S t a b l e I t e r a t e d F u n c t i o n S y s t e m s
Erland Gadde
Oj
s
University o f U m eå
Department of M athematics
Doctoral Thesis No 4, 1992
Stable Iterated Function Systems
Erland Gadde
University o f Umeå
Department of Mathematics Doctoral Thesis No 4, 1992
ISBN 91-7174-688-9
Akademisk avhandling som med tillstånd av rektorsäm betet vid Umeå Universitet för avläggande av filosofie doktorsexamen framlägges till offentlig granskning måndagen den 1 juni 1992 klockan 10.15 i Hörsal A, Sam hällsvetarhuset, Umeå Universitet.
S ta b le I t e r a t e d F u n c tio n S y s t e m s
Do c t o r a l Di s s e r t a t i o n
by
Er l a n d Ga d d e
Doctoral Thesis No 4, D epartm ent of M athem atics, University of Umeå, 1992 ISBN 91-7174-688-9
To be publicly discussed in the Lecture Hall A o f Samhallsvetarhuset, University o f Umeå, on M onday, June 1, 1992, at 10.15 am fo r the degree of Doctor o f Philosophy.
A b s t r a c t
T he purpose of this thesis is to generalize the growing theory of iterated function system s (IFSs). Earlier, hyperbolic IFSs with finitely many functions have been studied extensively. Also, hyperbolic IFSs with infinitely many functions have been studied. In this thesis, more general IFSs are studied.
The H ausdorff pseudom etric is studied. This is a generalization of the Hausdorff m etric. Wide and narrow lim it sets are studied. These are two types of limits of sequences of sets in a complete pseudometric space.
Stable Iterated Function S ystem s, a kind of generalization of hyper
bolic IFSs, are defined. Some different, bu t closely related, types of stability for the IFSs are considered. It is proved th a t the IFSs with the most general type of stability have unique attractors. Also, invariant sets, addressing, and periodic points for stable IFSs are studied.
H utchinson’s m etric (also called V aserhstein’s m etric) is generalized from being defined on a space of probability measures, into a class of norms, the £ -n o rm s, on a space of real measures (on certain m etric spaces). Under rather general conditions, it is proved th a t these norms, when they are restricted to positive measures, give rise to complete m et
ric spaces with the metric topology coinciding with the weak*-topology.
Then, IFSs w ith probabilities (IFSPs) are studied, in particular, stable IFSPs. The £ -n o rm -resu lts are used to prove th a t, as in the case of hyperbolic IFSPs, IFSPs with the most general kind of stability have unique invariant measures. These measures are ” attractiv e” . Also, an invariant measure is constructed by first ”lifting” the IFSP to the code space. Finally, it is proved th a t the Random Iteration Algorithm in a sense will ” work” for some stable IFSPs.
K e y w o rd s: Hausdorff m etric, iterated function system (IFS), a ttra c tor, invariant set, address, H utchinson’s metric, we a k* -topology, IFS with probabilities, invariant measure, the Random Iteration Algorithm.
Stable Iterated Function Systems
Erland Gadde
University o f Umeå
Department of Mathematics Doctoral Thesis No 4, 1992
ISBN 91-7174-688-9
Akademisk avhandling som med tillstånd av rektorsäm betet vid Umeå Universitet för avläggande av filosofie doktorsexamen framlägges till offentlig granskning måndagen den 1 juni 1992 klockan 10.15 i Hörsal A, Sam hällsvetarhuset, Umeå Universitet.
Tryckt vid UmU Tryckeri
S ta b le I t e r a t e d F u n c tio n S y s t e m s
Do c t o r a l Di s s e r t a t i o n
by
Er l a n d Ga d d e
Doctoral Thesis No 4, D epartm ent of M athem atics, University of Umeå, 1992 ISBN 91-7174-688-9
To be publicly discussed in the Lecture Hall A of Samhållsvetarhuset, University o f Umeå, on Monday, June 1, 1992, at 10.15 am fo r the degree of Doctor o f Philosophy.
A b s tr a c t
The purpose of this thesis is to generalize the growing theory of iterated function system s (IFSs). Earlier, hyperbolic IFSs with finitely many functions have been studied extensively. Also, hyperbolic IFSs with infinitely m any functions have been studied. In this thesis, more general IFSs are studied.
The H ausdorff pseudometric is studied. This is a generalization of the Hausdorff metric. W ide and narrow lim it sets are studied. These are two types of limits of sequences of sets in a complete pseudom etric space.
Stable Iterated Function S ystem s, a kind of generalization of hyper
bolic IFSs, are defined. Some different, but closely related, types of stability for the IFSs are considered. It is proved th a t the IFSs with the most general type of stability have unique attractors. Also, invariant sets, addressing, and periodic points for stable IFSs are studied.
H utchinson’s m etric (also called Vaserhstein’s m etric) is generalized from being defined on a space of probability measures, into a class of norms, the £ -n o rm s, on a space of real measures (on certain m etric spaces). Under rather general conditions, it is proved th a t these norms, when they are restricted to positive measures, give rise to complete m et
ric spaces with the metric topology coinciding with the weak*-topology.
Then, IFSs with probabilities (IFSPs) are studied, in particular, stable IFSPs. The £ -n o rm -resu lts are used to prove th a t, as in the case of hyperbolic IFSPs, IFSPs with the most general kind of stability have unique invariant measures. These measures are ” attractiv e” . Also, an invariant measure is constructed by first ”lifting” the IFSP to the code space. Finally, it is proved th a t the Random Iteration Algorithm in a sense will ”work” for some stable IFSPs.
K e y w o rd s: Hausdorff m etric, iterated function system (IFS), a ttra c tor, invariant set, address, Hutchinson’s metric, we a k* -topology, IFS with probabilities, invariant measure, the Random Iteration Algorithm.
A C K N O W L E D G E M E N T S .
I would like to thank my supervisor, Professor Hans Wallin, w ithout whose aid and inspiration this thesis would not have been w ritten.
Also, I would like to thank Professor Roland Häggkvist, Jan Gelfgren, Mikael Stenlund, and Ulf Backlund for help with T^X-problem s, and H ubert Shutrick for the opportunity to use his Macintosh II for working a long way from Umeå.
Also, I would like to thank Urban Cegrell for helping me w ith some measure theoretic problems.
Finally, I would like to thank Debra Milton for checking the English in the m anuscript.
0. S U M M A R Y .
This thesis is mainly a study of stable iterated function system s (stable IFSs). There are some different, bu t related, kinds of stability for these IFSs, see Definitions 2.4.
Stable IFSs are a generalization of a type of IFSs introduced by Wallin and Karlsson ([26], p. 11), which they used to study continued fractions.
All hyperbolic IFSs, studied by Hutchinson, Barnsley, Demko, Lewellen, and others ([14], [3], [4], [16]) are boundedly stable (Definitions 2.4).
This thesis is mainly concerned with convergence properties of these IFSs and not with the dimensions and geometric properties of the at
tractors of the systems.
For these convergences, we use the H ausdorff pseudom etric (see Defi
nition 1.9), a generalization of the Hausdorff m etric which is defined on all kinds of sets, not only closed and bounded ones. In C hapter 1, we first give a brief introduction to pseudom etric spaces. Then, we use this knowledge to study the Hausdorff pseudometric. We conclude C hapter 1 by studying wide and narrow lim it sets, which are two related types of limits of sequences of sets.
In C hapter 2, we define (X, Xo)-stable IFSs, and some special cases of these systems (Definitions 2.4). This is the most general kind if stability studied in this thesis. We show th a t all hyperbolic IFSs which satisfy a certain boundedness condition are boundedly stable (Proposition 2.9).
Bounded stability is one of the special cases mentioned.
