VNU JOURNAL OF SCIENCE Mathematics - Physics T XVIII, N()3 - 2002 ON T H E LO CAL DIM EN SIO N S OF F R A C T A L M EASU RES Le V an T h a n h , N g u y e n V an Q u a n g D e p a r tm e n t o f M a th e m a tic s , U n iv e rsity o f V in h , V ie tn a m Let X o , X i , - - - be a sequence o f independent, identically distributed random variables each taking values r o ,r j , • • • , r m with equal probability p = A b str a c t Lei Ị1 be the probability measure induced by s = Y ^ t o P lX i The aim of this paper is to study som e properties o f support o f ịi and the local dim ension o f ỊJL at elements s € supp/u in the case: ro = , 7*! < 7*2 < • • • < rrn and q are integers such that % 1=0 t =0 Let // and /in d en o te the p rob ab ility d istrib u tio n s o f and S n) resp ectively T h en /i is called th e fractal measure a sso c ia ted w ith the p r o b a b ilis t ic sy ste m R ecall th at for s Ç su p /I, th e lower local dim ension a*(.s) o f ụ, at s is defined by a * (s) = lim i n f ~ ~ ~ ~ ~ ——, /1-+0+ log h w h ere J3(s, /i) = [5 — h f s + h], We sim ilarly d ifin e th e upper local dim ension u sin g the u p per lim it a n d d en o te it by a* ( ) If the tw o lim its are equal, th en th e coriim on valu e is called th e local dim ension of Ị1 at s and is d en o ted by a(s) R oughly sp eak in g, if a(.s) ex its, then p rop ortion al to h a ^ , h)) is ap p roxim ately for sm all h T h u s /J can be view ed a s a p ro b a b ility m easu re of degree o f sin gu larity a ( s ) In th is sen se, the local d im en sion m ea siư es th e d egree o f sin gu larities o f \x locally In [4], T H u considered th e local d im en sion s o f fractal m easu re /X in the case m = ro = , r\ = and p ~ l = (th is num ber is sa id to b e th e g o ld en num ber) Ill [5], T H u an d N N guyen stu d ied th e problem in th e ca se ro = 0, rj = , • • * , r m = 771, Po = Pi = = p m = — Ị^ì p = “ » < q < m , q is a n integer It is v e ry d ifficu lt to stu d y th e ab o v e problem in th e ca se that th e d ista n ces b etw een n , 7*2, • • • , T are n ot equal In [6] on lv sid ered the problem in sp ecia l case: 771 = 2, ro = ,r i = ,T = 3; Po — P\ — P2 — a n d p = ị = ỵ 't y p e s e t b y 49 Le 50 V an Thanhy N g u y e n Van Q uang T h e a im o f th is p ap er is to s tu d y som e p ro p erties o f su p p /i an d th e lo ca l d im en sion o f /2 a t elem en ts s G su p p /i in m ore general case T h e m ain resu lts o f th is paper are the theorems 2.7 and 3.1 Our results here extent some results in [5] and [6 ] (see proposition 2.4 in [5], proposition 2.1 and M ain Theorem in [6]) A ll n o ta tio n s and d e fin itio n s o f th is paper w e refer to [2], [4] and [5] S o m e p r o p e r tie s o f f r a c ta l m e a s u re T h ro u g h o u t o f th is p a p er th e follow in g a ssu m p tio n s are m ade: r = , 1"! < r -2 < • p= • < r m are in teg er s such th a t ^ < < 771 + 1, r m - g € D = { r o ,r i , • • • , r m } and T h e fo llo w in g p ro p o sitio n w a s proved in [5 ] P r o p o s i t i o n s Let s n ( 0) < s n ( l ) < • • • < s n ( kn ) denote the set o f all distinct values of suppfin Then we have s n ( 0) = a nd s n+ i(fc n + i) = s n (kn ) + m q ~ n~ for every n € N The distiince between two consecutive points in su p Ị1 is a t least q~n suppfXn c suppfj.n+ an d suppụ, = U^L 0su PP^n- Let P r o p o s itio n 2.2 n = I (.To, J■ } •Eji) ^ ^ ^9 i =0 ,5n Then we have ụ-n(Sn) = # < Sn > (m + l ) - " ” , where # < n > denotes th e cardinality o f sn Proof For x ( n ) = ( ỉ o ,£ i, • • • >x n) £ < 5n > , p u t n Ạr(r.) = p |{ w : X i ( u ) = X, } 1=1 It is ea sy to se e th a t if x ( n ) , y ( n ) € < s n > , x ( n ) Ỷ y ( n ) th en / l x(n) n ^ly(n) = a n d n : *5(o;) = s n } = P {iJ : Mn(lSn) “ g = 5n i =0 n = p{u> : Xi(uj) = Xi Vi = ,n ; =i>( u ^ w ) = i(n )€ < s„ > = E = x (n )£ < sn > y q~i x i = sn } } i= i E ^ w x (n )€ < sn > ) = £ i’( r > :r(n)€ »=1 ( 1=n ^ : L im + V -1 x(n)€ = # < S n > ( m + l ) - n- : * < (" )= * < ) O n t h e lo c a l d i m e n s i o n s o f f r a c t a l m e a s u r e s 51 Sn+1 € su pp /zn+ i , w e say th a t «n + i is represented through sn if there exists x n+\ € Dm such th a t $n+i = sn - ợ- 11"*1 Xn+1 I t is easy to see th a t if s n r i is re p re se n te d th ro u g h n , th e n D e f i n i t i o n L et $n € su p p /i, # < «n > < # < Sf»+ > I f svi-t-1 € suppfip +1 then there is s n e \i n such th a t n + i is represented through sn and < Sn +1 — s n < 2q~n L e m m a Proof I f Sn + € supPM n - th e n th ere e x is ts x (n + ) = (xo, X \ , • • • , x n+ i ) G D n *2 su ch that n-M Sn + = 'S j T q ~ i x i = Y j = )n g ~ ’z , + - n - l a;„ +1 = s„ + _n_1 izcO ( and < Sn+J - Sn = _ n _ ^ n + i < ~ n - r m < q ~ n~ l2q = 2q~ n I f s n+\ € suppUn-* 1then there arc at m o st two p o in ts s n and s'n in suppfxn such th a t Sn+1 is represented through them In this case s n >s'n a re two consecutive p o in ts in su p p /in L e m m a P roof S u p p o se th a t th e re a re th re e p o in ts t n < l'n < t ” in s u p p fx a n d th re e e lem en ts Xrx5 x'n) x'n in D m such th a t ~ ®n-M Sn+l = t ’n + i g_n_l< +l s n + l = c + " 1a'n+ 1• T hen $ n + l ^ in in + n Thus 8n+l - c > 2g“ n , w h ich is im p o s ib le (b y Le m m a 2.4 ) a n d the first p a rt o f the le m m a is p ro ved N ow , su p p o se th a t Sn + is re p re se n te d th ro u g h s n a n d 5^, sn < q~n s'n) we have < |Sfi - *nl < 5n-t-l - «n < 2ợ“n w h ich follow s th a t Is n - 5^1 = q ~ n an d 5n, s'n are tw o c o n se c u tiv e p o in ts in support- I f Sn + € suppfJin + is represented through Sn £ suppfin a n d £n € suppfjin such that sn < tn < Sn+ then tn = n + Sn+1 - s n > t n - sn > T h is im plies ‘^71 = Q Ĩ w hich co m p letes th e proof T h e m a in re s u lt of th is se c tio n is follow ing th e o re m Le 52 Van Thanh, N g u yen Van Q uang T h e o re m 2.7 If sn)$'n are two consecutive points in $uppfjLn then < n + Mn(«n) Proof W e prove th e in eq u ality by in d u ction C learly th e in eq u a lity h o ld s for n = S u p p ose th a t it is true for Ti < k L et Sk-f i > s'k^ l b e tw o arb itrary c o n se c u ta tiv e p o in ts in supp/ifc+i s - k + l = sk + q ~k~ l x k+ ! s' - k + = Sk + q~k ~ l x'k+i, w h ere Sfc, s'k e supp^fci xfc+ )x'fc+1 € D m T h en < + < ®fc+i- Wo consider th ree case: a If s'k > Sfc then > s'k > Sk' Using lemma 2.6 we get Sfc = Sfc + -fc My lem m a 2.5, Sfc+ has at m o st tw o rep resen tation s th rou gh Sk an d s'k It follow s that, # < Sfc+l > < # < Sfe > + # < Sfc > • Thus W c+ ijsk+ i) _ # M fc+IK+I) ~~ = 1+ # < < Sk+I > < # #4+1 > = 1+ < Sfc > + # < > # < sk > AifcVfc) < + (* + ) = fc + b If.s'k = Sjfc th en by lem m a 2.5, there ex ists at m ost on e p o in t tfc € supp/ifc, Ể/c 7^ Sfc su ch that Sfc+ is represented th ro u g h tịc (s/fc a n d are two c o n se c u tiv e p o in ts ) It follow s th at # < Sjb+1 > < > + # < > # < Thus Ịik+[(Sk + l) _ Mfc+l ( 5fc-fl ) # < Sfc+1 > < # < Sfc > + # < £fc > MJc+l(5fc4-i) = 1+ $ < tk > < + (fc + ) = * + c If Sfc < 5it th en w e sid er tw o cases _ O n t h e lo c a l d i m e n s i o n s o f f r a c t a l m e a s u r e s 53 C\ If th ere e x is ts t'k € su p p fik such th a t s'k < L'k < Sk th en from th e in equality < s'k+l ~~ s 'k - sk - s'kf w e have < s k < 8k+1- 4+1 O n th e o th e r han d, sin ce Sk+\ and Skare tw o co n secu tiv e p o in ts, w e have Sfc+\ = Sit SfcH-1 = ^ + 2(7 ,SK-f 1= s /c + (i > s'k + q k > 4+1 > £ • U sin g lem m a w e g e t + ợ*”* T h is im p lies 4-1 = + < r f c _ l 'm = Since r m — r/ € Dm y we have ifc - ># We now prove th a t ^ and 5/e are consecutive points S u p p o se th a t th ere e x ists in s u p p /Z f c < Sky th en sjp+j > € supp/Zfc such th a t (b