DSpace at VNU: On the local dimensions of fractal measures

JOURNAL OF SCIENCE.

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 r s ity o f V in h, V ie tn a m

A b s t r a c t Let X o , X i , - - - be a sequence o f independent, identically distributed random variables each taking values r o , r j , • • • , r m w ith equal probability p =

Lei Ị1 be the probability measure induced by s = Y ^ t o P lX i The aim of this paper

is to stu d y som e properties o f support o f ịi and the local d im en sio n o f ỊJL at elem ents

s € supp/u in the case: ro = 0,7*! < 7*2 < • • • < rrn and q are integers such that

< q < m + 1, rm- i | e i J = {r0,r1, ,r ra); p

-1 I n t r o d u c t i o n a n d n o t a t i o n s

B y a probabilistic system w e m ean a seq u en ce Xq, X I, • • o f in d e p e n d e n t, id en tic a lly

d istr ib u te d ra n d o m variables c a d i ta k in g valu es ro , r \, • • • , r m w ith re sp e c tiv e probabilit ies

POiPi ) • • • Pm- W e say th a t th e s y s te m is u n iform ly d istr ib u te d i f Pi = For 0 < p < 1,

p u t

s = £ > ■ * , Sn = Y > %

Let // an d /in d e n o te the p ro b a b ility d istr ib u tio n s o f 5 an d S n) resp ectiv ely T h en /i is

ca lled th e fractal measure a sso c ia te d w ith th e p r o b a b ili s t i c sy ste m

R eca ll th a t for s Ç su p /I, th e lower local dim ension a * (.s) o f ụ, at s is defin ed by

a * ( s ) = lim i n f ~ ~ ~ ~ ~ ——, w h e re J3(s, /i) = [5 — h f s + h],

/1-+0+ log h

W e sim ila rly d ifin e th e upper local dim ension u sin g the u p p er lim it a n d d e n o te it by a* ( 5)

If th e tw o lim its are eq u al, th en th e coriim on v a lu e is ca lled th e local dim ension o f Ị1 a t s

an d is d e n o te d by a( s) R ou gh ly sp ea k in g , if a (.s) e x its, th en , h)) is a p p ro x im a tely

p ro p o rtio n a l to h a ^ for sm a ll h T h u s /J can b e v iew ed a s a p ro b a b ility m ea su re o f degree

o f sin g u la rity a ( s ) In th is sen se, th e local d im en sio n m e a siư e s th e d egree o f sin g u la rities

o f \x locally.

In [4], T H u con sid ered th e local d im en sio n s o f fractal m ea su re /X in th e case

m = 1 ro = 0, r\ = 1 and p ~ l = (th is n u m ber is sa id t o b e th e g o ld e n n u m ber) Ill [5], T H u a n d N N g u y en stu d ie d th e p rob lem in th e ca se ro = 0, r j = 1, • • * , r m = 771,

Po = Pi = = p m = — Ị^ì p = “ » 2 < q < m , q is an integer It is v e ry d ifficu lt

to stu d y th e a b o v e p rob lem in th e c a se th at th e d ista n c e s b e tw e e n n , 7*2, • • • , T r a are n o t

eq u al In [6] o n lv co n sid ered th e p rob lem in sp ecia l case: 771 = 2, ro = 0 , r i = 1,T2 = 3;

Po — P\ — P2 — 3 a n d p = ị = ỵ.

' t y p e s e t b y

49

T h e a im o f th is p a p er is t o s t u d y so m e p r o p e r tie s o f su p p /i a n d th e lo c a l d im en sio n

o f /2 a t e le m e n ts s G su p p /i in m ore gen eral case T h e m a in re su lts o f th is p ap er are the

theorem s 2.7 and 3.1 Our results here extent some results in [5] and [ 6 ] (see proposition 2.4

in [5], proposition 2.1 an d M ain Theorem in [6]).

A ll n o ta tio n s an d d e fin itio n s o f th is p ap er w e refer to [2], [4] an d [5]

2 S o m e p r o p e r t i e s o f f r a c t a l m e a s u r e

T h r o u g h o u t o f th is p a p er th e fo llo w in g a s su m p tio n s are m ade: r0 = 0,1"! < r-2 <

• • < r m are in te g e r s su ch th a t ^ < 9 < 771 + 1, r m - g € D = { r o , r i , • • • , r m } and

p = T h e fo llo w in g p r o p o sitio n w a s proved in [5].

