SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER THE INFINITE CONFLICT INTERACTION

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1 SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER THE INFINITE CONFLICT INTERACTION SERGIO ALBEVERIO 1,2,3,4, VOLODYMYR KOSHMANENKO 5, MYKOLA PRATSIOVYTYI 6, GRYGORIY TORBIN 7 Abstract. We ntroduce the conflct nteracton wth two postons between a couple of mage probablty measures and consder the assocated dynamcal system. We prove the exstence of nvarant lmtng measures and fnd the crtera for these measures to be a pure pont, absolutely contnuous, or sngular contnuous as well as to have any topologcal type and arbtrary Hausdorff dmenson. 1 Insttut für Angewandte Mathemat, Unverstät Bonn, (Germany); 2 SFB- 611, Bonn; BBoS, Belefeld - Bonn; 3 IZKS, Bonn; 4 CERFIM, Locarno and Acc. Arch. (USI) (Swtzerland). E-mal: albevero@un-bonn.de 5 Insttute of Mathematcs, Kyv (Urane). E-mal: osh@math.ev.ua 6 Natonal Pedagogcal Unversty, Kyv (Urane). E-mal: prats@urpost.net 7 Natonal Pedagogcal Unversty, Kyv (Urane). E-mal: torbn@math.ev.ua AMS Subject Classfcatons (2000):11A67, 11K55, 26A30, 26A46, 28A80, 28D05, 60G30 Key words: probablty measure, mage measure, conflct nteracton, stochastc matrx, sngular measure, Hausdorff dmenson 1. Introducton Let (Ω, F) be a measurable space, let P be a class of probablty measures on F, and let be a noncommutatve bnary algebrac operaton defned for elements of P. A measure µ P can be nterpreted as a measure of nfluence on controversal terrtory for some subject of controversy. If two non-dentcal measures µ and ν are not mutually sngular, then they are called conflct measures and an operaton represents the mathematcal form of the conflct nteracton between µ and ν. Gven µ, ν P let us consder a sequence of pars µ (n), ν (n) P of measures defned as follows: µ (1) = µ ν, ν (1) = ν µ; µ (2) = µ (1) ν (1), ν (2) = ν (1) µ (1) ;... µ (n+1) = µ (n) ν (n), ν (n+1) = ν (n) µ (n) ;... By ths each operaton defnes an mappng g( ) : P P P P and generates a certan dynamcal system (P P, g( )). The followng problems are of nterest: 1) exstence of nvarant ponts and nvarant sets of (P P, g( )); 2) descrptons of the lmtng measures µ = lm n µ(n) ; ν = lm n ν(n) ; 1

2 2 S.ALBEVERIO, V.KOSHMANENKO, M.PRATSIOVYTYI, G.TORBIN 3) topologcal, metrc, and fractal propertes of the lmtng measures and dependence of these propertes on the conflct nteracton. In [6, 7] a varant of a conflct nteracton for dscrete measures on fnte and countable spaces was dscussed. In ths paper we nvolve n consderaton the case of contnuous measures. More precsely, here we handle wth measures µ, ν whch are mage measures of nfnte products of dscrete measures. We prove the exstence of the lmtng nvarant measures µ, ν and show that they are mutually sngular f µ ν. We fnd necessary and suffcent condtons for these measures to be pure absolutely contnuous, pure sngular contnuous or pure dscrete resp. Metrc, topologcal and fractal propertes of the supports of the lmtng measures are studed n detals. We show that by usng rather smple constructon one can get a sngular contnuous measure µ wth desrable fractal propertes of ts support, n partcular, wth any Hausdorff dmenson 0 dm H (suppµ ) Sub-class of mage measures Let M([0, 1]) denote the sub-class of probablty Borel measures defned on the segment [0, 1] as follows (for more detals see [1, 2]). Let Q {q } be a sequence of stochastc vectors n R2 wth strctly postve coordnates, q = (q 0, q 1 ), q 0, q 1 > 0, q 0 + q 1 = 1. We wll refer to Q as the nfnte stochastc matrx ( ) (1) Q = {q } q01 q = 02 q 0 q 11 q 12 q 1 Gven Q we consder a famly of closed ntervals 1, 1 2,..., 1 2,... [0, 1], ( 1, 2,...,... are equal to 0 or 1) wth lengths 1 = q 1 1, 1 2 = q 1 1 q 2 2, = q 1 1 q 2 2 q, 1, and such that [0, 1] = 0 1, and so on for any, (2) Assume 1 = , = max{q } = 0. Then any x [0; 1] can be represented n the followng form x = 1(x)... (x) =: 1(x)... (x).... Moreover, (2) mples that the Borel σ-algebra B on [0; 1] concdes wth the σ-algebra generated by the famly of subsets { 1... }.

