�9 1 0 8 " Ftng Dcjun tt alz Dimensioa~ and Gauges

DIMENSIONS AND GAUGES

FOR SYMMETRIC CANTOR SETS

Feng Dejun, Rao Hui, Wen ghiying, Wu Jun" (W uhan University, China)

Received Dec. 1, 1995

1, Some definitions and notions

1.1. Hansdorff gauge and Packing gauge

A fundton h: [0,1"] --~ 8 is said to be a dimension function if it has the following

properties:

(1) h(t) is continuous and increasing, h(0) = 0, h(t) > 0 when t > 0.

(2) there exists a constant M > 0, h(2t) < Mh(t) for any t > 0.

For any set E C 8" and dimension function h, the Hausdorff h-measure for E is defined as

HA(E), = iim inf( )-'~, A(IE, [), where (E~} is a ~'-cover for g}. e-.o The Packing h-measure for E is

P~ (s s ffi in!{ ~ ~ ( ~ ) , where {K-} is a cover for E},

where, for any set F,

(F): = limsup( ~ , h ( IF, I), {V, } is a ~-Packing for F }. &--o

If 0 < H ~ (E) < + co, then we call h a Hausdorff gauge for E. If 0 < / ~ (E) < + oo,

we call h a Packing gauge for E.

1.2. Symmetric Cantor sets

Given a sequence of integers {n~ } and a sequence of positive numbers {r } satisfying ~c~

< 1, r~ ~ 2, lim~-.~c2""c~ = 0, we construct sets {Ei}~oas following:

For any i, E~ is a union of disjoint finite closed intervals contained in [0 '1] , which will be

called i-th basic intervals.

P~ = [0,13.

�9 Research support by the NNSF of China.

Approx. T heo ry b - / I s Appl. , I1:4 , Dec. , 1995 �9 109 �9

E; is a union of 'h closed intervals contained in F~, each interval has length c~, and they

situate space-symmetrically in E o , one of them has left end point O, and another has right end

point 1. Now, suppose E~ has been defined well for for some t , we begin to construct E~+l : for

any i-th basic interval J~, there exists ,~+t closed intervals contained space -symmetrically in

Js, each interval has length C,+1 IJi ], one of them has the same left end point as Jk, another

has the same fight end point as 3~. Thus we get (t + 1)-th basic intervals contained in J~.

When J~ goes over all Lth basic intervals, we get all (t + 1)-th basic intervals, so get the set

E t + ! �9

Let E({~,}, {c, }) = ~ E,, we call it a one-dimension symmetric Cantor set. Let t - - I

E" ({n, }, {c, }) = ] ' ~ " E({., }. {c, }), and we call it a m- dimension symmetric Cantor set.

2 Main Results

Given a one-dimention symmetric Cantor set E = E({n, }, {c~ }), let pz be a natural m~,ss

distribution on E, i. e , /~ (3~) = (n~nz...n,) -~ for any Lth basic interval 3,. We have the fol-

lowing resuits:

I ,mma 2. I. (1) for any x E [ 0 , 1 ] , t'f>O,p~([x,z-l-t])~6p~([O,t]).

(2) ~ ( [o ,2: ] )~7~ ([o,t]).

(3) for any xE E, t>O, I~ ( [ x - t , x + t ' ] ~ l P ~ ( [o,t]) .

Let uO) = / ~ ( [ 0 , 0 , we have,

Theorem 2. 2. The function u is a Hausdorff &auge and a Paci~ing gauge for E.

Generally, we have

Theorem 2. 3. For a dimension function h, let q = 5m h(t) , ~ ~ ( [o , t ] ) ' then

(i) I f O<:rl<,+oo, then h is a Hausdorff gauge for E.

(ii) If q - 0 , then It t (E)=0. (iii) If qf+eo, then H ' ( E ) = + o o .

Iog/~ ([O,t]) Corollary 2.4. dime - ,-.olim logt , where dim(E) denotes the Hausdorff di-

mension of E.

Dime = ll~ Iog~([0, t ] ) where Dim(E) denotes the Packing dimension of E. ,-.o Iogt '

Corollary 2. 5. dime = lira IogOh ~ " ' n , ) Dime -- ~ log('h'h""~) .L)" - log(cx%...c,)' ,_._ _ Iog(clc~...c,_t

�9 110 �9 Feng Deju. ct ai, Dim~sions and Gauges

Using the properties about symmetric Cantor sets, we get the following theorem z log(h(t)) b log(h (t)) Theorem 2.6 For any dimension function h, let a = lira ~ , = I ~

,'~" ,...o l og ( t ) " Then for every s6 [a,b], there exist nEN and ECR', such that dim(E)=s, and h is a Haus-

dorff gauge for E.

Department of Mathematics

Wuhan University

Wuhan 430072

PRC