A Characterization of Hyperbolic Affine Ifs

8/10/2019 A Characterization of Hyperbolic Affine Ifs

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TOPOLOGY

PROCEEDINGSVolume 36 (2010)Pages 189-211

http://topology.auburn.edu/tp/

E-Published on April 2, 2010

A CHARACTERIZATION OF HYPERBOLIC

AFFINE ITERATED FUNCTION SYSTEMS

ROSS ATKINS, MICHAEL F. BARNSLEY, ANDREW VINCE,AND DAVID C. WILSON

Abstract. The two main theorems of this paper provide acharacterization of hyperbolic affine iterated function systemsdened on Atsushi Kameyama [Distances on topologicalself-similar sets, in Fractal Geometry and Applications: AJubilee of Benoit Mandelbrot. Proceedings of Symposia inPure Mathematics, Volume 72, Part 1, 2004, pages 117129]asked the following fundamental question: Given a topolog-ical self-similar set, does there exist an associated system ofcontraction mappings? Our theorems imply an affirmativeanswer to Kameyamas question for self-similar sets derivedfrom affine transformations on .

1. Introduction

The goal of this paper is to prove and explain two theorems thatcharacterize hyperbolic affine iterated function systems dened on. One motivation was the following question: When are thefunctions of an affine iterated function system (IFS) on con-tractions with respect to a metric equivalent to the usual Euclideanmetric?

Theorem 1.1 (Characterization for Affine Hyperbolic IFSs). If = (; 1 2) is an affine iterated function system, thenthe following statements are equivalent.

2010 Mathematics Subject Classication. Primary 54H25, 26A18, 28A80.Key words and phrases. affine mappings, contraction mapping, hyperbolic

IFS, iterated function systems.c2010 Topology Proceedings.

189


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190 R. ATKINS, M. F. BARNSLEY, A. VINCE, AND D. C. WILSON

(1) is hyperbolic.(2) is point-bered.(3) has an attractor.(4) is a topological contraction with respect to some convex

body .(5) is non-antipodal with respect to some convex body

.

Statement (1) is a metric condition on an affine IFS, statements(2) and (3) are in terms of convergence, and statements (4) and (5)are in terms of concepts from convex geometry. The terms con-

tractive, hyperbolic, point-bered, attractor, topologicalcontraction, and non-antipodal are dened in denitions 2.2,2.3, 2.5, 2.7, 5.8, and 6.5, respectively. This theorem draws to-gether some of the main concepts in the theory of iterated functionsystems. Banachs classical Contraction Mapping Theorem statesthat a contractionon a complete metric space has a xed point0and that0= lim

(), independent of, where denotesthe iteration. The notion of hyperbolic generalizes the contrac-tion property to an IFS. Namely, an IFS is hyperbolic if thereis a metric on , Lipschitz equivalent to the usual one, such thateach is a contraction. The notion of point-bered, intro-duced by Bernd Kieninger [10], is the natural generalization of the

limit condition above to the case of an IFS. While traditional dis-cussions of fractal geometry focus on the existence of an attractorfor a hyperbolic IFS, Theorem 1.1 establishes that the more geo-metrical (and non-metric) assumptions topologically contractiveand non-antipodal can also be used to guarantee the existence ofan attractor. Basically, a function : is non-antipodalif certain pairs of points (antipodal points) on the boundary ofare not mapped by to another pair of antipodal points.

Since the implication (1) (2) is the Contraction Mapping The-orem when the IFS contains only one affine mapping, Theorem 1.1contains an affine IFS version of the converse to the Contraction

Mapping Theorem. Thus, our theorem provides a generalization ofresults proved by Ludvk Janos [8] and Solomon Leader [12]. Sucha converse statement in the IFS setting has remained unclear untilnow.


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AFFINE ITERATED FUNCTION SYSTEMS 191

Although not every affine IFS = (; 1 2) is hyper-bolic on all of , the second main result states that if has acoding map (Denition 2.4), then is always hyperbolic on someaffine subspace of.