3 SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER CONFLICT INTERACTION 3 To a fxed Q, we assocate a sub-class of measures M([0; 1]) wth a famly of all sequences of stochastc vectors ( ) P = {p } p01 p = p 0..., p 11 p p 1... where p 0, p 1 0, p 0 + p 1 = 1. Namely, we assocate to each such matrx P a Borel measure µ M([0; 1]) defned as follows. We consder a sequence of probablty spaces (Ω, A, µ ), where Ω = {0; 1}, A = 2 Ω, µ () = p. Let (Ω, A, µ ) be the nfnte product of the above probablty spaces. We defne a measurable mappng f from Ω nto [0; 1] n the followng way: for any ω = (ω 1, ω 2,..., ω,...) Ω we set f(ω) = ω1ω 2...ω... = ω1...ω, and, fnally, we defne the measure µ as the mage measure of µ under f,.e., : for any Borel subset E we put µ(e) = µ ( f 1 (E) ), where f 1 (E) = {ω : f(ω) E}. The followng results (see Theorem 1 below) on mage measures are well nown (see e.g. [5, 4, 2]). In order to formulate them we need some notatons. We wrte, µ M pp, M ac, M sc f the measure µ s pure pont, pure absolutely contnuous, or pure sngular contnuous, resp. Further, for the above Q and P we defne and ρ(µ, λ) := P max (µ) := max{p } ( p 0 q 0 + p 1 q 1 ). Theorem 1. Each measure µ µ P M([0, 1]) s of pure type: (a) µ M pp ff P max (µ) > 0, (b) µ M ac ff ρ(µ, λ) > 0, (c) µ M sc ff P max (µ) = 0 and ρ(µ, λ) = 0. In the contnuous case the measure µ can also be defned n the followng smple way. We defne a measure µ on the sem-rng of subsets of the form 1... = [a; b), where, a = 1... (0); b = 1... (1): we put ( ) µ 1... = p p. The extenson µ of µ on any Borel subset of [0; 1) s defned n the usual way. We put, fnally µ(e) = µ(e [0; 1)) for any Borel subset of [0; 1]. It s not hard to prove that µ µ. 3. Conflct nteracton between mage measures We defne the non-commutatve conflct composton wth two postons for a couple of stochastc vectors p, r R 2 as follows: p 1 := p r, r 1 := r p, where the coordnates of the vectors p 1, r 1 are gven by the formulae: (3) p (1) := p (1 r ) 1 (p, r), r(1) := r (1 p ), = 0, 1, 1 (p, r)

