PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 7, July 1997, Pages 2119{2125 S (97) THE STRONG OPEN SET CONDITION

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1 PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 125, Number 7, July 1997, Pages 2119{2125 S (97) TH STRONG OPN ST CONDITION IN TH RANDOM CAS NORBRT PATZSCHK (Communcated by Palle. T. Jorgensen) Abstract. To descrbe some fractal propertes of a self{smlar set or measure, such as the Hausdor dmenson and the multfractal spectrum, t s useful that t satses the strong open set condton, whch means, there s an open set satsfyng the open set condton and, addtonally, a part of the self{smlar set must meet the open set. It s known that n the non{random case the strong open set condton and the open set condton are equvalent. Ths paper treats the random case. If the open set condton s assumed, we show that there s a random open set satsfyng the strong open set condton. Further, we gve an applcaton to multfractal analyss of the random self{smlar fractal. 1. Notatons In ths secton we recall the denton of self{smlar sets and self{smlar measures, and gve some propertes. A more detaled ntroducton and the proofs of the propertes may be found n [PZ] and n [AP]. Let K R d be a xed compact set wth K = nt K. We are gven a postve nteger N 2 and a probablty measure on Sm N [0; 1] N, where Sm s the space of all smlartes of R d equpped wth the usual topology of unform convergence on compact sets. In ths paper we assume the followng. Assumpton 1. (I) Z N p (d(s 1 ; : : : ; S N ; p 1 ; : : : ; p N )) = 1, (II) S (nt K) nt K for all = 1; : : : ; N and S (nt K) \ S j (nt K) = ; for all 6= j for {a. a. (S 1 ; : : : ; S N ; p 1 ; : : : ; p N ), (III) there exst p mn > 0 and r mn > 0 such that p p mn and Lp S r mn for all = 1; : : : ; N and {a. a. (S 1 ; : : : ; S N ; p 1 ; : : : ; p N ). Condton (II) s known as the open set condton. In the sequel we often make use of symbolc dynamcs. Let = f1; : : : ; Ng N be the code space over the ndces S 1; : : : ; N, n = f1; : : : ; Ng n the space of all 1 sequences of length n, and = n=0 n. For 2 n denote by jj = n the length of, and by jk the truncaton of to the rst k entres, k n. For 2 and 2 [ we wrte f there s a 0 2 [ wth = 0. Further, let [] = f 2 : g be the cylnder sets n, 2. Receved by the edtors January 16, 1996 and, n revsed form, February 7, Mathematcs Subject Classcaton. Prmary 28A80, Secondary 60D05, 60G57. Key words and phrases. Random fractals, (strong) open set condton, multfractals c1997 Amercan Mathematcal Socety

2 2120 NORBRT PATZSCHK Dene the space =? Sm N [0; 1] N. Let F be the product {algebra on. Takng P the product measure wth on each component we get our prmary probablty space (; F; P). For! 2 and 2 wrte!() = (S 1 (!); : : : ; S N (!); p 1 (!); : : : ; p N (!)) and S ; (!) = d; p ; (!) = 1. By F k we denote the {algebra generated by all S and p wth jj k. For brevty wrte S = S j1 S jjj r = Lp S r = Lp S = r j1 r jjj p = p j1 p jjj K = S K for 2. In [AP] t s shown that the random varables = lm n!1 2 n p (j1) p (jn) exst for all 2 and!([]) = p (!) (!) extends to a random measure! on for P{a. a.! 2, wth () = ; = 1 and ; 2 < 1. Let us dene random mappngs! :! K by! () = lm n!1 S jn(!)(x 0 ): Ths lmt exsts for P{almost all! and does not depend on the choce of x 0. The random measure wth! =!!?1 s called the random self{smlar measure, and the random set wth! =! () the random self{smlar set. We call a random subset? a Markov stoppng, f (I) for each 2 and each! 2 there s a unque 2?(!) wth, and (II) f! 2 : 2?(!)g 2 F jj for all 2. A smple example s the random set? r = f 2 : r < r r jjj?1 g for r 2 (0; 1). If? s a Markov stoppng then, by the open set condton, fnt K (!) : 2?(!)g s a famly of mutually dsjont sets for almost all! 2. Let F? be the sub{{ algebra of F generated by fs ; p : there s a 2? wth g. Further, let us ntroduce shft operators : [! [ by () = and wrte T (!) =!. Then S (T!) = S (!) and p (T!) = p (!). We wll denote the objects generated by T wth a superscrpt, e. g. S (!) = S (T!);! = T!, and so on. The measures and and the set fulll the followng nvarances (cf. [AP, PZ]).

