The Chaos Game on a General IFS CUP

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Ergod. Th. & Dynam. Sys.(2011),31, 10731079 cCambridge University Press, 2010doi:10.1017/S0143385710000428

The chaos game on a general iterated

function system

MICHAEL F. BARNSLEY and ANDREW VINCE

Department of Mathematics, Australian National University, Canberra, ACT, Australia(e-mail: [email protected])

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA(e-mail: [email protected])

(Received3May2010and accepted in revised form26May2010)

Abstract. The main theorem of this paper establishes conditions under which the chaosgame algorithm almost surely yields the attractor of an iterated function system. Thetheorem holds in a very general setting, even for non-contractive iterated function systems,and under weaker conditions on the random orbit of the chaos game than obtainedpreviously.

1. IntroductionThere are two methods for computing pictures of fractals that are attractors of iteratedfunction systems: the deterministic algorithm and the more efficient chaos gamealgorithm [2]. This paper concerns the chaos game on a general iterated function system(IFS) Fdefined on a complete metric space(X,d). The IFS is general in the followingsense: the only restriction placed on X is that it is proper, i.e. closed balls are compact,and the only restriction on the functions in F is that they are continuous. In particular,they need not be contractions on X with respect to any metric giving the same topology asthe original metric d. A general IFS may possess more than one attractor, one attractor,or no attractor. Examples of iterated function systems that are non-contractive yet possessattractors are given in 4.

The main result, Theorem1, is new in that it shows that the chaos game algorithm,applied to such a general IFS, almost always yields an attractor. More precisely, if the IFShas an attractor A, then a random orbit starting with a point in the basin of attraction ofAconverges with probability one to the attractor. We show this under weaker conditionsthan have heretofore been described. In all other papers on this topic of which we areaware, for example[35, 79, 11, 13, 14], it is required that the IFS be either contractiveor contractive on the average. It is also required that the process by which the functions areselected to generate the orbit is stationary and that the selection process depends Hlder

continuously on the initial point; see, for example, [12]. For our result, none of theseconditions are required.
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1074 M. F. Barnsley and A. Vince

Section 2 of this paper contains basic definitions, in particular the definition of anattractor of an IFS. Lemma 2 in 2 provides an expression for an attractor, of someindependent interest, that will be used to prove the main result. The main result on thechaos game is Theorem1 in 3. Section4contains examples that illustrate the practical

value of Theorem1.

2. General iterated function systemsThroughout this paper,(X,dX)is a complete metric space.

Definition 1. If fm : X X, m=1,2, . . . , M, are continuous mappings, then F=(X; f1, f2, . . . , fM)is called an iterated function system(IFS).

With a slight abuse of terminology, we use the same symbol Ffor the IFS, the set offunctions in the IFS, and the following mapping: letting 2X denote the collection of subsetsofX, define F :2X2X by

F(B)=

fF

f(B)

for all B2X. Let H = H(X) be the set of non-empty compact subsets ofX. SinceF(H) H, we can also treat Fas a mapping F: H H. Let dH denote the Hausdorffmetric on H, defined in terms ofd:= dX.

A metric spaceX is locally compactif every point has a compact neighborhood, andit is proper if every closed ball {y: d(x ,y )r} is compact. Proper spaces are locallycompact, but the converse is not true in general. The proof of the following result is notdifficult. We use the notation

S+r= {y X :d(x ,y )rfor some x S}

with S X andr>0.

LEMMA1 .(1) A metric space is proper if and only if C+r is compact whenever C X is compact

and r is a positive real number.

(2) IfX is proper, then F : H(X) H(X)is continuous.

ForB X, let Fk(B)denote thek-fold composition ofF, the union of fi1 fi2

fik(B)over all finite words i1i2 ikof lengthk.DefineF0

(B)= B.Definition 2. A non-empty compact set A X is said to be an attractorof the IFS F if:(1) F(A)= A; and(2) there is an open set U X such that AU and limk Fk(B)= A for all

B H(U), where the limit is with respect to the Hausdorff metric.The largest open set Ufor which (2) is true is called the basin of attraction for the

attractor Aof the IFS F.

The notation S is used to denote the closure of a set S, and, when U X is non-empty,H(U)= H(X)2U. The quantity on the right-hand side of the equation below

is sometimes called thetopological upper limitof the sequence{Fk(B)}k=1and is relatedto other definitions of attractors of generalizations of the notion of an IFS; see, for example,



The chaos game on a general iterated function system 1075

McGehee[6]and Lesniak[10], as well as the references in these two papers. We will useLemma2in the proof of Theorem1,our main result.

LEMMA2 . IfFis an IFS that has an attractor A with basin of attraction U, then

A= limK

kK

Fk

(B)

for all B H(U).