4 4 S.ALBEVERIO, V.KOSHMANENKO, M.PRATSIOVYTYI, G.TORBIN where (p, r) stands for the nner product n R 2. Obvously we have to exclude the case (p, r) = 1. The teraton of the composton generates a dynamcal system n the space R 2 R 2 defned by the mappng: ( ) ( p N 1 p N (4) g : r N 1 r N ), N 1, p 0 p, r 0 p, where the coordnates of p N, r N are defned by nducton, (5) p (N) := p(n 1) (1 r (N 1) ) z N 1, r (N) wth z N 1 = 1 (p N 1, r N 1 ) > 0. := r(n 1) (1 p (N 1) ) z N 1, = 0, 1, Lemma ([6, 7]). For each par of stochastc vectors p, r R 2, followng lmts exst and are nvarant wth respect to : p = lm N pn, Moreover one has: p = (1, 0), r = (0, 1) ff p 0 > r 0, p = (0, 1), r = (1, 0) ff p 1 > r 1, p = r = (1/2, 1/2), ff p 0 = r 0, p 1 = r 1. r = lm N rn. (p, r) 1, the We wll ntroduce now a non-commutatve conflct nteracton (n the sense of [6, 7]) between mage measures from the sub-class M([0, 1]) and use the just presented facts for the analyss of the spectral transformatons of these measures. Let µ and ν be a couple of mage measures correspondng to a par of sequences of stochastc vectors P 0 = {p 0 } and R0 = {r 0 }, resp.,.e., µ = µ P 0, ν = ν R 0. The conflct nteracton between µ and ν, denoted by, s by defnton gven by the couple µ 1, ν 1 : µ 1 := µ ν, ν 1 := ν µ, where s defned by usng the above defned conflct compostons for stochastc vectors n R 2. Namely, we assocate a new couple of measures µ 1, ν 1 M([0, 1]) wth sequences P 1 = {p 1 } and R1 = {r 1 }, where the coordnates of vectors p 1, r1 are defned accordng to formulae (3),.e., (6) p (1) := p (1 r ) 1 (p, r ), r(1) := r (1 p ), = 0, 1, = 1, 2,... 1 (p, r ) where p p (0), r r (0) and r r 0, r r 0. Of course we assume that (7) (p 0, r 0 ) 1, = 1, 2,... By nducton we ntroduce the sequences P N = {p N } and RN = {r N } for any N = 1, 2,..., where the stochastc vectors p N = pn 1 r N 1, r N = r N 1 p N 1 are defned as N-tmes teratons of the composton ; the coordnates of the vectors p N, rn are calculated by formulae le (5). Further, wth each par P N, R N we assocate a couple of mage measures µ N µ P N and ν N ν R N from the class M([0, 1]). Therefore the mappng g generates

5 SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER CONFLICT INTERACTION 5 the dynamcal system n the space M([0, 1]) M([0, 1]): ( ) ( ) µ N 1 µ N U(g) : ν N 1 We are nterestng n the exstence and structure of the nvarant ponts of the so defned dynamcal system. Theorem 2. For each couple of mage measures µ µ P 0, ν ν R 0, under condton (7), there exst two lmtng nvarant measures, µ = lm N µn, ν = lm N νn. The measures µ, ν are mutually sngular ff P 0 R 0, and µ, ν are dentcal ff P 0 = R 0. Proof Ths follows easly from Lemma. Our goal here s to nvestgate the propertes of the lmtng measures. ν N 4. Metrc propertes Let us ntroduce two sets for a couple of probablty mage measures µ, ν. N = := { : p = r }, and N := N \ N = { : p r }, and put Q = := [(1 2q 0 ) 2 + (1 2q 1 ) 2 ], W (µ) := q (), where q () = W (ν) := (1 q () ). { q0, f p 0 < r 0 q 1, f p 1 < r 1, Theorem 3. (a) µ M pp, ff N = <. (b) µ M ac, ff Q = < and W (µ) <. (c) µ M sc, ff N = = and at least one of the condtons, W (µ) =, or Q = =, s fulflled. Proof. (a) By Theorem 1 the measure µ belongs to M pp ff P max (µ ) := max{p } > 0. N Snce for each vector p R2 the coordnates p ( ) are equal to 0, 1/2, or 1, we have { max p ( ) 1/2, f N= = 1, f N. Hence µ M pp f and only f p r > 0 for some 0 N. Ths means that suppµ conssts of at most 2 N = ponts. (b) Let N = = and, therefore, µ M ac M sc. By Theorem 1, µ M ac ff (8) ρ(µ, λ) = ( p ( ) 0 q 0 + p ( ) 1 q 1) > 0.