Theorem 1.2. If = (; 1 2) is an affine IFS with acoding map : , then is hyperbolic on the affine hull of(). In particular, if () contains a non-empty open subset of, then is hyperbolic on.

Using slightly different terminology, Atsushi Kameyama [9] posedthe following fundamental question: Is an affine IFS with a coding

map :

hyperbolic when restricted to ()? An affir-mative answer to this question follows immediately from Theorem1.2.

Our original motivation, however, was not Kameyamas question,but rather a desire to approximate a compact subset as theattractor of an iterated function system = (; 1 2)where each:

is affine. This task is usually done usingthe collage theorem [1], [2] by choosing an IFS so that theHausdorff distance( ()) is small. If the IFS is hyperbolic,then we can guarantee it has an attractor such that ( ) iscomparably small. But then the question arises: How does oneknow if is hyperbolic?

The paper is organized as follows. Section 2 contains notation,terminology, and denitions that will be used throughout the paper.Section 3 contains examples and remarks relating iterated functionsystems and their attractors to Theorem 1.1 and Theorem 1.2. InExample 3.1, we show that an affine IFS can be point-bered, butnot contractive under the usual metric on . Thus, some kindof remetrization is required for the system to be contractive. InExample 3.2, we show that an affine IFS can contain two linearmaps each with real eigenvalues all with magnitudes less than 1,but still may not be point-bered. Thus, Theorem 1.1 cannot bephrased only in terms of eigenvalues and eigenvectors of the individ-ual functions in the IFS. Indeed, in Example 3.3, we explain how,

given any integer >0, there exists a linear IFS2; 1 2

such

that each operator of the form 12 , with 1 2 for = 1 2, and has spectral radius less than one, while12 has spectral radius larger than one. This is related to the


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joint spectral radius [16] of the pair of linear operators 1 and 2and to the associated niteness conjecture; see, for example, [4]. Insection 8 we comment on the relationship between the present workand recent results concerning the joint spectral radius of nite setsof linear operators. Example 3.4 provides an affine IFS on 2 thathas a coding map but is not point-bered on 2, and hence, byTheorem 1.1, not hyperbolic on 2. It is, however, point-beredand hyperbolic when restricted to the -axis, which is the affinehull of(), thus illustrating Theorem 1.2.

For the proof of Theorem 1.1, we provide the following road map.

(1) The proof that statement (1) statement (2) is provided

in Theorem 4.1.(2) The proof that statement (2) statement (3) is provided



in Proposition 6.6.(5) The proof that statement (5) statement (1) is provided

in Theorem 6.7.

Theorem 1.2 is proved in section 7.

2. Notation and definitions

We treat as a vector space, an affine space, and a metricspace. We identify a point = (1 2)

with thevector whose coordinates are 1 2. We write 0

forthe point in whose coordinates are all zero. The standard basisis denoted 1 2 . The inner product between

is

denoted by . The 2-norm of a point is 2=

,and the Euclidean metric :

[0 ) is dened by( ) = 2 for all

.The following notations, conventions, and denitions will also be

used throughout this paper.

(1) A convex body is a compact convex subset of with non-empty interior.

(2) For a set in , the notation () is used to denotethe convex hull of.


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(3) For a set , theaffine hull, denoted aff(), ofis thesmallest affine subspace containing , i.e., the intersectionof all affine subspaces containing .

(4) The symbol will denote the nonempty compact subsetsof, and the symbol will denote the Hausdorff metricon . Recall that ( ) is a complete metric space.

(5) A metric on is said to be Lipschitz equivalent to if there are positive constants and such that

( ) ( ) ( )

for all . If two metrics are Lipschitz equivalent,then they induce the same topology on , but the converseis not necessarily true.

(6) For any two subsets and ofthe notation := : and is used to denote the pointwisesubtraction of elements in the two sets.