3 TH STRONG OPN ST CONDITION IN TH RANDOM CAS 2121 Theorem 2. Let? be a Markov stoppng. Then () = 2? of F?. () = 2? F?. [ () = 2? p?1, where the are.. d. copes of and ndependent p S?1, where the are.. d. copes of and ndependent of S where the are.. d. copes of and ndependent of F?. ; and are characterzed by the propertes of the theorem above and by the requrement (R d ) = () = The Man Theorem Now we are able to state the man theorem of ths paper. The proof s analogous to one of Schef [S]. Theorem 3. Let the assumptons 1 be satsed. Then there s a random open set U wth U = nt U, such that the followng hold. () S (U ) U for all = 1; : : : ; N wth probablty one (where U (!) = U(T!)). () S (U ) \ S j (U j ) = ; for all 6= j wth probablty one. () \ U 6= ; wth probablty one. Proof. 1. Fx " > 0 and let K (") = S x2k B(x; "). For 2 wrte G (!) = S (!)(K (") ) and dene I()(!) = f 2? r(!)(!) : G (!) \ K (!) 6= ;g: Fx x 0 2 nt K. Snce nt K s open and bounded, there are 0 < r < R < 1 such that B(x 0 ; r) nt K K B(x 0 ; R). Let 2 I()(!). By denton of K and of? r (!)(!), L d (K (!)) r (!) d L d (B(x 0 ; r)) r (!) d r d mn Ld (B(x 0 ; r)): On the other hand, snce G (!) B(S (!)x 0 ; r (!)(R + ")), we have K (!) B(S (!)x 0 ; r (!)(3R + ")). By volume estmatng therefore #I()(!) r (!) d r d mn L d B(x 0 ; r) L d (K (!)) 2I()(!) L d? B(S (!)x 0 ; (3R + ")r (!)) = r (!) d L d (B(x 0 ; 3R + ")): That means the cardnalty of I()(!) s bounded above, ndependent of! 2 and 2. Let M = ess sup sup 2 #I(). Snce the cardnalty s a dscrete value, there s an 0 2 such that P(#I( 0 ) = M) > Take! 2 and 2 such that #I()(!) = M. Let 2 and ~! 2 T?1f!g. Then r (~!) = r (~!)r (~!) = r (~!)r (!) and K (~!) = S (~!)K (!). Hence, 2 I()(~!) for all 2 I()(!). By maxmalty of I()(!) ths mples I()(~!) = I()(!) and #I()(~!) = M.

4 2122 NORBRT PATZSCHK 3. Denote 0 = f! 2 : there s an 2 wth #I()(!) = Mg: Then q = P( 0 ) > 0. Furthermore, T?1 0 0 for all and, by denton of P, ft 1?1 0 ; : : : ; T?1 0g s an ndependent famly. Therefore, N q = P( 0 ) P = 1? P N[ N\ T?1 0 = 1? P( c 0) N = 1? (1? q) N T?1 c 0 whch mples (snce q > 0), that q = 1. Hence, for almost all! there s an (!) 2 such that #I()(!) = M. The mappng! 7! (!) may be assumed to be measurable. Wth step 2 we get I( (!))(!) = I ( (!))(!) for all 2. Denote = f! 2 : T! 2 0 for all 2 g: Snce s countable and P a product measure we nfer that P() = Dene U(!) = [ 2 S (!) S (!)(!)(K ("=2) ) for! 2. It remans to show that the assertons hold. () If! 2 then [ S (!)(U (!)) = S (!) S (!) S (!)(!)(K ("=2) ) [ 2 = S (!) S (!)(!)(K ("=2) ) 2 U(!) for all = 1; : : : ; N. () Assume there s an! 2 and a par 6= j wth S (U (!)) \ S j (U j (!)) 6= ;. By denton of U there are ; 2 and = (!) and = j (!) and y 2 S (!) S (K ("=2) ) \ S j (!) S j (!)(K ("=2) ). Wthout loss of generalty we assume r j (!) r (!). Choose 0 such that j 0 2? r (!)(!). By denton there are y 1 2 K (!) wth d(y 1 ; y) < r (!) "=2 and y 2 2 K j 0(!) wth d(y 2 ; y) < r j (!) "=2 < r (!) "=2. Hence, d(y 1 ; y 2 ) < r (!) "=2, whch mples K j 0(!) \ G (!) 6= ;; hence j 0 2 I( (!))(!), beng a contradcton to step 3. () Clearly, K (!) (!) U(!) for all! 2 and, hence, (!) \ U(!) 6= ;. 5. Replacng U by U 0 = nt U, we nfer that U 0 = nt U 0. Moreover, (), () and () reman vald for U 0, concludng the proof.