Proof. We first show that

kKFk(B)is compact for K sufficiently large. Let{Oi :i

I}be an open cover of

kKFk(B). Since

kK

Fk(B)

kK

Fk(B)

and A=limk Fk(B), we have that {Oi :i I}is also an open cover ofA. Because Ais compact,{Oi :i I}contains a finite subcollection, say {Om :m =1,2, . . . , M}, such

that A O :=

Mm=1 Om . Because a metric space is normal, there is an open set O

containing A such that O O. Again using that fact that Fk(B) converges in theHausdorff metric to A, there is an integer K such that Fk(B) O for all k K. Itfollows that

kK F

k(B) O and therefore

kK

Fk(B) O =

Mm=1

Om for all K K.

Thus

kKFk(B)is compact ifK K.

By the same argument used above to show thatkKFk(B) O, it can be shown thatthere is a K such that BK :=

kKF

k(B)U for K K. Since BK H(U), by thedefinition of attractor we have

A = limK

FK(BK )= lim

KF

K

kK

Fk(B)

= limK

kK+K

Fk(B)= limK

kK

Fk(B).

3. The chaos game algorithmThe following lemma will be used in the proof of Theorem1.

LEMMA3 . LetX be a proper complete metric space andF=(X; f1, f2, . . . , fN) anIFS which has an attractor A with basin of attraction U . For any >0, there is an integerM= M() such that for each x (A+) U there is an integer m = m(x, ) < M such

that

dH(A, Fm ({x})) < /2.

Proof. Because X is proper and Ais compact, A+ is also compact by statement (1) ofLemma1.There is no loss of generality in assuming to be small enough that A+U.Ifx A+ U, then there is an integer m (x , )0 such that

dH(A, Fm(x ,)

({x })) < /4. (3.1)This is because limk Fk({x})= A.




Since X is proper, it follows from statement (2) of Lemma 1 that F : H H iscontinuous, whence Fm(x ,) : H H is continuous. SinceFm(x ,) : H H is continuous,there is an open ball B({x },rx ) (in H) of radius rx >0 centered at {x } such thatdH(F

m(x ,){x}, Fm(x ,)(Y)) < /4 for all Y B({x },rx ). It follows, in particular, that

there is a ball B(x ,rx ) (in X) centered at x such that dH(Fm(x ,)

({x }), Fm(x ,)

({y})) 0, there is an integer K>0 such that

xk A+ (3.2)

for allk K. By Lemma2we have A=limL

j L Fj ({x0}).It follows that for any

>0, we can choose Kso that x k

j KFj ({x0}) A+for allk K, as claimed.

We next show that for any >0, there is an integer K >0 such that

dH(A,{xk}k=L ) <

with probability one, for all L K. This is equivalent to the assertion of the theorem.To prove this, let >0. If K is as specified in the paragraph above, then by(3.2)wehave xL A+for L K. The attractor A, being compact, is totally bounded. Let {aq :

i =1,2, . . . , Q }be a set of points such that AQ

q=1 B(ai , /2), where B(aq , /2)isthe ball of radius /2 centered at aq . Note that each aq and the integer Q depend on .By Lemma3, there is an integer Msuch that for each x A+ there is an m< MwithdH(A, F

m ({x })) < /2.Hence

dH(A, Fm ({xL })) < /2




for some integer m < M. Therefore there is a sequence of symbolsL+1L+2 L+msuch that fL+m fL+m2 fL+1 (xL ) B(a1, /2). (We adopt the convention thatthe composition on the left equals xL ifm =0.) It follows that

B(a1

, /2) {xk

}L+M1

k=L = ,

or

B(a1, /2) {xk}L+M1k=L +.

The probability that this event occurs, i.e. that the particular sequence L+1L+2 L+mis chosen, is greater than pM. By repeating this argument, we deduce that the probabilitythat

B(aq , /2) {xk}L+q M1k=L+(q1)M+

is greater than pM >0, for each q {1,2, . . . , Q}, regardless of whether or not the

preceding events occur (which is not to say that the events are independent). It followsthat the probability that all of these events occur is greater than p Q M. Consider the eventE1defined by

Qq=1

B(aq , /2) {xk}L+Q M1k=L +.

The probability ofE1is less than 1 pQ M. By a similar argument, the probability of the

eventEr forr1, defined by

Qq=1

B(aq , /2) {xk}L+r Q M1k=L+(r1)Q M+,

is less than 1 p Q M, regardless of whether or not the previous events E1,E2, . . . , Er1(for r=2,3, . . .) occurred. It follows that the probability of the eventE1 E2 Eris less than(1 p Q M)r, for allr=1,2, . . .. This inequality holds regardless of the factthat the Erare not independent. To simplify notation, lets =1 p Q M




It follows from Theorem1thatK1

{xk}k=K =

K1

kK

Fk(B)

almost surely, for x0

U and B H

(U), e.g. for B= {x0

}. We draw attention to thisequality because it seems surprising when F contains more than one function; the set{xk}

Kseems sparse in comparison to

kKF

k(B).