(7) For a positive integer the symbol = 1 2 willdenote the set of all innite sequences of symbols

=1

belonging to the alphabet 1 2 . The set is en-dowed with the product topology. An element of will also be denoted by the concatenation = 123 ,where denotes the

component of. Recall that since is endowed with the product topology, it is a compact

Hausdorff space.Denition 2.1 (IFS). If >0 is an integer and :

, = 1 2 are continuous mappings, then =(; 1 2) is called an iterated function system (IFS). Ifeach is an affine map on , then is called an affine IFS.

Denition 2.2(Contractive IFS). An IFS = (; 1 2)is contractive when each is a contraction. Namely, there is anumber [0 1) such that (() ()) ( ) for all , for all.

Denition 2.3 (Hyperbolic IFS). An IFS = (; 1 2)

is called hyperbolic if there is a metric on

Lipschitz equivalentto the given metric so that each is a contraction.

Denition 2.4 (Coding Map). A continuous map : iscalled a coding map for the IFS = (; 1 2) if, for each


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= 1 2 the following diagram commutes,

(2.1)

where the symbol: denotes the inverse shift map denedby() =.

The notion of a coding map is due to Jun Kigami [11] andKameyama [9].

Denition 2.5(Point-Fibered IFS). An IFS = (; 1 2)

is point-bered if, for each = 123 the limit on theright hand side of

(2.2) () := lim

1 2 ()

exists, is independent of for xed , and the map : is a coding map.

It is not difficult to show that (2.2) is the unique coding map ofa point-bered IFS. Our notion of a point-bered iterated functionsystem is similar to that of Kieninger [10, Denition 4.3.6, p. 97].However, we work in the setting of complete metric spaces whereasKieninger frames his denition in a compact Hausdorff space.

Denition 2.6 (The Symbol () for an IFS). For an IFS =(; 1 2) dene : by

() ==1

()

(The same symbol is used for both the IFS and the mapping.)For , let() denote the -fold composition of, i.e., theunion of1 2 () over all words 12 of length.

Denition 2.7 (Attractor for an IFS). A set is called anattractorof an IFS = (; 1 2) if

(2.3) = () and

(2.4) = lim

()


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the limit with respect to the Hausdorff metric, for all .

If an IFS has an attractor , then clearly is the unique attrac-tor. It is well known that a hyperbolic IFS has an attractor. Anelegant proof of this fact is given by John E. Hutchinson [7]. He ob-serves that a contractive IFS induces a contraction : ,from which the result follows by the contraction mapping theorem.See also [6] and [19].

Section 4 shows that a point-bered IFS has an attractor ,and, moreover, if is the coding map of, then = (). Often is considered as the address of the point () in the attractor.In the literature on fractals (for example, [11]) there is an approach

to the concept of a self-similar system without reference to theambient space. This approach begins with the idea of a continuouscoding map and, in effect, denes the attractor as ().

3. Examples and remarks on iterated function systems

This section contains examples and remarks relevant to Theo-rem 1.1 and Theorem 1.2.

Example 3.1 (A Point-Fibered, Not Contractive IFS). Considerthe affine IFS consisting of a single linear function on 2 given bythe matrix

= 0 218 0

Note that the eigenvalues of equal 12 . Since

lim

2 = lim

1

( 12 ) 0

0 ( 12 )

=

0 00 0

where is the change of basis matrix, this IFS is point-bered.However, since

01

=

20

the mapping is not a contraction under the usual metric on 2.Theorem 1.1, however, guarantees we can remetrize 2 with an

equivalent metric so thatis a contraction.Example 3.2 (An IFS with Point-Fibered Functions That Is NotPoint-Fibered). In the literature on affine iterated function sys-tems, it is sometimes assumed that the eigenvalues of the linear


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parts of the affine functions are less than 1 in modulus. Unfortu-nately, this assumption is not sufficient to imply any of the vestatements given in Theorem 1.1. While the affine IFS (; ) ispoint-bered if and only if the eigenvalues of the linear part ofall have moduli strictly less than 1, an analogous statement cannotbe made if the number of functions in the IFS is larger than 1.