P r o p o s i t i o n s 2 1 L et s n (0) < s n ( l ) < • • • < s n ( kn ) d en o te the se t o f all d istin ct values

of suppfin Then we have

1 s n (0) = 0 a nd s n+ i( f c n + i) = s n (kn ) + m q~ n~ 1 for every n € N

2 The distiince between two consecutive points in su p Ị1 is a t least q~n

3 suppfXn c suppfj n+ 1 a n d suppụ, = U^L 0s u

PP^n-P r o p o s itio n 2.2 Let

n

<c sn > = I (.To, J ■ } •Eji) ^ ^ ^ 9 ,5n

i=0

T hen we have

ụ-n(Sn) = # < Sn > (m + l) - " ” 1 ,

where # < 5n > d en o tes th e card inality o f s n

Proof For x ( n ) = ( æ o ,£ i, • • • >x n) £ < 5n > , p u t

n

Ạr(r.) = p |{ w : X i ( u ) = X, }.

1=1

It is e a s y t o se e th a t if x ( n ) , y ( n ) € < s n > , x ( n ) Ỷ y ( n ) th e n / l x(n) n ^ly(n) = 0 a n d

n Mn(lSn ) “ : *5(o;) = s n } = P {iJ : g = 5n

i=0

n

= p{u> : Xi(uj) = Xi Vi = 0 ,n ; y q~i x i = s n } }

i= i

= i > ( u ^ w ) = E ^ w ) = £ i ’ ( r > : * < ( " ) = * < )

i(n )€ < s „ > x (n )€ < sn > :r(n)€<arl> »=1

= E ( n ^ :

x (n )£ < s n > 1 = 1

= L i m + V - 1 = # < S n > ( m + l ) - n - 1

x(n )€< s(n )>

O n t h e l o 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

D e f i n i t i o n 2 3 L et $n € s u p p /i, Sn+1 € su p p /zn+ i , w e sa y th a t «n + i is represented

through sn if there exists x n+\ € Dm such th at $n+i = sn 4 - ợ - 11"*1 Xn+ 1

I t is easy to see th a t if s n r i is re p re s e n te d th ro u g h 5n , th e n

# < «n > < # < Sf» + 1 > .

L e m m a 2 4 I f svi-t-1 € s uppf i p+1 th en there is s n e \ i n such th a t 5n + i is represented through sn and 0 < .Sn +1 — s n < 2q~n

Proof. I f Sn + 1 € supPM n- 1 th e n th ere e x is ts x ( n + 1 ) = (xo, X\ , • • • , x n+ i ) G Dn 4 *2

su ch th at

n-M

Sn + 1 = 'S j T q ~ i x i = Y j = 0 ) n g ~ ’ z , + 7- n - l a;„ +1 = .s„ + 7 _n_1

and

0 < Sn+J - Sn = 9 _ n _ 1 ^ n + i < 9~ n - 1 r m < q ~ n~ l2q = 2 q ~ n

L e m m a 2 5 I f s n+\ € suppUn-* 1 then there arc at m o st tw o p o in ts s n a n d s'n in suppfxn

such th a t Sn+1 is represented thro ug h them In this case s n > s'n a re tw o consecutive p o in ts

in su p p /in

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 pfx a n d th re e e le m e n ts

Xrx5x'n) x'n in D m su ch th a t

Sn+l = t ’n + i g_n_l< +l

s n + l = c + 9 " 1 a'n+ 1 •

T h e n

$ n + l ^ in in + n

T h u s

8n+l - c > 2 g “ 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 + 1 is re p re se n te d th ro u g h s n a n d 5^, sn < s'n) we have

q~n < |Sfi - *nl < 5 n-t-l - «n < 2 ợ“n

w h ich follow s th a t Is n - 5^1 = q ~ n a n d 5n, s'n are tw o c o n s e c u tiv e p o in ts in

support-L e m m a 2 6 I f Sn + 1 € suppfJin + 1 is represented through Sn £ suppfin a n d £n € suppfjin such that s n < tn < Sn+ 1 then tn = 5n + <7- n

Proof W e h ave

2 ç “ n > Sn+1 - s n > t n - s n > 0.

