Topo Contractive

8/10/2019 Topo Contractive

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M. Barnsley, K. Igudesman

TOPOLOGICAL CONTRACTIVE SYSTEMS

Abstract. We are concerned with one of the fundamental problems of

fractal geometry: When the topological contractive system is a metric

contractive? We show that there exists a metric such that a topological

contractive system is nonexpansive and give a sufficient condition for

the contractive property of this system.

1. Introduction

Let (X, d) be a bounded complete metric space and {fj}Nj=1contractive

mappings ofX into itself; i.e.,

d(fj(x), fj(y)) d(x, y), where (0, 1), x , y X, j= 1, N.

Then it follows from a theorem due to Hutchinson [1] that the iterated

imagesfu1 fu2 fun(X), whereuk = 1, N, shrink to the fixed point

{xu1u2...}. This fact can be written in the form

(1)

n=1

fu1 fu2 fun(X) ={xu1u2...}.

Since this formula does not involve the metric and has a topological

character, it is therefore natural to ask the following question. Let X be a

compact metrizable topological space and {fj}Nj=1 continuous mappings

ofX into itself which have the property (1). Is it possible to find a metric

d(x, y) generating the given topology ofXsuch that the mappings fj are

contractive with respect to d?

2. Nonexpansive property

Let Nbe a positive integer larger than 1. We denote by n the set ofwords of length n, that is,

n={1, 2, . . . , N }n ={u= u1u2. . . un:uk = 1, 2, . . . , N }.

2000 Mathematical Subject Classification. 28A80, 54H20, 54H25.Key words and phrases. contraction, iterated function systems, metric

structures, pseudodistance, attractor.Supported by ARC Discovery Grant DP0984353.

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4 M. BARNSLEY, K. IGUDESMAN

The empty worde is the sole member of 0. We write =k=0k. We

also write |u|= n ifu n. Let

={1, 2, . . . , N }N ={=12. . .: k = 1, 2, . . . , N }

be the set of one-sided infinite sequences ofN symbols. For we

write |n = 12. . . n n. We also write u = u1u2. . . un12. . .,

where u n and . For u n let u ={u: }.

Let Xbe a compact metrizable topological space. We denote by Tthe

set of all metrics on X generating the given topology ofX. Let

fj : X X, j = 1, 2, . . . , N

be continuous mappings on X. For u n we write

fu = fu1 fu2 fun , fe= id.

Definition 1. We sayX, T, {fj}

Nj=1

is a topological contractive system

if for every

n=1

f|n(X) ={x},

wherex X is a fixed point.

The next theorem gives a metric d Tsuch that all mappings fj are

nonexpansive with respect to d.

Theorem 1.LetX, T, {fj}Nj=1

be a topological contractive system. There

exists a metricd Tsuch that forx, y X, d(fj(x), fj(y)) d(x, y) for

j = 1, 2, . . . , N .

Let us take any d Tand define d(x, y) as follows.

d(x, y) = supu

{d(fu(x), fu(y))}.

For j= 1, 2, . . . , N we have

d(fj(x), fj(y)) = supu

{d(fuj(x), fuj(y))} supu

{d(fu(x), fu(y))}= d(x, y).

For let x, y,z X. Since for any u

d(x, y) +d(y, z) d(fu(x), fu(y)) +d(fu(y), fu(z)) d

(fu(x), fu(z)),

it follows the triangle inequality. The other properties of metric are

evident.

From the definition we have d(x, y) d(x, y). Therefore in order

to prove that d T we have only to show that d(xn, x) 0 implies

d(xn, x) 0. Let us assume that d(xn, x) 0. Then there exists a

subsequence of{xn}, which we denote again by {xn}, for whichd(xn, x)>


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TOPOLOGICAL CONTRACTIVE SYSTEMS 5

, where is some positive number. From the definition ofdfollows the

existence ofun such that d(fun(xn), fun(x)) > > 0. We have to

consider two cases:(a) The set {un} is finite. Then there exists u in the sequence

{un} which is infinitely repeated and therefore for a suitably selected sub-

sequence we have d(fu(xn), fu(x))> which is a contradiction because

d(xn, x) 0 and mapping fu is continuous.

(b) The set {un} is not finite. Then there exist a symbol 1

{1, 2, . . . , N } and infinite subsequences, which we denote again by {un}

and {xn}, for which |un| 1 and un1 =1 for every u

n. Put v1 =u1.

On the next step we find a symbol 2 {1, 2, . . . , N } and infinite

subsequences, which we denote again by{un}and {xn}, for which|un|

2 and un2 =2 for everyun. Put v2 =u2.