Consider the affine IFS =

2; 1 2

where

1=

0 218 0

and 2=

0 182 0

As noted in Example 3.1,

lim

1 u = lim

2 u=

00

for any vector u. Thus, both 1 =

2; 1

and 2 =

2; 2

arepoint-bered. Unfortunately, their product is the matrix

1 2=

4 00 164

so that

lim

(1 2)

10

= lim

4

0

= +

Thus, the IFS = 2; 1 2 fails to be point-bered.Remark 3.3. While it is true that (1) (2) in Theorem 1.1 evenwithout the assumption that the IFS is affine, the converse is nottrue in general. Kameyama [9] has shown that there exists a point-bered IFS that is not hyperbolic. We next give an example of anaffine IFS with a coding map that is not point-bered. Thus, theset of IFSs (with a coding map) strictly contains the set of point-bered IFSs which, in turn, strictly contains the set of hyperbolicIFSs.

Example 3.4 (The Failure of a Finite Eigenvalue Test to ImplyPoint-Fibered). Consider the linear IFS =

2; 1 2

where

1=

0 218 0

and 2=

cos sin sin cos

=

where denotes rotation by angle , and 0<


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4. Hyperbolic implies point-fibered impliesthe existence of an attractor

The implications (1) (2) (3) in Theorem 1.1 are proved inthis section. We also introduce the notation =1 2 (). Note that, for xed, () is a function of both and.

Theorem 4.1. If= (; 1 2) is a hyperbolic IFS, then is point bered.

Proof: For , the proof that the limit lim exists andis independent of is virtually identical to the proof of the classical

Contraction Mapping Theorem. Moreover, the same proof showsthat the limit is uniform in .

With : dened by () = lim, it is easy tocheck that, for each = 1 2 , the diagram (2.1) commutes.

It only remains to show that is continuous. With xed, () is a continuous function of . This is simply because, if are sufficiently close in the product topology, then theyagree on the rst components. By Denition 2.5, the function is then the uniform limit of continuous (in ) functions dened onthe compact set . Therefore, is continuous.

Letbe a point-bered affine IFS, and let denote the set

:= ()According to Theorem 4.3, is the attractor of.

Lemma 4.2. Let = (; 1 2) be a point-bered affineIFS with coding map : . If is compact, then theconvergence in the limit

() = lim

()

is uniform in= 12 and simultaneously.

Proof: Only the uniformity requires proof. Express () =

+, where is the linear part. Then

(4.1)

() = () +1() +2(1) + +12 +1

=() +(0)


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From equation (4.1) it follows that, for any ,

(4.2) (() ()) =( )2

sup =1

2 ()2 : 11+ +

max

() (0)2where= 2 sup max :11+ + and where

=1 is a basis for

.Let > 0. From the denition of point-bered there is a ,

independent of, such that if > , then() ()2< 4 and(0) ()2 < 4

which implies() (0)2 < 2 . This and equation (4.2)

imply that if := max, then, for any we have

(4.3) (() ())<

2=

2

Let be a xed element of . There is a, independent of , such

that if > , then (() ())< 2 . If > ( ), then,

by equation (4.3), for any

(() ()) (() ()) +(() ())<

2

+

2

=

Theorem 4.3 (A Point-Fibered IFS Has an Attractor). If =(; 1 2) is a point-bered affine IFS, then has an at-tractor= (), where: is the coding map of.

Proof: It follows directly from the commutative diagram (2.1)that obeys the self-referential equation (2.3). We next show that satises equation (2.4).

Let > 0. We must show that there is an such that if > , then (

() ()) < . It is sufficient to let =max(1 2), where 1 and2 are dened as follows.

First, let be an arbitrary element of. Then there exists a such that = (). By Lemma 4.2, there is an 1 suchthat if > 1, then (() ) = (() ())< , for all

. In other words, lies in an -neighborhood of().


