m〠

d^j 47

j=1 j=1

Thesequence AP₁^○A_P2∘⋯∘A_P ^∞_n=1 is also decreas- Lettingngo to infinity gives(40). ing with respect to inclusion. Therefore, its limit is a

singleton, and by the definition ofJ P_n^∞, we have Asfor(41),inviewof(40)wecanwrite

the equality ⁿ⁼¹

ΓJ P_n^∞,J P_m∘⋯∘P₁

⋂

≤ ΓJ P_n^∞,Pm∘⋯∘P₁⁻¹BR P1 P2

P_n n

nn=1

+ΓP_m∘ ⋯ ∘ P₁⁻¹BR,J Pm∘ ⋯ ∘ P₁ Thus,S=J P^∞:P ^∞∈F^ℕ. J

P_n

S J P

2 M

From the ^{nn=1 nn=1} J , it is obvious that

≤ definitiono f trF

Rd^md−1

43 For a finiteF, Proposition5was shown in[17].

⋃S F⊂J_trF. Let us fix a common escape radius R>0 for allP∈F.

Now, takez∈J_trF. First, we claim that for anyn∈ℕ, there existsE_ncontained in the 1/n-dilation ofS Fand

<

m 1 R m 1

n

<

2

d

n=1 4

n

∞

ε

n

m 1

m

m 1

Γ

Rd^m d − 1

sup Γ E, P−1 E : P ∈ ℱ dm−1d − 1

Γ E, B^R + Γ B^R, P−1 BR + ΓBP^−1R , P−1E dm−1d − 1

d +1 Γ E, B^R + M/R d^md − 1 such thatz∈E_n. Indeed, givenn∈ℕ, one can choose

m∈ℕso that the inequality

M 1

49

of(40)(but withEreplacingB_R) combined with the triangle inequality and(36),we have the following estimates:

dist_Γπ₁∘⋯∘π_m⁻¹E,Sℱ₋₁ is satisfied withMdefined in Proposition5.Sincez∈J_trF,

bythedefinitionofthelatter,thereexistP₁,…,P_m∈Fsuch

thatz∈P_m∘ ⋯ ∘ P₁⁻¹B_RThe inequalities(49)and(40)imply theestimate

ΓP∘⋯∘P⁻¹B,J P∘ ⋯ ∘ P <1 ,

50 and this means thatE_n=P_m∘⋯∘P₁⁻¹B_Rfulfills our claim.

≤ Γπ₁∘⋯∘π_m

≤

≤sup

P∈ℱ

≤

E,Jπ₁∘ ⋯ ∘ π_m

⟶0 ifm⟶∞, To finish the proof, we want to show thatz∈ ⋃S F. 53

SinceS Fis compact, the sequenceEnhas an accu- mulation pointE∈S F. But then, since convergence of sets inR,Γmeans uniform convergence oft h e

whichiswhatisneeded,asdist_ΓR\U,SF >0.

bTakeE∈S Fandε>0such thatBΓE,ε⊂V. Fixn∈ℕ.

Without loss of generality, we may suppose thatd⁻ⁿdiam_ΓS F<ε/4 It follows from Theorem1 corresponding pluricomplex Green functions, we can thatE=JP_n^∞ forsome P_n^∞∈F^ℕ.M o r e o v e r , conclude thatV_Ez= 0, which means thatz∈E⊂ ⋃S F.

J P^{nn =1} = lim_m→∞ⁿ⁼¹ J P_m ∘⋯∘P ₁by( 4 1 ) . Therefore, weⁿ⁼¹ Remark1.ItisworthemphasizingthatallofthetypesofJulia sets defined in

this section correspond one way or anotherto sequences in the semigroupF. This is the reason why

can choosem>nsuch that ΓE,J P_m∘⋯∘P₁

4 54

conceptually it is natural to see the setS Fnot only as the

attractor associated with the semigroupFbut also as a DefineQ_j=P m+1−j forj∈1,…,mand let kind of atlas of all Julia sets associated with that semigroup.