Then, we give a non-trivial example of a boundedly stable no n - hyperbolic IFS (Example 2.10).
In the next section, we prove th a t the (X, xo)-stable IFSs have unique attra cto rs (Theorem 2.15). For the proof, we use a fixed point theorem (Theorem 2.12) in a sim ilar way as B anach’s Fixed Point Theorem is used for the proof of the corresponding theorem about hyperbolic IFSs ([3], p. 82 and [14], p. 728).
After illustrating with some examples and counterexamples, we in tro duce addresses for (T, æo)-stable IFSs (Definitions 2.25), following the studies of Barnsley and Lewellen ([3], [16]). We give some relations be
tween addresses, attracto rs, and invariant sets, which we then illustrate with some further examples. Then, we study periodic points (Defini
tions 2.36) and continuous addressing, giving further inform ation about the a ttra c to r (Theorem 2.43 and Corollary 2.44). Finally, we make Q°°
(see Definition 4.1) to a hyperbolic IFS (Definitions 2.47 and Rem arks 2.48).
In C hapter 4, we study the convergence of measures related to IFSs.
To do this, we use the £ -n o rm s, which we investigate carefully and
extensively in C hapter 3. The £ -n o rm s (Definitions 3.21) are gener
alizations of H utchinson’s m etric ([14], p. 732). We begin C hapter 3 by stating and proving some well-known results from topology, m easure theory, and functional analysis, whose proofs are not so easy to find in the standard monographs, bu t which we will need.
We introduce reasonable spaces (Definition 3.11), an invention of my own, which is a generalization of second countable spaces. Most re
sults about measures on separable m etric spaces can be carried over to reasonable m etric spaces by using Theorem 3.6.
v We then introduce (for a reasonable m etric space X ) the spaces B ( X )
v
and A i ( X ) (Definitions 3.17), the latter on which the £ -n o rm s are de
fined. We give some properties of these spaces and then we define the
£ -n o rm s (Definitions 3.21). After giving some simple properties of the
£ -n o rm s we come to the main theorem s, basically stated as: 3.30) for v
compact spaces A , the £ -n o rm s all give A 4 ( X ) the w eak*-topology and make it compact, 3.33) if we restrict the £ -n o rm s to positive measures, these two topologies coincide for a separable space A , and the conver
gence is the same in both topologies even if X is non-separable, and 3.34) each £ -n o rm makes this space of positive measures complete.
Finally, we give a result about the density of the Dirac measures (The
orem 3.38) needed in C hapter 4.
In C hapter 4, we sta rt by defining an IFS with probabilities (IFSP) (Definition 4.1), following the studies of Barnsley, Demko, Hutchinson, Elton, Yan and others ([3], [4], [14], [7], [8]), as an IFS with a probability measure on the function space Q. For easier handling of the theory, we add some additional conditions. The probability transfers are then in
troduced (Definitions 4.5 and 4.7) and some general results about IFSPs are given.
In the next section we study stable IFSPs. The main theorem (4.15) says th a t the stable IFSPs have unique invariant measures and th a t these invariant measures are ” attractive” . For the proof, we use the fixed point theorem 2.12 in a similar way as Hutchinson and Barnsley used Banach’s Fixed Point Theorem to obtain the corresponding result for hyperbolic IFSs ([3], p. 365 and [14], p. 733).
We then use a different approach, by ” lifting” the IFSP to the code space f2°° and constructing an invariant measure there.
Finally, we show th a t the Random Reration Algorithm ([3], p. 91) will often work for stable IFSPs (Theorem 4.23).
A list o f sym bols is included (pp. 69-70).
1. P S E U D O M E T R I C S P A C E S . H A U S D O R F F P S E U D O M E T R I C . P s e u d o m e tr ic sp a c e s
In the theory of iterated function systems, the Hausdorff m etric plays an im portant role. However, in our study of stable IFSs it is convenient to generalize it to a pseudom etric, defined on all nonem pty sets in a space, instead of only closed, bounded sets. Also, we need a generalization of B anach’s Fixed Point Theorem , which is formulated for a pseudom etric space. Therefore, we sta rt with a brief study of pseudom etric spaces.
I should m ention th a t m ost standard monographs do not trea t pseu
dom etric spaces, sometimes called semimetric spaces. W ilansky’s book [27] is an exception. Therefore, much of the terminology below is my own invention, although the results stated are not new.
1.1 D EFIN ITIO N . A p s e u d o m e tr ic sp a c e is an ordered pair (X , d), where A is a set and d : X x X —► [0,oo] is a function, called a p s e u d o m e tr ic , such th a t for any x, y, z G X the following conditions hold:
(1) d (x ,x ) = 0 (2) d( x , y) = d( y , x )
(3) d( x , z ) < d( x , y) + d(y, z).
In addition, if d( x , y ) = 0 ==> x = y, then d is a m e tr ic and (X ,d ) is a m e t r ic sp a c e .
If d(x, y) < oo for all x, y G X i then d is a fin ite (pseudo)m etric and (A, d) is a f in ite ly ( p s e u d o ) m e tr ic sp ac e.
Note th a t we allow d(x, y) to be infinite. This is not standard, but convenient here. Topologically, this convenience does not m atter.
The following definitions are obvious generalizations from the m etric case.
1.2 DEFINITIONS. Let (A, d) be a pseudom etric space. For any subset A C A , d ia m (A ) = supx y£A d (x ,y ). A is b o u n d e d if diam (A ) < oo.
For any x G X , d(x, A) = infye>i d(x ,y ). For any x G X and any r G (0 ,oo], £ ( x , r ) = {y G X |d (x ,y ) < r}, B ( x , r ) = {y G X \ d ( x , y ) < r}
and for any nonem pty A C A , A r = {x G X \ d ( x , A ) < r}. For any sequence {xn }^°=1 in X 1 x n —+ x G X as n —► oo if, for every e > 0, B( x j e) contains xn for all bu t finitely many n we say th a t xn c o n v e rg e s to x, x is then a lim it of {xn }^cL1. If {rn }^cL1 is a sequence in [0,oo]
converging to 0 such th a t d(xn ,x ) < r n for all n, then we say th a t x n converges to x with s p e e d {rn }. { x n } ^ ! is C a u c h y if, for every e > 0, there is an N such th a t d(xm ,x n) < e for all m ,n > N. ( A ,d) is c o m p le te if every Cauchy sequence in X converges. (The converse is
always true, as in the m etric case.) For e > 0, an e - n e t in A is a set S C X such th a t S £ = X . X is to ta lly b o u n d e d if it has a ßnite e-n et for any e > 0. If (Y, d') is another pseudometric space and / : X —> Y is a bijection such th a t d '( /( x ) , /( y ) ) = d (x , y) for all x, y G X , then / is an is o m e tr y and we say th a t the spaces (A ,d ) and (Y ,df) are is o m e tric .
As in the m etric case, we observe th a t the family of balls B ( x i r) constitutes a basis for a topology on X . We always assume th a t X has this topology. This topology is not Hausdorff (in fact, not even To) unless ( X, d) is m etric. Also, unless (A, d) is metric, some convergent sequences have no unique limits, because d(x, y) may be 0 although x ^ y.
From this fact, it follows th a t for a non-m etric pseudom etric space (A , d), the following statem ents are not true:
(1) For every S C A , if every convergent sequence in S has a limit in S', then S is closed.
(2) Every complete subspace of A is closed.
(3) Every compact subspace of A is closed.
Instead of (1), the following is true:
(1’) If all limits of all convergent subsequences in S C A are in 5 , then S is closed.
The relation on A given by d(x ,y ) = 0 is an equivalence relation.