6 6 S.ALBEVERIO, V.KOSHMANENKO, M.PRATSIOVYTYI, G.TORBIN Tang nto account that p ( ) 0 p 0 > r 0, p ( ) 0 = 0 ff p 0 < r 0, we have { p ( ) 1 q 1 1 = p ( ) 0 q 0 + Therefore, µ M ac ff 1 2 q 0 + ( = p ( ) 1 = 1 2 for all N =, and p ( ) q 1) 1 ( 2 q q 1, ff N =, 1 q(), ff N. 2 q 0 + ( 1 q () ) > q 1) > 0, ( 1 q () ) > 0. By usng smple arguments, we have ( 1 q () ) > 0 (1 q () ) > 0 q () <, and 1 1 ( 2 q q 1) > 0 ( ) q 0 q 1 > 0 ( 1 2 ) q 0 q 1 < (1 2 q 0 q 1 ) < ( 4q 0 q 1 > 0 1 (1 2q0 ) 2) > 0 (1 2q 0 ) 2 < (1 2q 1 ) 2 <. Therefore, { µ W (µ) <, M ac Q = <. = 1 ff (c) If N = =, then from (a) the contnuty of µ follows easly. If Q = (µ) = or W (µ) =, then from (b) t follows that µ λ, and, therefore µ M sc. Conversely, f µ M ac, then µ s a contnuous measure and µ λ. The contnuty of µ mples N = =. Snce µ λ, we have ρ(µ, λ) = 0 and, therefore Q = = or W (µ) =. Remars. (1) The Theorem holds for the measure ν f W (µ) s replaced by W (ν). (2) Both measures µ, ν belong to M ac only n the case N < provded that Q = <, because, f N = then at least one of values W (µ) or W (ν) s nfnte. (3) The condton Q = < s fulflled ff (1 2q 0 ) 2 <.

7 SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER CONFLICT INTERACTION 7 Example 1. Let q 0 = q 1 = 1/2 for all N. Then Q = = 0 for any N = and W (µ) < ff N <. Thus both measures µ, ν belong to M pp ff N = <. Moreover µ, ν M ac ff N <, and µ, ν M sc ff N = and N = =. Example 2. Let q 0 / ( 1 2 ε; ε) for some ε > 0. Then Q = < only f N = <. Thus µ, ν M pp ff N = <, and µ, ν M sc ff N = =, but one never has µ, ν M ac. 5. Topologcal propertes A Borel measure µ on R s of the S-type f ts support, suppµ S µ, s a regularly closed set,.e., S µ = (nts µ ) cl, where nta denotes the nteror part of the set A, and (E) cl denotes the closure of the set E. A measure µ s of the C-type f ts support S µ s a set of zero Lebesgue measure. A measure µ s of the P-type f ts support S µ s a nowhere dense set and S µ has a postve Lebesgue measure n any small neghborhood of each pont x from S µ,.e., x S µ, ε > 0 : λ (B(x, ε) ) S µ > 0. We shall wrte (cf. wth [1, 9]) µ M S, resp. M C, or resp. M P, f µ s of the S-, resp. C-, or resp. P-type. Theorem 4. The nfnte conflct nteracton between two mage measures µ, ν M([0, 1]) produces lmtng measures µ, ν of pure topologcal type. We have: (a) µ M S, ff N <, (b) µ M C, ff W (µ) =, (c) µ M P, ff N = and W (µ) <. Proof. (a) By Theorem 8 n [2] the measure µ s of the S-type ff the matrx P contans only a fnte number of zero elements. Ths s possble ff N <. (b) The measure µ s of the C-type (see Theorem 8 n [2]) ff the matrx P contans nfntely many columns havng elements p = 0, and besdes, ( q ) =, that s equvalent to N = and W (µ) =. Snce :p =0 max q = 0, we conclude that from W = t follows that N =. (c) Fnally the measure µ s of the P -type (see agan Theorem 8 n [2]) ff the matrx P contans nfntely many columns wth zero elements p, and moreover, ( q ) <,.e., N = and W (µ) <. :p =0 Remars. (1) The assertons of Theorem 4 are also true for the measure ν f one replaces W (µ) by W (ν). (2) It s not possble for the measures µ, ν to be both of the P -type. So f one of them s of the P -type, then the other s necessarly of the C-type. The combnaton of Theorem 3 and Theorem 4 leads to