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Second, let be an arbitrary element of and an arbitraryelement of . If := () , then there is an 2 such thatif > 2, then (() ) = (() ()) < . In other

words, () lies in an -neighborhood of.

5. An IFS with an attractor istopologically contractive

The goal of this section is to establish the implication (3) (4)in Theorem 1.1. We will show that if an affine IFS has an attractoras dened in Denition 2.7, then it is a topological contraction.The proof uses notions involving convex bodies.

Denition 5.1. A convex body is centrally symmetricif it hasthe property that whenever then .

A well-known general technique for creating centrally symmetricconvex bodies from a given convex body is provided by the nextproposition.

Proposition 5.2. If a set is a convex body in then the set = is a centrally symmetric convex body in.

Denition 5.3 (Minkowski Norm). Ifis a centrally symmetricconvex body in then theMinkowski normon is dened by

= inf 0 :

The next proposition is also well known.

Proposition 5.4. If is a centrally symmetric convex body in then the functiondenes a norm on

Moreover, theset is the unit ball with respect to the Minkowski norm.

Denition 5.5 (Minkowski Metric). If is a centrally symmetricconvex body in and is the associated Minkowski norm,then dene the Minkowski metricon by the rule

( ) :=

While R. Tyrrel Rockafellar [15] refers to such a metric as aMinkowski metric, the reader should be aware that this term is also

associated with the metric on space-time in the theory of relativity.Since, for any convex body , there are positive numbers andsuch that contains a ball of radius and is contained in a ballof radius , the following proposition is clear.


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Proposition 5.6. If is a Minkowski metric, then is Lipschitzequivalent to the standard metric on

.

Proposition 5.7. A metric : [0 ) is a Minkowskimetric if and only if it is translation invariant and distances behave

linearly along line segments. More specically,

(5.1) ( + + ) =( ) and ( (1 ) + ) =( )

for all and all [0 1].

Proof: For a proof, see [15, pp. 131132].

Denition 5.8 (Topologically Contractive IFS)

. An IFS =; 1 2 is called topologically contractive if there is a

convex body such that() ().

The proof of Theorem 5.10 relies on the following lemma whichis easily proved.

Lemma 5.9. If : is affine and then(()) =(()).

Theorem 5.10 (The Existence of an Attractor Implies a Topo-logical Contraction). For an affine IFS = ; 1 2,if there exists an attractor of the affine IFS = ;

1 2, then is topologically contractive.Proof: The proof of this theorem unfolds in three steps.

(1) There exists a convex body 1and a positive integer withthe property that (1) (1).

(2) The set 1 is used to dene a convex body 2 such that(2) (2) where () = + and =1 2 .

(3) There is a positive constant such that the set = 2has the property () ().

(1): Let denote the attractor of . Let = :

( ) denote the dilation of by radius >0. Since weare assuming lim(() ) = 0we can nd an integer

so that ((1) )< 1. Thus,

(5.2) (1) (1)


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If we let 1:= (1), then

(1) =1

2

(1 2 ) ((1))

=1

2

(1 2 (1))

(by Lemma 5.9)

1

2

((1))

=((1))

(by inclusion (5.2)) ((1)) = (1)

This argument completes the proof of (1).

(2): Consider the set

2 :=1

=0(((1) (

(1)))The set 2 is a centrally symmetric convex body because it is anite Minkowski sum of centrally symmetric convex bodies. If anyaffine map in is written () =+where :

denotes the linear part, then

(2) =

1

=0

(

(1) (

(1))

(since is a linear map)

=1=0

(

(1)

) (

(1)

)

(by Lemma 5.9)

=1=0

(

(1)

) (

(1)

)

(since the s cancel)

1=0

((+1) (1)) (

(+1)(1))


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=

((1) ((1))

+1=1

((1) (

(1))

( (1) (1))

+1=1

(((1) ((1)))

(by step (1))

=(2)

The second to last inclusion follows from the fact that

(1)

(+1) (1). The last equality follows from thefact that if and are symmetric convex bodies in then() + = ( + ). We have now completed the proof ofstep (2).