5 TH STRONG OPN ST CONDITION IN TH RANDOM CAS 2123 Remark 1. In [AP] we sad that the strong open set condton s satsed, f the assertons of the theorem above hold wth a non{random set U. But many proofs reman vald replacng ths non{random set by a random set. There are some derences between the sets U and nt K. Whle K s gven a pror, the random set U depends (n general) on the whole {algebra F. A dsadvantage s also the fact that there s no constant r 0 such that for almost each! there s a ball wth radus r 0 nsde U(!). 3. An Applcaton An applcaton s the determnaton of the multfractal spectrum. Let us consder the sets a = fx 2 : lm r#0 log (B(x; r)) log r exsts and equals ag of ponts wth local dmenson a. The multfractal spectrum s dened by the Hausdor dmensons of these sets. In [AP] we dened a functon : R! R by N p q r(q) = 1: In [F] some propertes of ths functon are lsted. Let (q) =? 0 (q). Then s ether constant or strctly decreasng. The multfractal spectrum s related to the Legendre transform f = of satsfyng f((q)) = q(q) + (q). The key of the proof was the fact that log s ntegrable f the strong open set condton s satsed (cf. [AP, 2.8]). An analogue s vald n our case. Lemma 4. Let U be the random open set from Theorem 3. Then Z j log (dx) < 1: Proof. By Theorem 3 there are an 2 and a > 0 such that P(d(K > =r mn ) > 0. Let r = ess nf r r mn > 0 and 0 < z = 1fd(K > r=r mn g ([]) < 1: Wrte?(n) =? r n and G n = f 2?(n) : d(k r n g: Let 2 G n. Then r r n+1. If d(k ) > r=r mn, then, snce 2?(n), d(k d(k U ) = r d(k ) > r n+1 :

6 2124 NORBRT PATZSCHK Hence " 2G n+1; = p " p 1? [] F jj # # []=p F jj " #! (K )=p F jj 2G n+1; 2?(n+1)nG n+1; p? 1? 1fd(K ) > r=r mn g = p (1? z) = (1? z) [ [] j F jj ]: [] Fjj Let 2 G n+1. Then there exsts a unque 2?(n) wth. Furthermore, d(k d(k r n+1 r n : Hence, 2 G n. Ths mples [] (1? z) [] 2G n+1 2G n and, by nducton, [] (1? z) n 2G n for all n. Snce?1 fx 2 : r n g S 2G n [], fx 2 : r n g whch mples the asserton. 2G n [] (1? z) n ; The proof of the next theorem s, wth lttle changes (replacng K by U f necessary), the same as n [AP, 3.10]. Let mn = nff : f((q)) > 0g and max = supf(q) : f((q)) > 0g. Theorem 5. Let the assumptons 1 be satsed. Then ether case I or case II holds. Case I: mn = max = D. Then dm = dm D = D wth probablty one and a = ; for all a 6= D wth probablty one. Case II: mn < max. Let a 2 R, then () f a < mn or a > max then a = ; wth probablty one, () f mn < a < max then dm a = f(a) wth probablty one, () dm = f((0)) wth probablty one. Analogous results as n [AP] hold for generalzed dmensons (cf. [AP, 4.11]) and tangental dstrbutons (cf. [AP, 5.2]).

7 TH STRONG OPN ST CONDITION IN TH RANDOM CAS 2125 References [AP] M. Arbeter and N. Patzschke, Random self{smlar multfractals, Math. Nachr. 181 (1996), 5 { 42. [F] K. J. Falconer, The multfractal spectrum of statstcally self{smlar measures, J. Theoret. Probab. 7 (1994), 681{702. MR 95m:60076 [PZ] N. Patzschke and U. Zahle, Self{smlar random measures IV. The recursve constructon model of Falconer, Graf, and Mauldn and Wllams. Math. Nachr. 149 (1990), 285 { 302. MR 92j:28007 [S] A. Schef, Separaton propertes of self{smlar sets, Proc. Amer. Math. Soc. 122 (1994), 111 { 115. MR 94k:28012 Fakultat fur Mathematk und Informatk, Fredrch{Schller{Unverstat Jena, D{ Jena, Germany -mal address: patzschke@mnet.un-jena.de