Indeed, this is exactly the meaning of Theorem2combined with the definition ofJ_trF.

6. Onthe Attracting Nature ofSF

RecallthatweusethesymbolB_ΓE,rt o denotetheopenball inR,Γwith center atE∈Rand radiusr>0.

The next theorem is a counterpart of Theorems2(a)and2(b)in[1]in the case of infinite compact regular families of polynomial mappings.

Theorem 2.LetFbe a nonempty compact family inP^⋆. (a) Letπn∞⊂F. If E∈RandU⊃SFis an

opensubsetofR,thenalmostallelementsofthesequence

U=F∈Rdist_ΓF,S F <ε 55

IfF∈U, then there existsG∈S Fsuch thatΓF,G<ε.

Therefore,

ΓQ₁∘⋯∘Q⁻¹F,Q∘⋯∘Q⁻¹G≤d^−mΓF,G<ε 56 Moreover,

ΓJPm∘⋯∘P₁,Q₁∘⋯∘Qm−1G

=ΓP_m∘⋯∘P₁⁻¹JP_m∘ ⋯ ∘ P₁,P_m∘ ⋯ ∘ P₁⁻¹G

≤d^−mΓJ P∘⋯∘P,G≤d^−mdiamSℱ<ε

57

E= A_π∘ ⋯ ∘ A_π1E:n≥1 51

Combining(54),(56),(57),and using the triangle inequality, we see that

belong toU. In particular, all accumulation points of this sequence are inS Fand soEis compactinR . (b) LetE∈SF.ForeveryneighbourhoodVofE,thereexists

an open setU⊃S Fand mappingsQ₁,

…,Q_m∈F,suchthat

ΓE,Q₁∘ ⋯ ∘ Q_m^{− 1}F

<ε, 58

as required.

7. Chaos Game and

ApproximationofAttractors

A_Qm ∘ ⋯ ∘ A_Q

1 U⊂V 52

Wewillstartwiththedefinitionofdisjunctivesequencesover Moreover, m can be made arbitrarily large.

Proof8.a F i x acommonescaperadiusR>0forallP∈F.

LetMbelikeinProposition5.FixP∈F.Inviewoftheproof

a finite or countable alphabet.

LetAbe a nonempty set which is at most count- able.

A sequence of elements ofA, that is, a functionτℕ→Ais said to bedisjunctive, if for anym∈ℕ

and anyf u n c t i o n θ 1,…,m→Athere existsn∈ℕ

d

l

d

l

4

d

m→∞ =

4

lim

d

m→∞

n

,

ℓ 1

ℓ

n=1n

ε 4

Em

E^m

ε 2

Em

such thatθj=τn+j forj∈1,

…,m.As i m p l e example of a disjunctive sequence

withA=ℕisgiven Q₁∘ ⋯ ∘ Q_ℓ⁻¹K∈B_ΓA , 63

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 soon.

IfAis regarded as thealphabetand functions likeθ as possiblefinite words over A, then the sequenceτis

Proposition3assures continuity ofP^⋆×R∋Q,K↦

Q⁻¹K∈RBy Proposition1,the mappingF^l∋q₁,…,q_l

↦q₁∘⋯∘q_ℓ∈P^⋆is continuous, too. Therefore, the mapping

disjunctivei f i t c o n t a i n s a l l fi n i t e w o r d s a s i t s fi n i t e s u b - sequences. Disjunctive sequences, usually over a finite

alphabet, have been used for a long time in study of for- mal languages, in automata theory and number theory (see[21]for an overview). More recently, disjunctive sequences turned out to be a natural tool for derandomi- zation of the chaos game (see[20]).

The next result is a generalization of Theorem2(c)in[1].

Theorem 3.LetFbe a nonempty compact subset ofP^⋆andF₀=π_nn∈ℕa dense countable subset ofF.Letτℕ→ℕbe a disjunctive sequence.