We continue in this fashion to obtain and the infinite se-

quences {vn} and {xn} for which vn|n = |n and d(fvn(xn), fvn(x)) >

. But fvn(xn), fvn(x) f|n(X) where f|n(X) is a nested sequence

of compact sets such that f|n(X) = {x}. It is easy to see that

d(zn, x) 0 for any sequence {zn} such that zn f|n(X). There-

fore d(fvn(xn), fvn(x)) 0 and we arrive again at a contradiction with

d(fvn(xn), fvn(x))> .

3. Contractive property

Definition 2. A topological contractive systemX, T, {fj}

Nj=1

is called a

metric contractive system if there exist a metricd Tand real numbers

0 < j < 1 (j = 1, 2, . . . , N ) such that d(fj(x), fj(y)) jd(x, y) for

everyx, y X.

We call a point = (1, 2, . . . , N) (0, 1)N a polyratio. Foru ,

we write

(u) =

|u|

k=1

uk .

Ifu= e, we set (e) = 1. For x, y X, we write

(x, y) = infu

{(u) :x, y fu(X)}.

Note that, for x = y, we have (x, y) > 0. This follows by the same

method as at the end of the proof of Theorem 1. Indeed, let us assume

that for some x = y there exist an infinite sequence un such that

x, y fun(X) for all n. Then there exist and a subsequence

vn for which vn|n = |n. Because x, y fvn(X) f|n(X) and

f|n(X) ={x}, we get x= y.


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6 M. BARNSLEY, K. IGUDESMAN

From now on, we fix a nonexpansive metric d T from Theorem 1.

For 0< 1, we set

(x, y) =(x, y) (d(x, y)) .

For x, y X, we set

d(x, y) = infl

i=1

(xi, xi+1)

where the infimum is taken over all finite systems of elementsx1, x2, . . . , xl+1

X such that x= x1 and xl+1 =y.

Proposition 1. Let = (1, 2, . . . , N) (0, 1)N be a polyratio, and let

0< 1. Thend is a pseudometric onX. Furthermore, forx, y X,d(fj(x), fj(y)) jd

(x, y) forj = 1, 2, . . . , N .

The triangle inequality follows from the fact that the infimum taken

over all chains connectingxand zless than or equal to the infimum taken

over all chains passing through y and connecting xand z.

Forx, y fu(X), we havefj(x), fj(y) fju(X). Therefore(fj(x), fj(y))

j(x, y). Since d(fj(x), fj(y)) d(x, y), we have

d(fj(x), fj(y)) infl

i=1

(fj(xi), fj(x

i+1))

= infl

i=1

(fj(xi), fj(x

i+1))

d(fj(xi), fj(x

i+1))

infl

i=1

j(xi, xi+1)

d(xi, xi+1)

=jd

(x, y).

We denote by A the set of all fixed points x, that is

A =

n=1

f|n(X) =

{x}.

We call A the attractor of the topological contractive systemX, T, {fj}

Nj=1

.

Repeating the last part of the proof of the Theorem 1, we conclude that

A is a compact set. Let

Xn =

un

fu(X).

Then Xn is a decreasing sequence of compact sets. It is clear that A =

n=0Xn.


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TOPOLOGICAL CONTRACTIVE SYSTEMS 7

Proposition 2. If d(x, y) > 0 for all x, y A, x = y, then it is

compatible with the original topology metric onX, that isd T.

Let x / A. Then there exists n >0 such that x / Xn. Put

p= infyXn

(d(x, y)) and q= inf un1

(u).

Letx= x1, x2, . . . , xl+1 =y be a chain betweenxand y. If everyxi / Xn,

thenl

i=1

(xi, xi+1)(d(xi, xi+1)) ql

i=1

(d(xi, xi+1)) q(d(x, y)) >0.

Otherwise, letk be a minimal number such that xk Xn. Then

l

i=1

(xi, xi+1)(d(xi, xi+1)) k1

i=1

(xi, xi+1)(d(xi, xi+1)) qp >0.

Since d(x, y) (d(x, y)) it remains to show that d(x

n, x) 0

impliesd(xn, x) 0. If this is not the case, then because of compactness

with respect to d there exists a subsequence, which we denote again by

{xn} such that d(xn, y) 0. x=y , and therefore also that d(xn, y)

0, which is a contradiction.

Note that ifN == 1 we have Janoss contractive metric on X (see

[2]). For = 0, we get Kameyamas pseudometric on A (see [3]).

Acknowledgment. We acknowledge and thank E. Sosov for helpfulcomments.

References

[1] John Hutchinson. Fractals and self similarity.Indiana Univ. Math. J., 30:713747,

1981.

[2] Ludvik Janos. A converse of Banachs contraction theorem. Proceedings of the

American Mathematical Society, 18(2):287289, 1967.

[3] Atsushi Kameyama. Distances on topological self-similar sets and the kneading

determinants.J. Math. Kyoto Univ., 40(4):601672, 2000.

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