A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS

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1 Acta Math. Hugar., 2007 DOI: /s A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720 Szeged, Hugary s: stacho@math.u-szeged.hu, lszabo@math.u-szeged.hu (Received Jauary 12, 2007; revised Jue 22, 2007; accepted July 2, 2007) Abstract. We prove that the family of all ivariat sets of iterated systems of cotractios R N R N is a owhere dese F σ type subset i the space of the o-empty compact subsets of R N equipped with the Hausdorff metric. A iterated fuctio system (IFS for short) is a fiite collectio (T 1,..., T ) of weak cotractios of a metric space X. By a weak cotractio we mea a mappig T : X X such that d ( T (x), T (y) ) < d(x, y) for all x, y X, where d is the metric o X. A subset A X is called a ivariat set for the system if A = T 1 (A) T (A). Give a real umber 0 < r < 1, a mappig T is called a r-cotractio if d ( T (x), T (y) ) < r d(x, y) for all x, y. The term cotractio without adjectives refers to r-cotractio for some 0 r < 1 accordig to the most widespread termiology i the literature. It is kow that if the space X is complete the, for ay IFS of cotractios, there exists a uique oempty compact ivariat set (see [1], [3], [4]). A geeral IFS may admit o ivariat sets, however, it is ot hard to see that if it has a compact ivariat set the this must be uique. It is also kow that ay compact set i the euclidea spaces R ca be arbitrarily closely approximated (i Hausdorff distace) by ivariat sets of suitably chose IFSs; Supported by the Hugaria research grat No. OTKA T/ Key words ad phrases: iterated fuctio system, fractal, ivariat set, weak cotractio Mathematics Subject Classificatio: 49F /$ c 2007 Akadémiai Kiadó, Budapest

2 2 L. L. STACHÓ ad L. I. SZABÓ see [1], [3]. Ideed, fiite sets are dese amog the compact subsets of X i Hausdorff distace ad the fiite set A = {a 1,..., a } is the ivariat set of the IFS (T 1,..., T ) with oe poit shrikigs T k : X {a k }. It is a atural questio to ask whether it is true that ay compact set is actually the ivariat set of some IFS. We show below that the aswer to this questio is o: we costruct compact sets i R that are ot ivariat sets for ay IFS. Startig from a oe-dimesioal example, oe ca obtai may more. We prove that the ivariat sets form a owhere dese set, which is of F σ -type if oly IFS of r-cotractios are take, amog the compact subsets of R, thus solvig a problem raised by Edgar [2]. Lemma 1. There is a compact subset A of R such that A is ot the ivariat set of ay iterated fuctio system. Proof. Defie recursively the idex sequece 1, 2,... with the relatios 1 := 1, k := (k + 1)( k 1 ) for k > 1. The there is a uique decreasig sequece a 0 such that, by settig 0 := 0, for each idex k 1 we have a k = 2 1 k, a i a i 1 = δ k := 2 1 k ( k k 1 ) 1 if k 1 < i k. I terms of this sequece, defie A := {0} {a : = 1, 2,...}, A k := {a i : k 1 < i k }, a := 0. Cosider ay weak cotractio T : A A ad let i be ay idex with k 1 < i k. Let also T (a i ) = a m ad T (a i 1 ) = a, where m, N { }. The we have a a m = T (a i ) T (a i 1 ) < a i a i 1 = δ k. It follows that either, m k or = m. Therefore (1) either T (A k ) {a k } j>k A j {0} or T (A k ) = {oe poit} = T {a k }. O the other had, sice weak cotractios are cotiuous, T (a ) T (0) as. Sice 0 is the oly accumulatio poit i A, either we have T (0) = 0 or 0 T (0) = a except for fiitely may idices. If T (0) = 0 the, give ay idex, the assumptio T (a ) T (0) < a 0 implies T (a ) = a +d() for some d() > 0. Thus we have also the alteratives (2) either T (A) is fiite or T (0) = 0 ad T (a k ) j>k A j {0} for all k.