F^l×E_m∋q₁,…,q_ℓ,K↦q₁∘ ⋯ ∘ q_ℓ⁻¹K∈R 64 is uniformly continuous. Thus, there existsη>0 such that ifp_j,q_j∈Fwithq_j−p_j<η,j∈1,…,ℓ, andK,L∈EmwithΓK,L<η, then

Γq₁∘ ⋯ ∘ q⁻¹K,p∘ ⋯ ∘ p⁻¹L

<ε

65 SinceF₀isdenseinF,thereexistP¹,…,P_ℓ∈F⁰such

thatP¹−Q₁<η,…,P_ℓ−Q_ℓ<η. Letθ1,…,ℓ→ℕbe chosen so thatP_j=π_θjforj∈1,…,ℓ.

Sinceτis disjunctive, for somen≥m, we haveθj=

τn+jforj∈1,…,ℓ. Consequently, if we putK=

Then,foranyE∈R, π_τ1 ∘ ⋯ ∘ π_τn ⁻¹E, we haveK∈E_m and

limΓJ F,

⋃

−1

∘ ⋯ ∘ π_θℓ K wher

e

E_m= π_τ1∘ ⋯ ∘ π^τn

−1E:n≥m 60

We know that

=P₁∘⋯∘P_ℓ⁻¹K

66

Proof9.Firstofall,itshouldbenotedthatTheorem2yields

ΓP₁∘ ⋯ ∘ P_ℓ ⁻¹K,Q₁∘⋯∘Q_ℓ ⁻¹K

<ε ,

67 compactness ofE_m. Furthermore, a countable

subsetF₀ofFexists because of the separability ofP_d. Recall also thatRis separable (see Theorem1(d)in[1]).

because of the choice ofη, and so it follows from(63)combined with the triangle inequality that Fix a norm · inP_d.

InviewofTheorem2(b)andfromTheorem1(b)in[1],it π_τ1 ∘ ⋯ ∘ π_τn+ℓ E∈B_ΓA 68

su ces to prove thatffi χ

m→∞

S F, =0, 61

And this concludes the proof.

The next statement is a probabilistic version of the abovet heorem .

whereχ_Γdenotes the Hausdor metric corresponding toff Γ.

Takeε> 0. In view of Theorem2(a), ifmis su cientlyffi

Corollary 2.LetFbe a nonempty compact subset ofP^⋆andF₀=π_nn∈ℕa dense countable subset ofF.

Letτℕ →ℕbegeneratedaccordingtoprobabilitiesp₁,p₂,…

large, then theε-dilation ofS FcontainsEm, and hence

alsoE. In order to prove that for su ciently largeffi m, the >0such that∑^∞p= 1, that is, the valuesτj ofτare

m

ε-dilation ofE_mcontainsS F, it is enough to show that any point from anε/2-dense finite subset ofS Fis withinε/2- distance from a point ofE_m.

LetA∈S Fbe an element of a fixedε/2-dense finite subset ofS F. By Theorem2(b),there existℓ∈ℕand

chosen at random, independent from each other, so that ℙτj=i=p_ifor i,j∈ℕ.

Then, for any E∈R, with probability 1,

limΓJ F,

⋃

Q₁,…,Q_ℓ∈Fsuch that forδ∈0,ε/2 the image of the

δ-dilation ofS Fvia the mapping whereE_m= π_τ1∘ ⋯ ∘ π^{τ −1}E:n≥m

F↦Q₁∘ ⋯ ∘ Q_ℓ⁻¹F 62

is a subset ofB_ΓA,ε/4 . Using Theorem2(a)again if necessary, we can increasemso that theδ-dilation ofS FcontainsE_m. In particular, ifK∈E_m, then

Proof10.Becauseofthestronglawoflargenumbersapplied to Bernoulli processes, we can conclude that, given a finite word over the alphabetℕ, the sequenceτcontains this word with probability 1. Hence, we can use the same reasoning as in the theoremabove.

−

Γ

d

n

n=

1

n=1

n=1

0 1

P 1

P_k

Z ^{k nn =1 +}

k

Complexity 11

We would like to finish the article with a general observation.