(3): It follows from step (2) and the compactness of2that thereis a constant (0 1) such that 2(() ()) < 2( )for all and all = 1 2 .

Let

>

(1 )

where = max2

(1

0) 2

(2

0) 2

(

0). If 2and () =+ is any function in the IFS , then

()2 =2(() 0)

=2(+ 0) 2(+) +2( 0)

=2( 0) +2( 0) (by equation (5.1))

< +2( 0) =+ 2 +


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interior of. The pair is usually referred to as the twosupporting hyperplanes of orthogonal to . (See [13, p. 14].)

Denition 6.1 (Antipodal Pairs). If is a convex bodyand 1then dene

:= () = ( ) ( ) and

:= () =

1

.

We say that ( ) is an antipodal pairof points with respect to if ( ) .

Denition 6.2 (Diametric Pairs). If

is a convex body,and 1 then dene the diameter of in the direction tobe

() = max 2 : =

The maximum is achieved at some pair of points belonging to because is convex and compact, and 2 is continuousfor ( ) . Now dene

= ( ) :() = 2 and

=

1

.

We say that ( ) is adiametric pairof points in the direction

of and that is the set of diametric pairs of points of

Denition 6.3 (Strictly Convex). A convex body is strictlyconvex if, for every two points , the open line segmentjoining and is contained in the interior of.

We write to denote the closed line segment with endpoints at and so that is the vector, in the direction from to whose magnitude is the length of.

Theorem 6.4. If is a convex body, then the set of antipo-dal pairs of points of is the same as the set of diametric pairs ofpoints of i.e.,

= Proof: First, we show that . If ( ) then

and for some 1. Clearly, any

chord ofparallel tolies entirely in the region between and


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and therefore cannot have length greater than that of. So( ) = and ( ) . For use later in theproof, note that if is strictly convex, then is the unique chordof maximum length in its direction.

Conversely, to show that , rst consider the case where is a strictly convex body. For each 1 consider the points and . The continuous function :1 1 dened by

() = 2

has the property that () > 0 for all . In other words, theangle between and () is less than 2 . But it is an elementaryexercise in topology (see, for example, [14, Problem 10, p. 367])that if :1 1 maps no point to its antipode , thenhas degree 1 and, in particular, is surjective. To show that ,let ( ) for some

1. By the surjectivity of thereis 1 such that () = . According to the last sentence ofthe previous paragraph, is the unique longest chord parallelto. Therefore, = and= and consequently, ( ) .

The case where is not strictly convex is treated by a standardlimiting argument. Given a vector 1 and a longest chord parallel to , we must prove that ( ) . Since is the

intersection of all strictly convex bodies containing , there is asequence of strictly convex bodies containing with thefollowing two properties.

(1) There is a longest chord of parallel to such thatlim 2 = 2 and the limits lim = and lim = exist.

By the result for the strictly convex case, there is a sequenceof vectors

1 such that = () and = (). By perhaps going to a subsequence

(2) lim = 1 exists.

It follows from item (1) that 2 = 2 and is

parallel to . Therefore, , as well as , are longest chords of parallel to . It follows from (2) that if and are thehyperplanes orthogonal to through and , respectively, then and are parallel supporting hyperplanes of . Therefore,


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necessarily and , and consequently, ( ) .

Denition 6.5 (Non-Antipodal IFS). If is a convexbody, then : is non-antipodal with respect to if() , and ( ) () implies (() ()) (). If = ; 1 2 is an iterated function system with theproperty that each is non-antipodal with respect to then is called non-antipodalwith respect to.

The next proposition gives the implication (4) (5) in Theo-rem 1.1. The proof is clear.