Let (A ) denote its quotient set, and let (x) denote the equivalence class containing x G A . We now make (A) into a m etric space by defining the m etric (d) on A by (d)((x), (y)) = d(x ,y ) for any (x), (y) G (A ). It is easy to check th a t (d) will be a well defined m etric on (A ).
1.3 DEFIN ITIO N . The m etric space ((A ), (d)) ju st defined is called the q u o tie n t s p a c e of (A, d).
We observe th a t if a sequence { x n} in A converges to x G A , then the equivalence class (x) is its set of limits. It follows th a t x n -> x in A if and only if (xn) —* (x) in (A ) as n —* oo and th a t { x n } is Cauchy if and only if {(xn )} is also Cauchy. Also, (A , d) is complete, compact, or totally bounded if and only if ((A ), (d)) has the same property.
If (A, d) is m etric, then ( A , d) is isometric to ((A ), (d)). The isom etry is given by x k (x). Hence, in th a t case, ((A ),(d )) can be identified with (A, d).
Now, let (Y, d') be another pseudom etric space and let / : A —> Y be a function such th a t d (x ,y ) = 0 = > d '( /( x ) , /( y ) ) = 0 for all x ,y G A . This condition is satisfied if / is continuous. This will induce a function ( /) : (A ) —► (Y) given by ( f ) ( ( x) ) = (/(* ) ) for every (x) G (A ). This function will be well defined.
1.4 D EFINITIONS. The function ( / ) ju s t defined is called the q u o tie n t f u n c tio n in d u c e d b y / .
/ is u n if o r m ly c o n tin u o u s if ( / ) exists and is uniformly continuous.
A family Q of functions from A to Y is e q u ic o n tin u o u s if its family of quotient functions exists and is equicontinuous.
It is easy to see th a t the quotient function ( /) exists and is continuous if / is continuous. If (A, d) and (Y, d') are m etric and if we identify their quotient spaces with the original spaces, as mentioned above, then ( / ) will be identified with / .
1.5 EXAM PLES. Let X be the complex plane C and put d( z , w) =
|Re z — Re w\, for all z , w G X . Then, (A, d) is a pseudom etric space.
The eqivalence class (z) is {u; G C |Im w = Im z}, and its quotient space (A ) can be identified with R, if we identify (z) G (A ) with Re z.
In this case, d(-, 0) is a seminorm on A . In general, i f p i s any seminorm on a real vector space A , then d( x , y) — p( x — y) is a pseudom etric on A .
Instead, let A be C \ {0} and put d (z , w ) = \ Arg(z/ w)\ for z, tu G A . Again, ( A ,d) is a pseudometric space. For z G (A ), (z) is the ray { rz |r > 0} and (A ) is isometric to the unit circle, which is m etrized by arc length.
We include a well-known result and the following definition.
1.6 D EFIN ITIO N . A sequence in a pseudometric space is called r e l
a tiv e ly c o m p a c t if each one of its subsequences has a convergent (sub)subsequence.
As in the m etric case, we observe th a t if A is complete, then it is compact if and only if it is totally bounded, and this holds if and only if every sequence in A is relatively compact.
1.7 LEMMA. I f (A n , dn ) is a sequence o f pseudometric spaces and if, for every n, { x 1j^}<^L 1 is a relatively compact sequence in X n , then there is a strictly increasing sequence o f positive integers such that for every n, the sequence converges in X n .
PRO O F: This is a standard diagonal argument.
By assum ption, there is a strictly increasing sequence of positive in
tegers such th a t converges in A i. This sequence has a subsequence such th a t converges in A2. Continuing in this m anner, we obtain a sequence of sequences of positive integers,
where each sequence is a subsequence of the preceding
one and converges for every n. For every n, the diagonal se
quence {kj}j?=1 is, disregarding the first n — 1 term s, a subsequence of { k ^ } j CL 1 and hence converges in X n . Thus, we can take kj = kj for all j .
We also include the following simple lemma.
1.8 LEMMA. I f { x n } is a relatively compact sequence in a pseudometric space (A , d) such that all convergent subsequences have a common lim it
X , then x n —► x as n —> oo.
PRO O F: Assume th a t {x n } satisfies the conditions but does not con
verge to x. Then, there is an e > 0 and a subsequence { x nk}^ L1 such th a t d{xnk, x) > £ for all k . This subsequence m ust have a convergent subsequence, which m ust converge to x. But this is impossible. Hence, x n —► x as n —► oo.
H a u sd o rfF p s e u d o m e tr ic
In this and the following section, (X , d) is always a complete pseudo- m etric space.
1.9 DEFINITIONS. 6 ( X ) denotes the family of all nonem pty subsets of X . For A , B G 6 ( X ) , we put S)(A, B) = inf{£ > 0|A C B t and B C A e}.
f) is called the H a u sd o rfF p s e u d o m e tr ic on 6 ( X ) .
£ (X ) denotes the family of nonem pty closed subsets of X . Ü)' is the restriction of S) to £ (X ). f)! is called the H a u sd o rfF m e t r ic on £ (X ).
23(X) denotes the family of nonem pty bounded subsets of X , we put
£2$(X ) = ^ (X ) fl 25(X ), and Ä(X ) denotes the family of nonem pty compact subsets of X .
£ (X ) denotes the family of singleton subsets of X .
1.10 REM ARKS. It is easy to verify th a t indeed is a pseudom etric on 6 ( X ) , so th a t ( 0 ( X ) ,i3 ) is a pseudometric space, and likewise th a t jo' metrizes £ (X ). We will always assume th a t these spaces have these pseudometrics. W hen confusion is impossible, we write f) instead of f ) ' . Also, for any A, A! G 6 ( A ) it is easy to see th a t (A) = {B G 6 ( X ) | B = A} and S ) ( A, A') = i5((A), (A7)). The latter Hausdorff pseudo- m etric is here defined on ((X ), (d)) with (A) denoting {(x)|x G A} and likewise for (A'). Notice th a t the notation is ambiguous at this point.
It follows th a t ( ( 6 ( X ) ) , (J5)) can be identified with ( £( X) , S) ' ) as well as ( ( 6 ( ( X ) ) ) , (Jo)) (with the latter notation).
Also it is easy to see th a t 2$(A ) is a closed subspace of © (A ) and th a t £2$(A ) is a closed subspace of £ (A ).
If S) is restricted to £ (A ), it is easy to see th a t (A , d) is isometric to (£ (A ),f]j). The isometry is given by x i-+ {x}.
Let Y be a nonem pty closed subspace of A . Then it is easy to see th a t S ( y ) is a closed subspace of © (A ).
The following result is well-known ([9], p. 37 and [3], p. 37), but we include a simple proof for it, which I have not seen before.
1.11 TH EO REM . The pseudom etric space © (A ) and the m etric space
£ (A ) are both complete.
PRO O F: Since ft (A, A ) = 0 for every A G © (A ), it suffices to prove the first statem ent.
So, let {An} be any Cauchy sequence in © (A ). It has a subsequence such th a t f t { Anki A n kJtl) < 2~k for all k. It suffices to show th a t this subsequence converges. Now, let 21 be the family of all se
quences {a:*} in A such th a t (i) Xk E A nk and (ii) d ( x jfe,£fc+i) < 2~k for all k.
21 is nonempty, in fact we will show th a t for every k and for every y E A nk there is a sequence { x j } G 21 such that xk = y .
To show this, pick k and y G A Uk. P u t Xk = y. Since fi{ A nk, A nk+1 ) <
2~k , we can take Zfc+i G Anfe+1 such th a t d{xkyx *+i) < 2~k . Likewise, we can take Xk+ 2 E A nk+2, such th a t d (x fc+i,^ +2) < etc. Sim
ilarly, if k > 1, we can ”go backwards” and find Xk- i E Anfc_j such th a t d ( x k - i j Xk ) < 2~(k~ 1) and so on down to an X\ G A ni such th a t d(x 1,2:2) < 2_1. Thus, we have constructed a sequence satisfying our conditions.