4. ExamplesExample 1. The IFS Fin this example has a unique attractor, yet each f Ffails to be acontraction with respect to any metric giving the same topology as the original metric. Withprobability one, the chaos game applied to this example draws a picture of the attractorof the IFS.

IfX = {(x ,y ) R2

:x2

+y2

=1} and d is the Euclidean metric, then (X,d) is acompact metric space. Let F=(X; f1, f2) where f1(x ,y )=(x ,y ) and f2(x ,y )=(xcos y sin , xsin + y cos), with / irrational. The map f1 is the identitymap, and f2 is a rotation through angle anticlockwise about the origin. Since neitherf1nor f2 has a unique fixed point, it follows that there exists no metric don X such that(X, d)is complete and either f1or f2is a contraction. On the other hand, Fhas a uniqueattractor A= X. To see this, first note that F(X)= X. Also, if(x0,y0) X, then

Fk(x0,y0)= { f

j

2(x0,y0)}kj =0

for allk. The right-hand side is well known to converge in the Hausdorff metric to X ask

tends to infinity. It follows that Fk(B)converges to X for all B H(X). By Definition2,the IFS Fhas a unique attractor, namelyX. By Theorem1, with probability one, the chaosgame applied to this example draws a picture of the unit circle, the attractor of the IFS.

Example 2. In the same spirit as Example1,this IFS on the real projective plane possessesan attractor, but there is no equivalent metric with respect to which the maps of the IFS arecontractive. Again, with probability one, the chaos game draws a picture of the attractorof the IFS.

This example appears in [1]. The metric space is(RP2,d), whereRP2 denotes real

projective two-dimensional space and d denotes the round metric; see [1]. Let F

=(RP2; f1, f2), where the projective transformations f1 and f2 (acting on homogeneouscoordinates) are represented, respectively, by the pair of matrices

1 0 00 2 00 0 2

and

1 0 00 2 cos 2 sin

0 2 sin 2 cos

,

with / irrational. In terms of homogeneous coordinates (x ,y ,z), the attractor ofF isthe linex =0. This can be proved by using an argument similar to the one in Example 1.

Such non-contractive IFSs occur often in real projective IFS theory. Theorem1tells us

that the chaos game algorithm can always be applied to compute approximate pictures ofattractors of real projective iterated function systems.


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REFERENCES

[1] M. F. Barnsley, A. Vince and D. C. Wilson. Real projective iterated function systems.Preprint, 2010.[2] M. F. Barnsley.Fractals Everywhere. Academic Press, Boston, MA, 1988.[3] M. F. Barnsley and S. G. Demko. Iterated function systems and the global construction of fractals.Proc.

R. Soc. Lond. Ser. A 399(1985), 243275.[4] M. F. Barnsley, S. G. Demko, J. H. Elton and J. S. Geronimo. Invariant measures arising from iterated

function systems with place-dependent probabilities.Ann. Inst. H. Poincare24(1988), 367394.[5] M. A. Berger. Random affine iterated function systems: curve generation and wavelets. SIAM Rev. 34

(1992), 361385.[6] R. P. McGehee and T. Wiandt. Conley decomposition for closed relations.Difference Equations Appl.12

(2006), 147.[7] J. H. Elton. An ergodic theorem for iterated maps.Ergod. Th. & Dynam. Sys. 7(1987), 481488.[8] J. Jaroszewska. Iterated function systems with continuous place dependent probabilities.Univ. Iagel. Acta

Math.40(2002), 137146.[9] B. Forte and F. Mendivil. A classical ergodic property for IFS: a simple proof.Ergod. Th. & Dynam. Sys.

18(1998), 609611.

[10] K. Lesniak. Stability and invariance of multivalued iterated function systems. Math. Slovaca53 (2003),393405.

[11] O. Onicescu and G. Mihoc. Sur les chanes de variables statistiques.Bull. Sci. Math. France59 (1935),174192.

[12] . Stenflo. Uniqueness of invariant measures for place-dependent random iterations of functions.Fractalsin Multimedia (IMA Volumes in Mathematics and its Applications, 132). Eds. M. F. Barnsley, D. Saupeand E. R. Vrscay. Springer, New York, 2002.

[13] I. Werner. Ergodic theorem for contractive Markov systems.Nonlinearity17(2004), 23032313.[14] R. Scealy.V-variable fractals and interpolation.PhD Thesis, Australian National University, 2008.

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The Chaos Game on a General IFS CUP