eca u se s'k J a n d Sfc+1 are tw o co n secu tiv e p o in ts in supp/ijfc+i) It follow s th a t J Sfc+1 *“ t V* lit J \ ^ ^ic - 5fc > «/ , + — / _ o — 2ọ —ik It is im p o s ib le (b y L e m m a 2.4) H e n ce ^ a n d SA: are two co n se cu tiv e p o in ts B y lem m a 2.5 , Sk+i(= Sk) h as a t m o st tw o rep resen ta tio n s th rou gh Sk and s'k It follow s that # < S k+1 > < # < sk > +# < t ’k > and M fc±i(ffc±il < # < Mfc+i(5 fc+i) C2 If d o es n o t e x is ts ^ > # < Sk > < ^ > = 4- # < f* < l + fc + l = f c + ^ < L'k > G supp/Zfc such th a t s* < < Sjfc, th en Sfc and Sjfc are two co n secu tiv e p o in ts in fik- B y lem m a 2.5, Sfc+ h as at m ost tw o rep resen ta tio n s through Sk and Sfc It follo w s th a t # < Sfc+1 > < # < Sfc > # < and < l ( 5fc+l) T b e th eorem is proved jfc + > Le 54 C o r o ll a r y Van T h a n h , N g u yen Van Q uang € s u p p f i n a n d Ịsn — s'n \ < cq~n then < (n + )‘ Proof L e t s'n = ÍQ < 1 < • • • < £jfc = T h en b y P ro p o sitio n 2.1 w e h a v e k < c and b e k + c o n secu tiv e p o in ts in su p p /in i^n{^k) _ /^n(^Ảr) _ Mn(^o) Unfak—l ) /^n(^l) —l) H'Txi't'k—2) 0) < ( n + l ) ( n + ) • • •( n + ) < ( n + l ) c L o ca l d im e n s io n s o f fr a c ta l T h e fo llo w in g th eo rem is an e x te n sio n o f P ro p o sitio n 2.1 in [6] T h e o r e m For S’ € su p p n , we have a (.s ) = lim log /z„(an ) n lo g g provided that th e limit, exists O therw ise, by taking th e upper and lower lim its , respec tively, we g et the form ulas for a*(s) and a * ( s ) Proof S u p p o se th a t th ere e x is t s the lim it at.) = lim AI-ÍÕ+ /i) = w h ere [5 log /l — ft, s 4* h] For /1 > ta k e II su ch th a t n —1 we ftave q (,s ) lo g (m + ) lo g # < * • „ > = — - + lim - ' — — , lo g q n-»oo n lo g ọ provided th a t the lim it exists O therwise, by taking the upper a n d lower lim its respectively wc get the form ulas for a* (s) a n d a * ( s ) C o ro lla ry 3.3 log q, where a = sup{a(â) : s € supp/i} a* = sup{a*(.s) : s € s u p p f.i} Le 56 Van T h a n h , N g u yen Van Q uang C o r o l l a r y (see [6] M a in T h e o re m ) For m = 2, ro = 0, r j = , 7"2 = 3, q = rn + = 3, we have ã = a* = A c k n o w le d g e m e n t s T h e a u th ors are grateful to Professor N g u y en T o N h u o f N ew M ex ico U n iversity and Professor N g u y en N h u y o f V in h U n iversity for th eir h elp fu l su g g estio n s and valuable d iscu ssion s du rin g th e preparation o f th is paper R eferen c es K J.F alconer, Techniques in Fractal Geometry , Joh n W iley a n d Son s, 1997 K J.F alcon er, Fractal Geometry, M athematical Foundations and Applications , Joh n W iley a n d Sons 1993 N.T.Nliu, Fractal and Non-Fractal D im ensions , Statistic Seminar, P a rt 3, New Mex ico S ta te U niversity, U S A , 2000 T H u, The Local D im ensions o f the Bernoulli convolution associated w ith the G o ld en n u m b e r , T n s A m er M a th Soc 349 (1997),2917-2940 T H u and N T N hu , Local D im ensions o f Fractal Measures Associated with Uni fo rm ly D istributed Probabilistic System s , to ap p ear in th e A M S P ro ceed in g s, 2001 T H u, N N guyen a n d T W aiig , Local D im ensions o f The Probability Measure A s sociated with The (0 , 1, 3) Problem , Preprint, 2001 ... iscu ssion s du rin g th e preparation o f th is paper R eferen c es K J.F alconer, Techniques in Fractal Geometry , Joh n W iley a n d Son s, 1997 K J.F alcon er, Fractal Geometry, M athematical... Foundations and Applications , Joh n W iley a n d Sons 1993 N.T.Nliu, Fractal and Non -Fractal D im ensions , Statistic Seminar, P a rt 3, New Mex ico S ta te U niversity, U S A , 2000 T H u, The. .. The Local D im ensions o f the Bernoulli convolution associated w ith the G o ld en n u m b e r , T n s A m er M a th Soc 349 (1997),2917-2940 T H u and N T N hu , Local D im ensions o f Fractal