Proposition 6.6 (A Topological Contraction Is Non-Antipodal).If = ; 1 2 is an affine iterated function systemwith the property that there exists a convex body such that() () for all = 1 2 then is non-antipodalwith respect to .

The next theorem provides the implication (5) (1) in Theo-rem 1.1.

Theorem 6.7. If the affine IFS = (; 1 2) is non-antipodal with respect to a convex body, then is hyperbolic.

Proof: Assume that is a convex body such that is non-antipodal with respect to for all . Let = and let () = + , where is the linear part of. ByProposition 5.2, the set is a centrally symmetric convex bodyand

() =() () =() () =

We claim that () (). Since is compact and islinear, to prove the claim, it is sufficient to show that () for all . By way of contradiction, assume that and () . Then the vector is a longest vector in in itsdirection. Since = , there are 1 2 such that

= 12, and (1 2) () = (), where the last equality isby Theorem 6.4. So (1 2) is an antipodal pair with respect to.Likewise, since is a longest vector in in its direction, there are1 2 such that = 1 2, and (1 2) () =().


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Therefore,

(2) (1) =(2) (1) =(2 1) = = 1 2

which implies that ((1) (2)) () = (), contradictingthatis non-antipodal with respect to .

Ifdenotes the Minkowski metric with respect to the centrallysymmetric convex body , then by Proposition 5.4, is the unitball centered at the origin with respect to this metric. Since iscompact, the containment () () implies that there is an [0 1) such that< for all

. Then

(() ()) = () ()=

= ( )< = ( )Therefore, is a metric for which each function in the IFS is acontraction. By Proposition 5.6, is Lipschitz equivalent to thestandard metric.

7. An answer to the question of Kameyama

We now turn to the proof of Theorem 1.2, the theorem that set-tles the question of Kameyama. If and =(; 1 2) is an IFS on , then the denitions of codingmap and point-bered forare exactly the same as denitions 2.4and 2.5, with replaced by. The proof of Theorem 1.2 requires

the following proposition.Proposition 7.1. If and = (; 1 2) is anIFS with a coding map : such that() = then ispoint-bered on.

Proof: By Denition 2.5, we must show that lim1 2 () exists, is independent of , and is continuousas a function of = 12 . We will actually show thatlim1 2 () =().

Since is a coding map, we know by Denition 2.4 that () = ()for all = 1 2 . By assumption, if is anypoint in then there is a such that () =. Thus,

lim 1 2 ()= lim 1 2 (())

(since () =)


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= lim

(1 2 )

(by diagram (2.1))

=( lim

1 2 )

(since is continuous)

=()

Theorem 7.2. If = (; 1 2) is an affine IFS with acoding map : , then is point-bered when restricted tothe affine hull of(). In particular, if() contains a non-emptyopen subset of then is point-bered on.

Proof: Let := (). Since () for all , the restric-tion of the IFS to , namely := (; 1 2 ), is welldened. It follows from Proposition 7.1 that is point-beredand, because the coding map for a point-bered IFS is unique,

() = lim

1 2 ()

for ( ) . It only remains to show that the restrictionaff() := (aff(); 1 2 ) of the affine IFS to the affinehull of is point-bered.

Let aff(), the affine hull of. It is well known that anypoint in the affine hull can be expressed as a sum,=

=0for

some 0 1 such that

=0 = 1 and 0 1 . Hence, for ( ) aff(),

lim

1 2 () = lim

1 2 (

=0

),

= lim

=0

1 2 ()

=

=0

() =()

Theorem 1.2 now follows easily from Theorem 7.2 and Theo-rem 1.1.

Proof of Theorem 1.2: Let := () and let dim aff() = . It is easy to check from the commuting diagram (2.1)


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that () for each implies that (aff())aff() foreach . Since aff() is isomorphic to , Theorem 1.1 canbe applied to the IFS aff() := (aff(); 1 2) to concludethat, since it is point-bered, aff() is also hyperbolic.

Note that the IFS (; ), where() = 2 + 1, is not hyperbolicon , but it is hyperbolic on the affine subspace 1 .