Let us assume that we have a probability measureWon someσ-algebra of subsets ofF, where as in Theorem3,Fis a compact subset ofP^⋆. We will follow the general set-up from[7].We will be concerned with a Markov chainZ^K, with initial stateK∈Ra n d

LetμbethepushforwardmeasureonRobtainedfromthemeasureWont hecodespaceF^ℕvia

themappingΠ.Then:

(a) Ifνis a Borel probability measure, thenFⁿν→μweakly. In particular, Fμ=μandμis the uniqueprobability measure invariant with respect toF.

(b) For all K∈Rand for a.e. P_n^∞⊂F

K ∞

k nn =1 K ifk=0,

= 70 〠δⁿ Z^KP^∞ ⟶μ 74

A∘⋯∘A_P1 K ifk≥1, n_k=1 k ⁿⁿ⁼¹

whereP_kare independently and identically distributed (IID) random elements inFwith probability distributionW.

LetWalso denote the induced probability measure on the code spaceF^ℕ.

Inamoregeneralsetting,theinitialstatecanbegivenby a random elementX₀inR, independent ofP_n^∞, andwith

the probability distributionν. Then, it is natural to definethe randomel em ents

weakly.

(c) The support ofμisS F; hence, this is theuniquefixed point of the iterated function systemAPP∈F. In particular, the support ofμiscompact.

Proof 11.(a) and (b) are straightforward consequences of [7](Theorem 8).

ν ∞

k nn =1

=Z^X0P^∞,

k∈ℤ (c) By Theorem 8 (15)from[7],there existsn₀, which may depend onP_n^∞andε, such that

So in particular,νis the probability distribution ofZ^ν. If we also defineFνas the probability distribution ofZ^ν, then the probability distribution ofZ^νisF^kν.

Pn∘ ⋯ ∘ P₁⁻¹BR∈suppμ^ε, ifn≥n₀ 75 Therefore,S F⊂suppμ. On the other hand, The reverse order chain isdefined to be asfollows:^k ΠF^ℕ=SF and hence suppμ⊂S F.

̂̂^{K ∞}

K ifk=0, It should be noted that the novel element in the above Z_kP_nn=1=

AP₁∘ ⋯ ∘ A_P K ifk≥1 72 observationisthecompactnessofthesupportofthemeasure inthecaseofinfinitefamilyanditsinvarianceundertheIFS inthiscase.Theorem8in[7]givesthisproperty,butonlyin BecauseoftheIIDproperty,bothZ^νandẐ^νhavethesame probability

distributionF^kν

Notethatalloftheabovedefinitionsmakesensebecause

Proposition3is guaranteeing appropriate measurability of thesets.

Letδ_adenote the Dirac measure concentrated ata, that is,δ_aE=1_Ea. Below, we use the mapping

the case of finite iterated function systems.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

ΠF^ℕ∋P_n^∞↦J P_n^∞ ∈SF 73

Acknowledgments

n=1 n=1

The research of the third author was partially supported by It is continuous because of the estimate(40)combined

with the definition(39)of the metricρonF^ℕ. Indeed, givenε>0 andQ=Qn∞∈F^ℕ, choosem

the NCN Grant no. 2013/11/B/ST1/03693 and that author is also grateful to Uppsala University for its hospitality. The authors wish to thank Margaret Stawiska-Friedland andt h e so thatM/Rd^md−1 < /4. IfP= ^{n=1 ∞}∈ _ℕ

is such anonymous referees for their helpful remarks.

ε P_nn=1F

thatρP,Q<ε/2^m+1,t h e n Pm−Q_mB<ε/2 andthus Γ ΠP,ΠQ<εin view of(40)triangle

inequality.

R

combined with the

References

[1]A. Alghamdi and M. Klimek,“Probabilistic approximation Z

k

71

d

10 Complexity

Proposition6.LetFbeanonemptycompactsubsetofP^⋆,letWb eaprobabilitymeasureonsomeσ-

algebraofsubsetsofFandletWalsodenotetheinducedprobability measureonF^ℕ.

of partly filled-in composite Julia sets,”Annales

PoloniciMathematici, vol. 119, no. 3, pp. 203–220, 2017.