By (ii) and the completeness of (A ,d ), every sequence in 21 converges.
Let A be the set of limits of sequences in 21. Then, A ^ 0 . We will show th a t S)(Ank, A) < 22~ k for all k , which will complete the proof.
Pick an arbitrary y G A nk. Then, there is a sequence {#j} G 2t with Xk = 2/. Let x G A be one of its limits. By (ii), d( yyx) < 2l ~k. Hence, A nk C A 2i-k .
Conversely, pick an arbitrary x G A. Then, there is a sequence { x j } G 21 converging to x. By (i) and (ii), Xk E A nk and d( x k i x) < 2l ~ k.
Hence, A C (A nk) 2i - k .
Hence, f ) ( A nki A) < 2l ~k < 22~k, which completes the proof.
I have not been able to decide whether or not the set A in the above proof must be closed.
We include two more well-known results with proofs.
1.12 TH EO REM . I f X is m etric, then &( X) is a closed subspace o f
€ ( X ) .
PRO O F: Let { A n } be a sequence in Æ(X) which converges to a set A G £ ( X ) . We m ust prove th a t A is compact. To do this, let {£*}
be an arbitrary sequence in A. We m ust prove th a t {a?*} has a conver
gent subsequence, of which the limit will be in A since A is closed. For every n, we can select a sequence in A n such th a t d( x^, Xk)
< S){An , A) for all k . Since A n is compact, {££}£?_! is relatively com
pact. By Lemma 1.7, there is a strictly increasing sequence { kj } JL1 of positive integers such th a t for every n, { x 1£j }jCL1 converges and thus is Cauchy. We will show th a t is Cauchy and hence converges.
Pick e > 0. Take n such th a t h { A n , A) < e/ 3 and take N such th a t d{x1l ^ x 1l .) < e /3 for all i , j > N . Then, for i , j > N , d(xki1Xkj ) <
d(x k i , x l ) + + d ( x l . , x kj ) < 3e/3 = e.
It follows th a t is Cauchy, which completes the proof.
Our next result is the well-known ”Blaschke selection theorem ”, which we form ulate for a compact space. We borrow the proof essentially from Falconer ([9], p. 37).
1.13 TH EO REM . I f X is compact metric, then so is £ ( X ) = Â ( X ) . PRO O F: Assume th a t X is compact metric, hence totally bounded. We will show th a t £ ( X ) = Æ(X) is totally bounded. Since it is complete, it is therefore compact (and of course m etric).
P ic k £ :> 0 . We must find a finite £-net in Æ(A). Let S = { a q ,. . . , be a finite e /2 -n e t in X . Then, £>(£) is a finite e-n et in Æ(X). The reason is th a t for any A G Æ(AT), S fl A ej 2 G 0 ( 5 ) . It is then easy to see th a t iV(A, S fl A £/ 2) < e/ 2 < e.
W id e a n d n a r r o w lim it s e ts
Barnsley and Demko ([4], p. 245) define the attractor of an iterated function system as the set with the property th a t every neighbourhood of every point in the set intersects infinitely many sets in a certain se
quence of sets. Two questions th a t arise are: W hat happens if ”infinitely m any” is changed to ” all b u t finitely m any”? W hat is the relation to convergence in the Hausdorff pseudometric?
In this section, we investigate this problem in a general setting.
1.14 DEFINITIONS. Let { A n } be a sequence in 6 ( X ) .
The w id e lim it s e t of { A n ) } denoted w l { A n }, is the set of all x G X such th a t every neighbourhood of x intersects A n for infinitely many n.
The n a r r o w lim it s e t of {An }, denoted n/{A n }, is the set of all x G X such th a t every neighbourhood of x intersects A n for all but finitely many n.
Notice th a t, by definition, these limit sets of a sequence m ust always exist, although one or both may be empty.
1.15 P R O PO SITIO N . I f {A n } and {-Bn } are sequences in & ( X ) such that A n C B n for all n, and i f { A Uk }^(L1 is a subsequence o f { A n ) } then:
(1) n l { A n ) C w l { A n }.
(2) wl { A n ) C w l { B n ) and n l { A n } C n l { B n }.
(3) w /lA n jJ g ij C w l { A n } and n l{ A n } C n l{ A nk}%D=1.
(4) wl { A n } and n l { A n } are both closed.
(5) wl { A n } = w l { A n } and n l { A n } = n l { A n }.
PRO O F: (1), (2), and (3) are obvious. (4) and (5) follow from the fact th at if A, U C X with U open, then A fl U = 0 <=> A f ) U = 0 . 1.16 EXAM PLES Let X be C.
1) Let A n = {re*n |0 < r < 1} for all n. Then, wl { An } is the closed unit disc {z\ \z\ < 1} and n l { A n } = {0}.
2) Let B n = Uk=iAk for all n. Then, both w l{B n } and n l{ B n } are the closed unit disc.
3) Let Cn = {re*n |l < r < 2} for all n. Then, w l{C n } is the annulus { z |1 < \z\ < 2} and n l{C n } = 0 .
4) Let D n = {z\ \z\ = n} for all n. Then, w l { Dn } = n l { Dn } = 0 . We will now investigate the connection with the Hausdorff pseudo
metric.
1.17 TH EO REM . I f {A n } is a relatively compact sequence in &( X ) , then the following conditions are equivalent:
(1) { An ) converges in © (A ).
(2) wl { A n ) = n l { A n }.
Furthermore:
(3) I f A e £ ( X ) C & ( X ) is a lim it o f { A n }, then A = w l { A n ) = n l { A n ).
PRO O F: (1 )= > (2 ). Suppose th a t {An } converges. By Proposition 1.15 (1), it suffices to prove w l { A n ) C n l { A n }-
So, pick X G w l { A n } and a neighbourhood U of æ. Take e > 0 such th a t B ( x , e ) C U. Then, A n fl B ( x i e / 2) ^ 0 for infinitely many n.
Since {An } is Cauchy, there is an N ' such th a t J5(Am,A n) < e / 2 for all m ,n > N ' . Hence, there is an TV > N ' such th a t A jv intersects B ( x ye / 2) at a point xjy. Hence, for any n > N , there is an x n G A n such th a t d( xn j x ^ ) < e/2. Thus, x n G B ( x , e ) C U. It follows th a t w l { A n } C n l { A n }.
Next, we prove (3). Assume th a t the hypothesis in (3) holds. By the previous case, w l { A n } = n l { A n }. Hence, it suffices to show A = n l { A n }.
First, pick X G A, a neighbourhood U of ar, and an e > 0 such th a t B( x , e ) C U. Since $)(AnyA) —» 0 as n —* oo and B ( x , e ) C U % U intersects A n for all but finitely many n. Hence, A C n l { A n}.
Conversely, pick x G n l { A n }. Now, for any k , A n fl B( x, 1/ k) / 0 holds for all but finitely many n. Hence, we can find a strictly increasing sequence of positive integers {rik}<^L1 such th a t, for all k } A n C\ B( x i 1/ k) / 0 for all n > ra*. Then, we take a sequence {#*} such th a t Xk G A n h n B ( x , 1/ k) for all k. Then, x^ —► x as k —y oo.
Now suppose th a t x £ A. Then, d { x i A) > 0, since A is closed. Then there is an N such th at, for all k > N } S)(Ank, A) < d ( x , A ) / 2, and hence d( x k , A) < d ( x , A) / 2 . But, if k > m ax(A , 2/ d(x, A)), then d(x, Xk) <
d ( x , A ) / 2, hence d(x, A) < d ( x , x k) + d( x k , A) < d(x, A ) / 2 + d{ x i A ) / 2
— d ( x i A) which is a contradiction. Hence, x G A. It follows th a t n l { A n } C A, which completes the proof of (3).
