ON THE LOCAL DIMENSIONS OF FRACTAL MEASURES

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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 Ị

1

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

X

q

, 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

oo n

s = £ > ■ * , Sn = Y > %

1=0 t=0

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

Ị

1 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\ — P

2

—

3 a n d

p

=

ị

=

ỵ.

' t y p e s e t b y

49

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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. O ur 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

Ị

1

is

a t

least q~n 3. suppfXn

c

suppfj

.n+ 1 a n d

suppụ,

= U^L0s 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 )>

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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,

S

n+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 a t $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

izcO (

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 S

n+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

~ 9 ®n-M

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

S

n + 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 I

s 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 S

n + 1 €

suppfJi

n + 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 .

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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 a lity by in d u ctio n . C learly th e in e q 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 a t m o s t tw o rep resen ta tio n s th ro u g h 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

tị

c (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.

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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 <

8

k+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 _

1

r 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.

# <

S

k+1 > < # <

s k

> + # <

t ’k >

and

S k >

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

Mfc+i(5fc+i) # < ^

< L'k

< 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 a t m o st tw o r e 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 .

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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 < 11 < • • • < £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.

AI-ÍÕ+ log /l

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 )) lo

gq~n ~~ log h

~

log q

~ n~ 1

s -

8

n\ <

i= n + l

o o *—71

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 )).

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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

n-i<x> —n log q n-4-oo — n log q

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

s u p p f . i } .

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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.

ON THE LOCAL DIMENSIONS OF FRACTAL MEASURES