T h is im plies

‘^71 = Q Ĩ

w hich c o m p le tes th e proof

T h e m a in re s u lt o f th is s e c tio n is follow ing th e o re m

T h e o r e m 2.7 If sn)$'n are two consecutive points in $uppfjLn then

Mn(«n)

< n + 1

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 = 0

S u p p o se th a t it is true for Ti < k L et Sk-f i > s'k^ l b e tw o arb itra ry c o n s e c u ta tiv e p o in ts

in supp/ifc+i

s - k + l = s k + q ~ k~ l x k+ ! s' - k + 1 = Sk + q~k ~ l x'k+i,

w h e re Sfc, s'k e sup p^fci xfc+ 1 )x'fc+1 € D m

T h e n

4 < 4 + 1 < ® f c + i

-Wo con sid er th ree case:

a If s'k > Sfc then > s'k > Sk' Using lemma 2.6 we get

Sfc = Sfc + 9 -fc

My lem m a 2.5, Sfc+ 1 has at m o s t tw o rep resen ta tio n s th rou gh Sk a n d s'k It follow s that,

# < Sfc+l > < # < Sfe > + # < Sfc > •

T h u s

W c + i j s k + i ) _ # < S k + I > < # < Sfc > + # < 4 >

M fc+IK+I) ~~ # 4 + 1 # < s k >

= 1 + = 1 + < 1 + ( * + 1) = fc + 2.

# < 4 > AifcVfc)

b If.s'k = Sjfc th en by le m m a 2 5 , th ere e x ists a t m o st o n e p o in t tfc € supp/ifc, Ể/c 7^ Sfc

su ch that .Sfc+ 1 is represented th ro u g h tịc (s/fc a n d are two co n se cu tiv e p o in ts ) It follow s

th a t

# < Sjb+1 > < # < > + # < 4 > .

T h u s

Ị i k + [ ( S k + l) _ # < Sfc+1 > < # < Sfc > + # < £fc > _ Mfc+l ( 5fc-fl ) MJc+l(5fc4-i) $ < t k >

= 1 + < 1 + (fc + 1) = * + 2.

c If Sfc < 5it th en w e co n sid er tw o cases.

O n t h e l o 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\ I f th ere e x is t s t'k € su p p fik su ch th a t s'k < L'k < Sk th en from th e in eq u ality

s'k + l ~~ s 'k < - sk - s'kf

w e have

4 + 1 < s k < 8k+1

-O n th e o th e r h a n d , sin ce Sk+\ an d Sk are tw o c o n secu tiv e p o in ts, w e h ave Sfc+\ = .Sit

SfcH-1 = ^ 4 + 2(7 > s'k + q k 1 > ,SK-f 1 = s /c + (i 1

4 + 1 > £ •

U sin g le m m a 2 6 w e g e t + ợ*”*.

T h is im p lies

4-1 = 4 + < r f c _ l 7 ' m = ifc - <7_fe + < / _ f c _1r m = ifc + g - fc- ' ( r m - 7)

Since r m — r/ € D m y we hav e is re p re se n te d th ro u g h £'fc I t follows t h a t

# < 4 + 1 > > # < i f c >

We now prove th a t ^ and 5/e are consecutive points in s u p p /Z f c

S u p p o se t h a t th ere e x is ts € supp/Zfc su ch th a t < Sky th en sjp+j >

(b e c a u se s'k J a n d Sfc +1 are tw o c o n secu tiv e p o in ts in supp/ijfc+i) It fo llo w s th a t

J t V* lit J \ «/ , — / _ o —ik Sfc+1 *“ ^ ^ic - 5fc > + 9 — 2ọ

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 le m m a 2 5 , Sk+i (= Sk) h a s a t m o st tw o r e p re se n ta tio n s th ro u g h Sk and s'k It

follow s that.

# < Sk+1 > < # < s k > + # < t ’k >

and

S k >

M fc±i(ffc±il < # < > < ^ > = 1 4- # < f*

< l + fc + l = f c +2

>

C2 If d o es n o t e x is t s ^ G supp/Zfc su ch th a t s* < < Sjfc, th e n Sfc an d Sjfc are tw o

co n se c u tiv e p o in ts in fik- B y le m m a 2 5 , Sfc+ 1 h a s at m o st tw o re p r e se n ta tio n s th rou gh Sk

and Sfc It fo llo w s th a t

# < Sfc+1 > < # < Sfc > # < 4 >

and

< jfc + 2

l (5fc+l)

T b e th eo rem is proved

C o r o l l a r y 2.8 € s u p p f i n a n d Ịsn — s'n \ < cq~n then

< ( n + 1)‘

Proof L e t s'n = ÍQ < 1 1 < • • • < £jfc = b e k + 1 c o n s e c u tiv e p o in ts in su p p /in

T h e n b y P r o p o s itio n 2.1 w e h a v e k < c and

_ /^n(^Ảr) _ i^n{^k) Unfak—l ) /^n(^l) Mn(^o) — l) H'Txi't'k— 2) 0)

< ( n + l ) ( n + 1) • • • ( n + 1) < ( n + l ) c

3 L o c a l d i m e n s i o n s o f f r a c t a l

T h e fo llo w in g th e o r e m is an e x te n s io n o f P r o p o s itio n 2.1 in [6]

T h e o r e m 3 1 For .S’ € s u p p n , we have

log /z„(an )

a ( s ) = l i m

n l o g g

provided that th e limit, exists O therw ise, by ta kin g th e upper a n d low er lim its, respec­ tively, we g et the form ulas for a*( s) a n d a * ( s )

Proof S u p p o s e th a t th e r e e x is t s th e lim it

at.) = lim.

w h e re /i) = [ 5 — ft, s 4* h].