3 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS 3 Assume that, i cotrast with the statemet of the lemma, A is the ivariat set of some IFS (T 1,..., T N ) cosistig of weak cotractios of A. Without loss of geerality, we may also assume that 0 = T 1 (0) = = T M (0) ad T m (A) is fiite for ay m > M, that is M<m N T m(a) A 1 A K 1 for some idex K with K > N. The, usig (1) ad (2) we get However, = A K = A K M m=1 j N { } N m=1 T m (A) = #A K = K K 1 > K j<k M m=1 [ Tm (A j ) A K ] M j K j<k [ Tm (A) A K ] m=1 j<k T m (A j ). #A j > M j<k #A j # M m=1 j<k [ Tm (A j ) ], a cotradictio. Usig the set just costructed i R, we ca obtai examples i R. Lemma 2. Let A [0, 1] be the set costructed i Lemma 1. Suppose ε > 0, u R N is a uit vector ad Q = {q 1,..., q m } is a subset of R N such that q l q 1, u > ε (l = 2,..., m) ad defie B := Q + εau. The B is ot the ivariat set of ay iterated fuctio system of weak cotractios. Proof. Let B l := q l + εau = {q l + εαu : α A}. Observe that B = m B l, B l = S l (εa) where S l : R R N, S l (λ) := q l + λu. l=1 O the other had, we have B l, u (:= { b, u : b B } ) = q l, u + εa for every idex l ad, by assumptio, B l, u q 1, u > ε (that is b q 1, u > ε for all b B l ) if l > 1. Hece P (B 1 ) = εa, P (B l ) = {ε} for l > 1 with the mappig P : R N R, P (v) := mi { ε, v q 1, u }.

4 4 L. L. STACHÓ ad L. I. SZABÓ Assume that the cotrary of the statemet of Lemma 2 holds, that is B = r T k(b) where T 1,..., T r : B B are weak cotractios of B. The B = r m l=1 T k(b l ). Therefore, sice 1 = a 1 = max A by costructio, we have r m εa = T kl (εa) where T kl := P T k S l. l=1 The mappigs P, S l are o-expasive ad hece each T kl : εa εa is a weak cotractio. This fact cotradicts Lemma 1 (with εa istead of A ad T kl istead of T k ). Defiitio 3. Give ay costat r > 1, by a r-fractal i R N we mea a o-empty compact subset B R N that is the ivariat set of a IFS (T 1,..., T r ) cosistig of (1 r 1 )-cotractios of B. We write F r (N) for the set of all r-fractals ad F (N) for the family of all ivariat sets of IFS by cotractios of R N. Notice that F (N) = r>1 F r (N). Lemma 4. The families K (N) := F (N) B (N) where B (N) := { B R N : B compact, sup B B B, B 1/2 } are compact i the Hausdorff distace d N amog the o-empty compact subsets of R N. Proof. Fix, N arbitrarily. Suppose B (1), B (2),... is a sequece i K (N). We have to see that some of its subsequeces coverges to a set B K (N). By assumptio, there are (1 1 )-cotractios T (i) k : B(i) B (i) (1 k, i = 1, 2,...) such that B (i) = T(i) k (B (i) ). Cosider the sets graph (T (i) k ) := {(B, T (i) k (B)) : a B(i) } R N R N R 2N. It is well-kow that the family B (N) is compact i d N. Also graph (T (i) B (2N) 2. Thus we may assume without loss of geerality that lim d N(B (i), B) = 0, lim d 2N( graph (T (i) k ), G k) = 0, 1 k for some o-empty compact sets B B (N) ad G 1,..., G B (2N) 2. It is also well-kow about limit sets i Hausdorff distace that here we have B = {b R N : b (1), b (2),... B (i) b (i) b (i )}, (3) G k = {(a, b) R 2N : b (1), b (2),... B (i) b (i) a, T (i) k (b (i) ) b (i )}. k )