[2] A.HinkkanenandG.J.Martin,“Thedynamicsofsemigroups of rational functions. I,”Proceedings of the London Mathe- matical Society, vol. s3-73, no. 2, pp. 358–384,1 9 9 6 .

[3] R.StankewitzandH.Sumi,“Backwarditerationalgorithmsfor Julia sets of

Möbius semigroups,”Discrete and

ContinuousDynamicalSystems,vol.36,no.11,pp.6475–6485,2016.

[4] H. Sumi,“Random complex dynamics and semigroups of holomorphicmaps,”ProceedingsoftheLondonMathematicalSociety,vol.1 02,no.1,pp.50–112,2011.

[5] H.Sumi,“Randomcomplexdynamicsanddevil’scoliseums,”

Nonlinearity, vol. 28, no. 4, pp. 1135–1161, 2015.

[6] M.Klimek,“Metricsassociatedwithextremalplurisubharmo-nic functions,”Proceedings of American Mathematical Society,vol.123,no.9,pp.2763–2770,1995.

[7] M. F. Barnsley, J. E. Hutchinson, and Ö. Stenflo,“V-variable fractals: fractals with partial self similarity,”Advances inMathematics,vol.218,no.6,pp.2051–2088,2008.

[8] M. Klimek,“On perturbations of pluriregular sets generated by sequences of polynomial maps,”Annales Polonici Math- ematici, vol. 80, pp. 171–184,2003.

[9] M. Klimek,“Pluripotential

theory,”inLondonMathematicalSociety Monographs. New Series, 6. Oxford Science Publica- tions, The Clarendon Press, Oxford University Press, NewYork,1991.

[10] W.Rudin,“InverseiterationsystemsinCⁿ.Reprintofthe1980 edition,”inClassicsinMathematics,Springer,Berlin,2008.

[11] M. Klimek,“Iteration of analytic

multifunctions,”NagoyaMathematicalJournal,vol.162,pp.19–

40,2001.

[12] M. Klimek and M. Kosek,“Composite Julia sets generated by infinite polynomial arrays,”Bulletin des Sciences Mathe- matiques, vol. 127, no. 10, pp. 885–897,2 0 0 3 .

[13] J.Siciak,“Onmetricsassociatedwithextremalplurisubharmo- nic functions,”Bulletin of the Polish Academy of Sciences- Mathematics,vol.45,no.2,pp.151–160,1997.

[14] R. Brück and M. Büger,“Generalized iteration,”Computa- tional Methods and Function Theory, vol. 3, no. 1, pp. 201–

252,2003.

[15] J. E. Fornaess and N. Sibony,“Random iterations of rational functions,”Ergodic Theory and Dynamical Systems, vol. 11, no.4,pp.1289–1297,1991.

[16] R. Engelking,General Topology, PWN-Polish Scientific Publishers,Warsaw,1977.

[17] M.Klimek,“InverseiterationsystemsinCⁿ,”ActaUniversita-tis Upsaliensis, Scr. Uppsala Univ. C Organ. Hist, vol. 64, pp.206–214,1999.

[18] K. R. Wicks,Fractals and Hyperspaces, Lecture Notes inMathematics, 1492, Springer-Verlag, Berlin,1 9 9 1 .

[19] J. E. Hutchinson,“Fractals and self

similarity,”IndianaUniversity Mathematics Journal, vol. 30, no. 5, pp. 713–747,1981.

[20] K. Leśniak,“Random iteration for infiniten o n e x p a n s i v e iterated function systems,”Chaos, vol. 25, no. 8, article083117, 2015.

[21] C.S.Calude,L.Priese,andL.Staiger,DisjunctiveSequences:An Overview. Centre for Discrete Mathematics and Theoretical Computer Science, Research Report Series, No. 063, Universityof Auckland,1997.