8. Concluding remarks

Recently it has come to our attention that another condition,equivalent to conditions (1) (5) in Theorem 1.1, is (6) has joint

spectral radius less than one. (We dene the joint spectral radius(JSR) of an affine IFS to be the joint spectral radius of the set oflinear factors of its maps.) This information is important becauseit connects our approach to the rapidly growing literature aboutJSR; see, for example, [3], [5], and works that refer to these.

Since Example 3.3 and the results presented by Vincent Blondel,Jacques Theys, and Alexander Vladimirov [4] indicate there is nogeneral fast algorithm which will determine whether or not thejoint spectral radius of an IFS is less than one, we believe thatTheorem 1.1 is important because it provides an easily testablecondition that an IFS has a unique attractor. In particular, thetopologically contractive and non-antipodal conditions (conditions

(4) and (5)) provide geometric/visual tests, which can easily bechecked for any affine IFS. In addition to yielding the existence of anattractor, these two conditions also provide information concerningthe location of the attractor. (For example, the attractor is a subsetof a particular convex body.) We also anticipate that Theorem 1.1can be generalized into other broader classes of functions, wherethe techniques developed for the theory of joint spectral radius willnot apply.

References

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[3] Marc A. Berger and Yang Wang, Bounded semigroups of matrices, LinearAlgebra Appl. 166(1992), 2127.

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[5] Ingrid Daubechies and Jeffrey C. Lagarias, Sets of matrices all inniteproducts of which converge, Linear Algebra Appl. 161(1992), 227263.

[6] Masayoshi Hata,On the structure of self-similar sets, Japan J. Appl. Math.2 (1985), no. 2, 381414.

[7] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J.30(1981), no. 5, 713747.

[8] Ludvk Janos, A converse of Banachs contraction theorem, Proc. Amer.

Math. Soc. 18(1967), 287289.[9] Atsushi Kameyama, Distances on topological self-similar sets, in Fractal

Geometry and Applications: A Jubilee of Benoit Mandelbrot. Ed. MichelL. Lapidus and Machiel van Frankenhuijsen. Proceedings of Symposia inPure Mathematics, Volume 72, Part 1. Providence, RI: Amer. Math. Soc.,2004. 117129

[10] Bernd Kieninger,Iterated Function Systems on Compact Hausdorff Spaces.Aachen: Shaker Verlag, 2002.

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[12] Solomon Leader, A topological characterization of Banach contractions,Pacic J. Math.69(1977), no. 2, 461466.

[13] Maria Moszynska, Selected Topics in Convex Geometry. Translated andrevised from the 2001 Polish original. Boston, MA: Birkhauser Boston,Inc., 2006.

[14] James R. Munkres, Topology. 2nd ed. Upper Saddle River, NJ: PrenticeHall, 2000.

[15] R. Tyrrell Rockafellar, Convex Analysis. Princeton Mathematical Series,No. 28. Princeton, NJ: Princeton University Press, 1970.

[16] Gian-Carlo Rota and Gilbert Strang, A note on the joint spectral radius,Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 379381.

[17] Rolf Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclo-pedia of Mathematics and its Applications, 44. Cambridge: CambridgeUniversity Press, 1993.

[18] Roger Webster, Convexity. Oxford Science Publications. Oxford: OxfordUniversity Press, 1994.

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(Barnsley) Department of Mathematics; Australian National Uni-versity; Canberra, ACT, Australia

E-mail address: [email protected]; [email protected]: http://www.superfractals.com

(Vince) Department of Mathematics; University of Florida; Gainesville,FL 32611-8105, USA

E-mail address: [email protected]: http://www.math.ufl.edu/vince/

(Wilson) Department of Mathematics; University of Florida; Gainesville,FL 32611-8105, USA

E-mail address: [email protected]: http://www.math.ufl.edu/dcw/

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A Characterization of Hyperbolic Affine Ifs