8 8 S.ALBEVERIO, V.KOSHMANENKO, M.PRATSIOVYTYI, G.TORBIN Corollares. (a) The set M pp M S s empty. µ M pp M C ff N = <. The set M pp M P s empty. (b) µ M ac M S ff N <, and Q = <. The set M ac M C s empty. µ M ac M P ff N = =, N =, but Q = < and W (µ) <. (c) µ M sc M S ff N <, and Q = =. µ M sc M C ff W (µ) =. µ M sc M P ff N =, W (µ) < and Q = =. Proof. (a) M pp M S = snce N < and N = < are mutually exclusve condtons. So, f N = <, then N =, and µ M C. M pp M P =, snce W (µ) < wth N = < mean that q > 0, but ths contradcts our assumpton (2). (b) The frst asserton s evdent snce N < mples W (µ) <. Further M ac M C = snce the condtons W (µ) < and W (µ) = can not be smultaneously fulflled. Fnally, n the case N =, µ M ac ff µ M P. (c) In spte of that W (µ) < f N <, t s stll possble that Q = (µ) =. Thus, n general, we have M sc M S. Moreover, f N =, then W (µ) = mples µ M sc M C. But f N = and W (µ) <, then t stll possble that Q = (µ) =, or equvalently, µ M sc M P. Remar. Of course, all corollares are true for the measure ν f one replaces W (µ) by W (ν). The measures µ, ν n general have a rather complcated local structure and ther supports mght posses arbtrary Hausdorff dmensons. Let us denote by dm H (E) the Hausdorff dmenson of a set E R. Suppose q 0 = q 1 = 1/2. Theorem 5. Gven a number c 0 [0, 1] let µ M([0, 1]) be any probablty mage measure. Then there exsts another probablty mage measure ν such that (9) dm H (suppµ ) = c 0. Proof. Let N, = N {1, 2,..., } denote the cardnalty of the set N, := {s N : s }. Clearly N, + N =, =, where N =, := {s N = : s }. By Th.2 [8] the Hausdorff dmenson of the set suppµ may be calculated by the formula: dm H (suppµ ) = lm nf N =,. Gven the stochastc matrx P correspondng to the startng measure µ one always can chose (n a non-unque way) another stochastc matrx R (unquely

9 SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER CONFLICT INTERACTION 9 N assocated wth a measure ν) such that the condton lm =, = c 0 wll be satsfed. Acnowledgement Ths wor was partly supported by DFG 436 UKR 113/53, DFG 436 UKR 113/67, INTAS , SFB-611 projects and by UFFI 01.07/00081 grant. References [1] S. Albevero, V. Koshmaneno, G. Torbn, Fne structure of the sngular contnuous spectrum, Methods Funct. Anal. Topology., 9, No. 2, (2003). [2] S. Albevero, V. Koshmaneno, M. Pratsovyty, G. Torbn, e Q representaton of real numbers and fractal probablty dstrbutons, subm. to Infnte Dmensonal Analyss, Quantum Probablty and Related Topcs. [3] M.F. Barnsley, S. Demo, Iterated functonal system and the global constructon of fractals, Proc. Roy. Soc. London, A 399, (1985) [4] S.D. Chatterj, Certan nduced measures on the unt nterval, Journal London Math. Soc., 38, (1963). [5] S. Kautan, Equvalence of nfnte product measures, Ann. of Math., 49, (1948). [6] V. Koshmaneno, The Theorem on Conflct for a Par of Stochastc Vectors, Uranan Math. J., 55, No. 4, (2003). [7] V. Koshmaneno, The Theorem of Conflcts for a Couple of Probablty Measures, accepted to publ. n Math. Meth. of Operatons Research 59, No. 2, (2004). [8] M.V. Pratsovyty, Fractal, superfractal and anomalously fractal dstrbuton of random varables wth a fxed nfnte set of ndependent n-adc dgts, Explorng stochastc laws., VSP, (1995). [9] M.V. Pratsovyty, Fractal approach to nvestgaton of sngular dstrbutons, (n Uranan) Natonal Pedagogcal Unv., Kyv, 1998.