Finally, we prove (2 )= > (1 ). So, assume th a t w l { A n } = n l { A n }.
By Lemma 1.8, it suffices to prove th a t any convergent subsequence { ^ n fc}jfcLi has the limit wl { A n } = n l { A n }. By the above cases, {Anfc}^_1 has the limit ' ^ i { Ank} ^ Ll = n l { A nk}^L1. But, by the assum ption and Proposition 1.15 (3), ^ /{ A rXfc}^L1 C w l { A n ) - n l { A n } C n l { A nk}kLi = w l { A nh }^_1. It follows th a t these inclusions must be equalities, and the conclusion follows.
1.18 COROLLARY. I f X is com pact, then the conclusions in Theorem 1.17 hold for any sequence in 0 ( Y ) .
PRO O F: Since X is compact, (X ) is compact m etric. By Theorem 1.13 and Remarks 1.10, £ ((X )) = (© ((A ))) = ( 6 ( A ) ) is compact, so every sequence in ( 6 ( A ) ) is relatively compact. Hence, every sequence in
© (A ) is relatively com pact. Thus, Theorem 1.17 applies.
From the above theorem and corollary, the following result can be derived:
1.19 COROLLARY. Suppose that { A n} is a monotonie (i.e m onoton- ically increasing (An C A n+\ for all n) or m onotonically decreasing (A„ + 1 C An for all n )) and relatively compact sequence in S ( A ) .
In particular} i f X is com pacty {An } could be any m onotonie sequence in &( X ) .
T hen , { A n} is converges in & ( X ) to A = UnA n i f it is m onotoni
cally increasing, and it converges to A = C\nA n i f it is monotonically decreasing.
PRO O F: It is easy to see th a t, in both cases A = w l { A n } = n l { A n }.
Hence, Theorem 1.17 or Corollary 1.18 applies.
2. S T A B L E I T E R A T E D F U N C T I O N S Y S T E M S . D e fin itio n s a n d e x a m p le s
2.1 D EFIN ITIO N . An i t e r a t e d f u n c tio n s y s te m (IFS) is an ordered triple (X , d, i?), where (A , d) is a complete pseudometric space and f i is a nonem pty family of functions s : X —► X .
If f i consists of one single function w, we write (X ,d , w) instead of (X, d, {w}). We say th a t (X , d, w) is a sim p le IFS.
The word ” iterated ” indicates th a t we are interested in how these functions behave when they are applied several times.
There is a growing theory of IFSs today, and our investigation is a contribution to this. We will generalize the theory of hyperbolic IFSs developed by Hutchinson, Barnsley and Demko, Le wellen, and others ([14], [4], [16], [13]). In particular, much of the investigation in this chapter will overlap with Lewellen’s work [16].
2.2 DEFINITIONS. Let (X ,d , fi) be an IFS. For any n, f i n denotes the family of ordered n -tu p les ( $ i , . . . , sn ) of functions in fi. i?°° denotes the family of infinite sequences {sn }^Li of functions in i?. fi : © (X ) —►
6 ( X ) denotes the m ap given by fi( B ) = Usen S( B) for all B G ® (X )- Note, in these definitions, s, s i, $2> etc. denote function variables.
We choose this way of notation rather th an indexing the functions.
2.3 D EFIN ITIO N . Let (X ,d ,f2 ) be an IFS and let A G 6 ( X ) . For any n > 0, the num ber sup{d2am ({s! o • • • o sn ( A ) |( s i,. . . , sn ) G f2n })} G [0, oo] is denoted A n (A).
We use the convention th a t the case n — 0 in this definition corresponds to the set A. Thus, A$( A) — di am( A) .
2.4 DEFINITIONS. Let (X ,d , Q) be an IFS. Let z 0 G X and let T be a family of nonempty, closed subspaces of X , which all contain x 0l such that:
(1) For any B G 93(X) such th a t x Q G B , B C A for some A G T.
(2) For any A G X and any s G s(A) C A.
(3) For any A G T, limn_oo A n (A) = 0.
Then, (X, d, i?) is called (T, x 0) - s ta b le .
(X, d, fi) is called T - s t a b l e if, for every x G X , it is (T, y )-stable for some y G B ( x , oo).
If it is {X }-stable, it is called to ta lly s ta b le and if it is T -stable for some T C ® (X ), it is called b o u n d e d ly s ta b le .
(2) above is equivalent to the following condition: f2(&(A)) C ©(>1) for all A 6 T.
Our definition is adm ittedly technical. Actually, the only cases which we are really interested in are totally and boundedly stable IFSs. B ut, in the proof of the fundam ental Theorem 2.15, we need some auxiliary spaces which may be neither. Since we want a theory as general as possible, we choose the above definition.
The notion of a totally stable IFS was introduced by Wallin and Karls
son ([26], p. 11). They used it to reformulate a classical theorem about continued fractions.
2.5 PRO PO SITIO N . I f ( X , d , Q ) is a ( %, x ^ - s ta b le IFS, then the fol
lowing statem ents hold:
(1) I f d is finite, then (X , d, i?) is %-stable.
(2) I f X G Ï , then (X , d, 12) is totally stable.
(3) I f A £ T and if d and the functions in Q are resticted to A , then (A, d, 12) is totally stable.
(4) I f B £ £{A) for some A £ %, x$ £ B and Ù( B ) C B, then (X , d, 12) is (T U {B} , #o)-stable.
(5) I f B is as in (4) and i f d and the functions in Q are restricted to B , then ( B, d, f 2) is totally stable.
PRO OF: (1), (2), (3), and (4) are obvious, and (5) is a consequence of (3) and (4).
In the sequel, with B as in (5) above, we write ( B, d, 12) and tacitly understand th a t d and the functions in i? should be restricted to B.
The simplest case of all is the following:
2.6 COROLLARY. Let ( X, d, 12) be an IFS. L et x q £ X be any point, and let T be a fam ily o f nonem pty, closed subspaces o f X satisfying (2) o f Definitions 2.4.
Consider the following conditions:
(1) ( X, d , 12) is (T ,x o )-sta b le . (2) (X, d , 12) is ‘Z-stable.
(3) (X , d, 12) is totally stable.
(4) (X , d, 12) is boundedly stable.
I f X is bounded, then (1) (2) = > (3) 4=>* (4). In addition, i f
xq £ A for all A £ T, then (1) (2), and i f also { X } £ T, then the four conditions are all equivalent.
PR O O F : These are easy consequences of Definitions 2.4 and Proposition 2.5.
Our next proposition gives an im portant example, which has been extensively studied for finitely many functions ([14], [4], [3]), but also for infinitely many functions [16].
2.7 D EFIN ITIO N . A h y p e r b o lic IFS is an IFS ( A , d ,ß ) with a finite m etric d , such th a t there is a constant c G [0,1) such th a t d(s(x), s ( y)) <
cd( x, y) for all s G 12 and for all x, y G A .
2.8 REM ARK. By induction, it follows th a t A n ( B) < cn di am( B) for all B G ® (A ) and all n > 0.
2.9 PR O PO SITIO N . L et (A, d, 12) be a hyperbolic IFS. Assum e th a t, for some x Q G A , U5e/2{s(x0)} is bounded. T hen, (A , d, 12) is boundedly stable.
PRO O F: Let c be as in Definition 2.7 and let xo G A satisfy the assum p
tion. P u t d = sup56ß d (s(z 0), xo) < oo. P u t X = { 5 ( s 0, Ä )|d/(1 - c) <
Ä < oo}. Then, X is a family of closed bounded subspaces of A , all containing xo. We will show th a t (A , d, i?) is (X, a?o)~stable, hence it is boundedly stable by Proposition 2.5 (1).