5 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS 5 Usig the coordiate projectios Π 1, Π 2 : R 2N R N, Π 1 (a, b) := a, Π 2 (a, b) := b, it is immediate that B = lim B (i) = lim Π 1 graph (T (i) k ) = Π 1 G k, 1 k, B = lim = Π 2 lim T (i) k (B (i) ) = lim graph (T(i) k ) = Π 2 Π 2 graph (T (i) k ) G k = Π 2 G k where the limits are take i the respective Hausdorff metrics. O the other had, from (3) ad the fact that the mappigs T (i) 1,..., T(i) are (1 1 )- cotractios, we see that (a, b), (a, b ) G k b b, b b 1/2 [1 1 ] a a, a a 1/2. Therefore each set G k is the graph of some (1 1 )-cotractio T k : B B ad T k(b) = Π 2G k = B. That is B F (N) which completes the proof. Theorem 5. The set F (N) of all ivariat sets by IFS of cotractios R N R N is a owhere dese F σ -set amog the compact subsets of R N with respect to the Hausdorff metric d N. Proof. We have F (N) = =1 K(N). Accordig to Lemma 4, each family K (N) is closed with respect to d N. Thus F (N) is of F σ type i the d N - metric. It is well-kow that the family Q (N) of all fiite subsets of R N is dese i C (N) := {compact subsets of R N }. Give ay set Q Q (N) ad a poit q Q, we ca choose a uit vector u Q,q R N such that ε Q,q := mi p Q\{q} p, u Q q, u Q > 0. By Lemma 2, for the sets Q ε := Q + mi{ε, ε Q,q /2}Au Q,q with ε > 0 we have Q ε F (N). However, d N (Q ε, Q) < ε for ay ε > 0. Thus the set {Q ε : Q Q (N), ε > 0} cotaied i the complemet of F (N) is dese i C (N). Cosequetly, sice F (N) is of F σ - type, it must be owhere dese with respect to d N. Remark. Actually, by Lemma 2, the sets Q ε are ot ivariat sets of ay IFS eve by weak cotractios. Therefore the family F (N,w) of all ivariat sets of IFS by weak cotractios R N R N is also ot dese i Hausdorff distace i the space of all compact sets. However, F (N,w) has a more sophisticated structure tha beig F σ -type.

6 6 L. L. STACHÓ ad L. I. SZABÓ: A NOTE ON INVARIANT SETS... We give aother example of a subset of R that is ot the ivariat set of ay IFS cosistig of r-cotractios. We claim that the set K = { 0, (l 3) 1, (l 4) 1, (l 5) 1,... } has this property. Notice that for the proof of Theorem 5 it suffices to use such a set oly istead of that i Lemma 1 with the additioal restrictictio of ot beig the ivariat set of ay IFS of weak cotractios. Let us assume, to the cotrary, that K = f 1 (K) f (K) for some r(< 1)-cotractios f 1,..., f. Let a k = ( l(k + 2) ) 1, k = 1, 2,.... Defie the desity d(b) of a subset B K i the followig way: d(b) = lim sup N {k : 1 k N ad a k B}. N Clearly, we have d ( f 1 (K) ) + + d ( f (K) ) 1. Next we show that d ( f(k) ) = 0 for ay weak cotractio f : K K, thus obtaiig a cotradictio. First suppose f(0) 0. The, by the cotiuity of f, the set f(k) is easily see to be fiite. It follows that d ( f(k) ) = 0. Now let us assume that f(0) = 0. Let 0 < r < 1 deote the cotractio factor of f, that is, we have f(x) f(y) r x y for every x, y. If 1 k N ad a k f(k) the a k = f(a j ) for some j. The we have a k = a k 0 r a j 0 = r a j, that is, ( l(k + 2) ) 1 r ( l(j + 2) ) 1, so j + 2 (k + 2) r (N + 2) r. It follows that ad therefore d ( f(k) ) = 0. d ( f(k) ) lim if N (N + 2) r 2, N Ackowledgemet. The authors wish to thak Gerry Edgar for raisig the questio aswered i Theorem 5 above. Refereces [1] G. A. Edgar, Measure, Topology ad Fractal Geometry, Spriger (New York Berli, 1990). [2] G. A. Edgar, Private commuicatio, [3] K. J. Falcoer, Fractal Geometry. Mathematical Foudatios ad Applicatios, Wiley ad Sos (Chichester New York, 1990). [4] J. E. Hutchiso, Fractals ad self similarity, Idiaa Uiv. Math. J., 30 (1981),