For /1 > 0 ta k e II su c h th a t

n —1

< h < q

T h en

S in ce

w e havr

j i ( H{ s q n ' ) ) < n ( B ( s , h)) < ị.i(B (s, q n ))

s - 8n\ <

i= n + l

V - , r n ợ

> q r n = ■■

0 — 1

f i ( B ( s , q - n )) < ftn ( B( s , cq~n )) < ụ ( B ( S>2 c q - n )),

where c = - + 1 is a constant, depending only on IĨ 1 and q Similarly, we have

n { B { s , q~ n )) > f i n { B ( s , c ' q ~ n )).

T hus

ụ n(B (s,c 'q- n) ) < ị i { B { s, q - n)) < fj.n(B(s, cq~n)).

Tliis implies

" /" " ) ) < log ỊJ.(B(s,q~n )) < l o g ịin { f í ( s , c ' q - n ))

- n lo g < / " - n l o g ç “ - n lo g q

For t € B ( s } cq) n su pp /j, w e h ave

I i n - sn \ < 2 cq~n

By corollary 2.8, we have

M n(tn) < ( n + l )2c//n (S n ).

T h is im p lies

Hn(B (s ,c q ~n )) < (2c + l)(n + l)2c/in(sn), and

lim j g j O f W f l g l Z ) ) ) > lim i g g M f n ) = lirn | l o g ^ n ( * n ) | n-4oo — n l o g q n->oo —71 lo g <7 n-+oo —n l o g r / Similarly, we have

lim < lim l lo g M n (a" ) l

T his completes the proof.

T h e follow ing c o ro lla rie s c a n b e p ro v e d by th e s a m e te c h n iq u e a s th o s e in [6] (b y using theorem a 3.1 and proposition 2.2).

C o r o l l a r y 3 2 For s = € SUPPM> we ftave

lo g (m + 1) lo g # < * • „ >

q (, s ) = — - + lim - ' — — ,

lo g q n-»oo n l o g ọ

provided th a t the lim it exists O therw ise, by ta kin g th e upp er a n d lower lim its respectively

wc g et th e form ulas for a* (s) a n d a * ( s )

C o r o lla r y 3.3

lo g q,

where

a = sup{a(â) : s € supp/i} a* = sup{a*(.s) : s € s u p p f i }

C o r o l l a r y 3 4 (see [6] M a in T h e o re m ) F or m = 2, ro = 0, r j = 1,7"2 = 3, q = rn + 1 =

3, we have

ã = a * = 1

A c k n o w l e d g e m e n t s T h e a u th o rs are gratefu l to P rofessor N g u y e n T o N h u o f N ew M ex ­ ico U n iv ersity an d P rofessor N g u y e n N h u y o f V in h U n iv ersity for th eir h elp fu l su g g estio n s and valu ab le d iscu ssio n s d u rin g th e p rep aration o f th is paper.

R e f e r e n c e s

1 K J.F alcon er, Techniques in Fractal G eom etry, J o h n W ile y a n d S o n s, 1997.

2 K J F a lco n er, Fractal G eom etry, M athem atical Foundations and A pplications, J o h n

W iley a n d Sons 1993

3 N.T.Nliu, Fractal and Non-Fractal D im ensions , S tatistic Seminar, P a rt 3, New Mex­

ico S ta te U n iversity, U S A , 2000.

4 T H u, The Local D im ensions o f the B ernoulli convolution associated w ith the

G o ld e n n u m b e r, T ra n s A m er M a th Soc 349 (1997),2917-2940

5 T H u an d N T N h u , Local D im ensions o f Fractal M easures A ssociated w ith Uni­

fo rm ly D istributed Probabilistic S ystem s, to ap p ea r in th e A M S P ro c eed in g s, 2001.

6 T H u, N N g u y en a n d T W a iig , Local D im ensions o f The Probability Measure A s­ sociated with The (0, 1, 3) Problem, P reprin t, 2001.