In Definitions 2.4, (1) is obvious and (3) follows from Remark 2.8.
To prove (2), take R G [d/( 1 — c),oo). Then, d + cR < R. Hence, if X G B (xq, R ) and s G 12, then s(æ) G B( s ( x o ) i cR) C J9(:ro,i7). Hence, s(J5(xo,Ä )) C B ( x o i R) for every s £ 12, so (2) holds.
Of course, if f2 above is finite, then the boundedness condition above is satisfied, bu t if 12 is infinite, it may fail.
As an example of the latter, take A = R with the usual metric, and put 12 = > 0}. This IFS is obviously hyperbolic, but since, for any x G A , is unbounded, the condition is not satisfied, and it is easy to see th a t (A , d, 12) is not boundedly stable.
The theory in the following sections is developed for stable IFSs (of different kinds), but for hyperbolic systems, the theory becomes sim
pler, as we will see. In order to justify our study of stable systems, we should therefore find examples of stable non-hyperbolic IFSs. As we have remarked, an example has been studied in the theory of continued fractions ([26], p. 15). Also, it is not hard to find simple stable systems, some of these are studied in the following sections.
Here, we give a rather im portant, nontrivial class of examples.
2.10 EXAM PLE. We will need some prerequisites.
We define a function / : [0,oo) —► [0,oo) as f ( t ) = m ax ( t - Ì ± Ì t ( l - , t I 1 -
1 + f ' V (1 +
i f) ) '
f ( i ) is continuous and strictly increasing on [0,oo), which could be ver
ified by differentiation, for example. Also, for all 2 > 0, 0 < f ( t ) < t.
It follows th a t, for any to > 0, the sequence { / n (*o)}£Lo decreases strictly to a limit s > 0. B ut if s > 0, then f ( s ) < s, and by the continuity of / , f ( t ) < s for all t in some interval [s, s + e) for some
£ > 0. But this interval contains / n (^o) for some n. Hence f n+1(to) < s , which is impossible. Hence, 5 = 0.
Now, we tu rn to our example.
Let X be a Banach space and take any A G ® (A ). To each a G A, we assign an arbitrary norm preserving map Ta on X (i.e. an isometry on X fixing 0, which is linear by M azur-U lam ’s Theorem ([2], p. 166)), and a map wa on X defined by
“ ‘W = T‘ ( 1+Ì | l ! a | | ) +a
for all X G X . P u t Q = { wa\a G A}. Pick xo G A. P u t X = {jB(xo, R) \ Zdi am( A) + 1 < R < oo}.
We will show th a t (X, || • — • ||>f2) is a (X, xo)-stable non-hyperbolic IFS. Thus, it is boundedly stable by Proposition 2.5 (1).
Since, for any a G A and any x G X \ { a } , ||i/;a (x) — wa(a ) ||/||£ — a|| = 1/(1 -1- \\x — a||) - ^ l a s x - ^ a , the system is not hyperbolic.
X is obviously a family of nonempty, closed subspaces of A , of which all contain xo and satisfy (1) of Definitions 2.4.
To prove (2), take any R G [3di am( A) + l,o o ). Pick a G A and x G B ( x 0, R) arbitrarily. We will show th a t wa(x) G B ( x QiR). Hence, (2) will hold.
If \\x — a\\ < 1, then
||u;a(x) - Zoll < ||^ a (z ) - a|| + ||a - x0|| < ||w a(z) - Wa(a)|| + di am( A)
< ||£ — a|| + d i am( A) < 1 + di am( A) < R.
If, instead, ||x — a|| > 1, then
||u>a(*) - x Q\\ < ||^ a (z ) - a\\ + ||a - x 0\\ < ^ ^ + di am( A) s | | » - . o H + | | . o - . | | + Jìam{A) s R + Zdiam(A) ^ R
Z Zi
Thus, in both cases, wa(x) G B ( x o i R).
Finally, we will show th a t, for any B (xq, R) G X, A n (B(xo, R)) <
f n (<2R) for all n, with / as above. Hence, (3) will hold.
By induction, using th a t / is strictly increasing on [0, oo), it suffices to show th a t for any a £ A and an y x , y G X , ||wa(x) — wa(y)\\ < f ( \ \ x —y\\).
So, pick a G A, x, y G X , put d = \\x — y ||, and assume, w ithout loss of generality, th a t ||x — a|| > ||y — a||. Then,
x — a y — a
I K W - « . . W I I = 1 + | | l _ a | | i + | | „ - . | | 11(1 + t o - « | | ) ( * ( t o - «II - t o - « I I M » - «111
< d
(1 + I k — a | | ) ( l + || y — a|
1 + 2||j/ — a||
„1 + II* -
°ll(1 + I k - «IDC1 + llv
< m in ^->1 + l k - a ll H ( , _
- m { 1
+ | | x - a | | ’V (l + llw-alDvJ
If ||y — a|| < d/3 , then ||x — a|| > 2d/3. Hence, in this case
i + I k — a ll ' " 1 + “ On the other hand, if ||y — a\\ > d/3, then
V (i + to-«ll)Vs ^ (i + g)V
since the function 1 — t 2/ ( 1 -f-1)2 decreases on [0,oo). Hence, in both cases, ||u;a(x) — wa(y)\\ < /( d ) , and we are finished.
A t t r a c t o r s fo r s ta b le IF S s
In this section, we prove a generalized version of B anach’s Fixed Point Theorem , and then use this version to prove the existence of an attractor for a stable IFS.
2.11 LEMMA. Let (X , d, Q) be a totally stable IFS and let {sn } G ß ° ° . For any n and any y £ X , p u t S n (y) = si o • • • o sn (y), ( S0(y) = y)-
Then, there is a point x G X such that:
(1) For any B G &( X ) , the sequence { S n ( B ) j converges to {x} (in
& ( X ) ) with speed {zln (X )}.
(2) For any y £ X , the sequence {5n (y)} converges to x (in X ) with speed (Z\n (X )}.
PRO O F: The sequence {5„(X )} is monotonically decreasing. It is then easy to see th a t for any N and any m ,n > N f ) ( Sm ( X) , S n ( X ) ) <
d i a m( S N ( X ) ) < An( X) . By (3) of Definitions 2.4, {5„(X )} is Cauchy.
Hence, it converges in 6 ( A ) , by Theorem 1.11. By Corollary 1.19, a limit is C = By (^) °f Definitions 2.4, diam {C ) = 0.
C ^ 0 , so pick X G C. ______
Now, for any B G © (A ) and any n, S n ( B) C S n ( X) . Since x G S n ( X) , we Set Ä (5 n (J3),{®}) < di am( Sn ( X) ) < A n ( X) . Hence, (1) follows.
For any y G X and any n, { S n (y)} G £ (A ). By (1) and Rem arks 1.10, (2) follows.
2.12 TH EO REM . Let ( X , d , w ) be a simple (X, zo)-stable IFS. T h en , there is an x G such that the following statem ents hold:
(1) For any yo G LUesA, the sequence {wn (yo)} converges to x with speed { A n (A) } i f y0 G A G X.
(2) I f w(y) = y for some y G UAe Z ^ , then y G (x).
(3) If, for some A G Ï , d \ ^ XA *s a m etric and w\a is continuous, then x is a unique fixed point o f w in Ua£%A.
(4) I f d is finite or i f (X , d , w ) is totally stable, then in (1)— (3), X m ay be su b stitu ted for
PRO O F: By Proposition 2.5 (3), (A, d, w) is totally stable for every A G T. Hence, for every A G X, there is an x a G A such th a t for any y G A, the sequence {wn (y)} converges to xa with speed {zln (A)}, by Lemma 2.11 (2). This holds in particular for y = xq, since by definition, xo G A for all A G T. Hence, d ( x A1XAi) = 0 for all A ,A ' G X. By definition, each A G X is closed. Hence, we can take a common xa for all A. P u t x as this common x a• This gives (1).
To prove (2), assume th a t w(y) = y for some y G Ua ç%A. Then, by induction, wn ( y) = y for all n. Hence, w n(y) —► y as n —+ oo. By (1), y g {x).
I f the assum ptions of ( 3 ) hold, then ( 1 ) gives w( x) = lim n _ o o wo wn (x)
= limn_ 00 wn (x) = x , and by (2), the fixed point x is unique in U So, (3) holds.
If d is finite, then (1) of Definitions 2.4 gives Ua^ A = X . Also, the same is obviously tru e if ( X , d , w) is totally stable. Hence, (4) holds.
2.13 REM ARK. Using Proposition 2.9, we see th a t the classical B anach’s Fixed Point Theorem is a special case of Theorem 2.12.
In the same way as B anach’s Fixed Point Theorem is used to prove the corresponding result for hyperbolic IFSs with finitely many functions
([3], p. 82 and [14], p. 728), we use Theorem 2.12 to prove our next theorem.
2.14 D EFIN ITIO N . Let ( A , d ,ß ) be an IFS. A set B E © (A ) is called 12—in v a r ia n t if f2(B) = B.
If it is clear from the context which Q we mean, B is ju st called in v a ria n t.
2.15 TH EO REM . L et (A, d, 12) be a (T, zo)-stable IFS. T hen, there is a set S E C (rU e îA ) such that:
(1) For any B E L U eî© (A ), { Ùn ( B) } converges to S (in © (A )^ with speed {zAn (A)} if B E ©(A).
(2) If, for some B E L U eî® (A ), ß (-ö) = B , then B = S .
(3) If, for some A E X, ß (£ ( A )) C £(A ) and i?|e(A) IS continuous, then S is a unique closed f1-invariant set in LUe3;©(A).
(4) I f (A , d , Q) is totally stable, then, in (1)— (3), © (A ) m ay be su b stitu ted for LU€s;©(A).
PRO O F: P u t X = {© (A )|A E X}. We will show th a t the simple IFS ( © ( A ) ,ij,i2 ) is (X, {xo})-stable with zAn (© (A )) < A n(A) for all n and
all A E X. _
By Remarks 1.10, each ©(A) E X is a closed subspace of © (A ) con
taining { x Q}.
Now, let 21 be a bounded subset © (A ) containing {#o}- Then, for any B E 21, S ) ( B , { x 0j ) < d ia m (21) < oo. Hence, B C B(xo, diam (2l)), which is bounded. Since B(xo, diara(2l)) is contained in some A E X, H E ©(A). It follows th a t 21 C ©(A) E X. Hence, (1) of Definitions 2.4 holds.
(2) of Definitions 2.4 follows easily from the corresponding property of the (X, xo)-stability of (A, d, Q).
Next, pick A E X, an n, and two sets B , B ' E !2n (© (A )). Then, there are two sets E , E ' E ©(A) such th a t f2n (E) = B and Qn ( E /) = B ' . Now, pick y E B. Then, there is a z £ E and an n -tu p le ( s i , . . . , s n ) E i?n such th a t s io- • *osn (z) = y. Pick z ' E E ' and put y' = sio- • 'Osn (z').
Then, y' E B ' and d(y, y') < zln (A). It follows th a t 5 C B'A ^ Ay By the same argum ent, B f C # 4 n(A)- Hence, Ü)( B , B f) < A n {A). It follows th a t d i a m( Qn (&( A) ) ) < A n (A) and (3) of Definitions 2.4 holds.
We have now proved the statem ent above.
We can now apply Theorem 2.12 to the simple (7 , {ro})-stable IFS (© (A )), f), i?). Since to every B E © (A ) there is a unique closed C E
© (A ) such th a t S}( B, C) = 0, namely C = B, and since all ©(A) E X
are closed subspaces of © (X ), (1) follows from (1) of Theorem 2.12 as does (2).
If the assum ptions of (3) hold, then on the m etric quotient space (£ (A ),i} ) (see Remarks 1.10), the quotient function (f2) exists, is con
tinuous, and can be identified with i?| c(A)-
Then, it is easy to see th a t (£(A ), f), ß|c(>i) ) 1S a simple, totally stable IFS.
Now, (3) follows from (3) of Theorem 2.12 and (2).
Finally, (4) is obvious.
2.16 PR O PO SITIO N . Let (A ,d ,f2 ) be a (X ,x 0)-stable IFS. If, for some A £ X, the fam ily Q \a = iS equicontinuous, then I2|6(a) iS uniformly continuous.
PRO O F: Assume th a t i? | , 4 is equicontinuous for some A £ X. Pick e > 0. Then, there is a S > 0 such th at, for any y , z £ A with d (y , z) < 6, d(s(y), s ( z )) < e/ 2 for all s £ 12. Hence, for any s £ 12 and any B, B f £
©(A) w ith f ) ( B , B ' ) < 6, we get s( B) C s ( B' ) £ / 2 and s( B' ) C s(J5)f / 2- It follows th a t S){fi(B), Q ( B 1)) < e/ 2 < e. Hence, is uniformly continuous.
We now get the corresponding result for hyperbolic IFSs as a corollary (see [3], p. 82 and [14], p. 728): :
2.17 COROLLARY. Let ( X, d , f 2 ) be a hyperbolic IF S, with c as in Definition 2.7, which satisfies the assumption o f Proposition 2.9. A ssum e that for some C £ £ (A ), i?(C (C )) C £ (C ) (this holds i f f2 is ßnite, C is com pact, and s(C) C C for all s £ 12, for example).
Then there is a unique S £ £25(X ) such that f2(S) = S. Furthermore, for this S , all sequences { Ù n ( B) } with B £ © (A ) converges to S in
© (A ) with speed { cn D }, for some D depending upon B .
PRO O F: By Proposition 2.9, (A, d, 12) is boundedly stable, th a t is, it is T -stab le for some % C ® (A ). For each A £ %, Q\a is equicontin
uous (with Q\a as in Proposition 2.16). Hence, f2|©(A) is uniformly continuous by Proposition 2.16.
Now, with C as above, pick xq £ C, remove from T all sets A such th a t xq £ A , and for some A still in T (such an A exists, since {xo} is bounded), adjoin CC\ A to T, using Proposition 2.5 (4). Then, (A, d, f2) is (X, zo )-stable with this new X.
Now the assum ptions of (3) of Theorem 2.15 hold. Hence, using th a t theorem and Rem ark 2.8, we obtain the desired results.
2.18 D EFIN ITIO N . T he set S in Theorem 2.15 is called the a t t r a c t o r of the (T, x o)-stable IFS (X, d, i?).
In the next section, we will investigate attractors and sets with Hausdorff- pseudodistance 0 from them .
A d d re s s in g a n d e x a m p le s
First, we take a closer look at the Theorems 2.12 and 2.15, and point out w hat they say and don’t say. This will also show the relevance of the technical Definitions 2.4.
2.19 EXAM PLE. Let (X ,d , w) and ( X ',^ ',« / ) be two simple totally stable IFSs such th a t X fl X ' = 0 , d and d' are metrics, and w and wf are continuous.
We now define d" as the only pseudometric on X " = X U X ' such th a t dn \ x x x = dy dn I x ' x X ’ = d! and d ! \ X x X ') = {oo}. We also define w n as the unique function on X " such th a t wn \x = w and w'^x* = w*• w n is then continuous.
Then, (X " , d", it/') is a simple IFS which is both {X }-stable and {X '}-stab le, bu t not totally stable. Theorem 2.12 gives two different fixed points in X " , x £ X and x ' G X ', with d " ( x , x ') = oo, such th a t {w"n (y0)} converges to x for all y0 G X and {u//n(yo)} converges to x' for all y'Q G X .
This is a reason why we need a family T and a point Xo in Definitions 2.4. We need a point to specify in which finitely-m etric subspace we are (compare Proposition 2.5 (1)).
Now, the reader might think: Infinite pseudometrics ju st seem to com
plicate the theory. Why allow them in the first place?
The answer is th a t we want to define the Hausdorff pseudometric not only on ® (X ) but also on © (X ) for unbounded spaces X . Otherwise we cannot use Theorem 2.12 in the proof of Theorem 2.15, for example in the case of a totally stable IFS on an unbounded space X .
2.20 EXAM PLE. This example is perhaps more n atural than the pre
ceding one.
Consider (R, | • — • |, w), where | • — • | is the usual metric on R and w(x) = x / 2 for all x G R. This is a simple hyperbolic IFS. Hence, it is boundedly stable, by Proposition 2.9 and the rem ark succeding it. {0} is its a ttra c to r and it is invariant. B ut R is closed and invariant too. This does not contradict (2) and (3) of Theorem 2.15 since R is not bounded, and therefore, R ^ U i f T C ^3(X). Again, this shows the necessity of considering a family T.
2.21 EXAM PLE. Consider the first pseudometric space (C ,d) of Exam
ples 1.5. P u t w (x + iy ) = \ x -f i(y + 1) for all x , y £ R. Then, w is continuous and (C,d, tu) is a simple, boundedly stable IFS with the im aginary axis as the (invariant) attracto r. B ut, w has no fixed point in C. This does not contradict (3) of Theorem 2.12 since d is no m etric.
2.22 EXAM PLE. Let X be [0,1] with d as the usual m etric. P u t iu(0) = 1 and w(x) = x / 2 for x G (0,1]. Then, ( X , d , w) is a simple totally stable IFS with {0} as the attra cto r. But, neither {0} nor any other set is invariant. The reason is th a t w is neither continous at 0, nor does it take closed sets to closed sets in any neighbourhood of 0, so the hypotheses of (3) of Theorem 2.15 are not satisfied.
I have not been able to decide whether or not the continuity assum ption of (3) of Theorem 2.15 can be dispensed with.
However, the assum ption of taking closed sets to closed sets cannot:
2.23 EXAM PLE. Let X be the closed unit disk in C with d as the usual metric. P u t f i = {(z + e2l*d)/2 \0 G Q}. Then, ( X , d , f i ) is a hyperbolic and totally stable IFS.
P u t Y = X ° U {e2i*$\0 G Q}. We will show th a t f i ( X ) = Y = fi{ Y ) . Then, Y is invariant and, by (2) of Theorem 2.15, X is the attra cto r although X is not invariant. f i is uniformly continuous, but it m aps the closed set X onto the non-closed set Y .
First, put z — e2lir6, with 0 £ Q. Then, z ^ s ( X ) for all s £ fi. Hence, f i ( X ) c Y .
Hence, it suffices to show Y C f i ( Y ) . Take z = e2t*s with 6 G Q.
Then, (z + e2iwe)/2 = z. Hence, z G f i ( Y ) . Next, since (1 -f el7r)/2 = 0, then 0 G f i ( Y ) .
Finally, take z = re2tn9 with r G (0,1) and 6 G R. If r > 1/2, put 0' = 6 and r ' = 2r — 1, and if r < 1/2, put 6' = 0 -h 1/2 and r ' = 1 — 2r.
P u t z' = r'e2i*e' G X ° . P u t «'(z") = (z" + e2#>*)/2 for all z" G X ° . Then, z = s '( z ;) € « '(X 0). Since s '( X ° ) is open, it contains re2l*<f> for some <j) G Q. P u t <f>f = <j) if r > 1/2 and put = </>+l/2 if r < 1/2. P u t s(z") = (z" + e2i^ ) / 2 for all z" G X . Then, s G ß , s(r'e 2i* / ) = re2iir<t>
and by symmetry, z G s(X °) C «(Y). It follows th at Y C f i ( Y ) .
2.24 EXAM PLE. Let X be [0,1] with d as the usual m etric. P u t f i = { | , Then, (X ,d , fi) is a hyperbolic and totally stable IFS with X as invariant attracto r. However, [0,1) and (0,1] are invariant too (their closures are the a ttra c to r [0,1], of course). Hence, there can be several invariant sets, but at most one is closed, namely, the attracto r.
Instead of finding the attra cto r by constructing a simple IFS and then using Theorem 2.12, as we did in the proof of Theorem 2.15, we can use
Lemma 2.11 directly.
We will generalize the addresses studied by Barnsley and Lewellen ([3], ch. 4, [16], ch. 2) for hyperbolic systems.
2.25 D EFINITIONS. Let ( X , d ,f 2 ) be a (X, x 0)-stable IFS. We define the a d d r e s s m a p A d r : fi°° —► (X ) by p utting A d r ( { s n }) = limn_oo (si o • • • o sn (xo)) for all {sn } G ß ° ° . The convergence of this sequence is guaranteed by (2) of Lemma 2.11 and (3) of Proposition 2.5. If x G A d r({sn }) for some {sn } G ß ° ° , we say th a t {sn } is an a d d r e s s of x and th a t x is an a d d r e s s a t e of {sn }- The set of all addressates, denoted Adr(I2°°) (somewhat im properly), is called the a d d r e s s a t e s e t.
If d is a m etric, then we consider X instead of (X ) as the target of the address map. In this case, every {sn } € !?°° has a unique addressate.
2.26 REMARK. In this definition, we chose xo as the starting point for the sequence above. But by Lemma 2.11, we could have chosen any y £ Ua ç zA w ithout changing the result since each (A ,d, i?) is totally stable. In fact, by th a t lemma, we could have started with any B G LU g $ 6 (A ) and then taken a corresponding sequence in 6 ( A ) instead and then arrived at an equivalence class (x) of addressates (a singleton set if d is a m etric).
2.27 PR O PO SITIO N . Let (A, d, Q) be a (X, x 0)-stable IFS with attrac
tor S. Then, Adr(I2°°) = S.
PRO OF: Pick A G X. T hen (A ,d, 12) is totally stable. We will show th a t n/{i?n (A)} = Adr(I2°°). Since the convergent sequence { Q n (A)}
in G (A) is relatively com pact, the conclusion then follows from (3) of Theorem 1.17.
So, pick x G n l{ Ö n (A )} and pick e > 0. Take N such th a t A n (A) <
e/2 and B ( x , e / 2) intersects i?n (A) for all n > N . Then, for any n > N , there is an ( s i , . . . , s n) G Lin and a y G A such th a t s\ o • • • o sn (y) G B ( x , e / 2). Now, lengthen ( s \ , . . . , s n) to a sequence {s*} G ß ° ° . By Lemma 2.11 (2), (d)((s!o- • -osn (y)), A d r({sk })) < A n {A) < e/2. Hence, B { x , e ) n A d r ( Q ° ° ) ± 0 .
It follows th a t n l{ Ù n (A )} C Adr(I2°°).
Conversely, pick x G Adr(I2°°) and e > 0. Then, there is an {sn } G Q°° such th a t {$i o • • • o s n (x 0)} converges to x with speed {zln (A)}.
Take N such th a t A n {A) < e for all n > N . Then, for any n > N , B (x , e) fl Q n {A) / 0 . ________
Using Proposition 1.15 (4), A dr(fi°°) C n l{ Q n (A)} follows, which completes the proof.
