A note on equivalent interval covering systems for Hausdorff dimension on ℝ

VOL.

II

NO. 4

(1988)

643-650

A NOTE ON

EQUIVALENT

INTERVAL

COVERING

SYSTEMS

FOR HAUSDORFF DIMENSION ON

R

C.D. CUTLER

Department

of Statistics and Actuarial Science Universityof Waterloo

Waterloo,Ontario Canada N2L 3G1

(Received September 30,

1987)

ABSTRACT. The Hausdorff dimension ofasetin R isusuallydefined by consideringcountablecoveringsof thesetby generalintervals.

In

thisnote weestablish sufficient conditions under whichcoveringswhose members arerestrictedto aparticularfamily g ofintervalswillproducethesamevalue fordimension. Aresult of Bil-lingsleyisthen employedtoobtaina general techniqueforcomputingthe dimensions ofsetsdefinedbycertain types ofgeneralized expansions.

A

specificexampleis included.

KEY

WORDS

AND

PHRASES. Hausdorffdimension,Vitalicovering,self-similar. 1980AMS

SUBJECT

CLASSIICATION. 28A75,28A12.

1. INTRODUCrION.

Let s_>0.

The s-outerHausdorff measure ofa set E C_ R isusuallydefinedby

H"(E

lim inf d

(lj)o

d_{(tj)_<}

where

lj

is an interval in R and d

(lj)

denotes the diameter of lj.

It

iseasy tosee that the value of H

(E

isunchangedwhenthecoveringsof E are restrictedtolacingclosedintervals.

(It

will be convenient for us to consideronlyclosed intervals in Section

2.)

It

iswell known that there existsaunique point s0 such that

H(E )=

for

s<s

and

H(E

)=

0 for

s>s

0. Thisvalue s iscalled the Hausdorff

dimension of

E

(denoted

by

dim(E

)).

generalconditions on interval families which wouldguarantee preservationof thecorrectdimensionvaluefor all sets.

We

address thisproblemin Section2.

2.

COVERING RESULTS.

Let

F C_ R be a closed interval.

A

collection g of intervals is a Vitalicoveting of F if for each e>0 and each xeF there exists ! eg suchthat x d and d

(I)<e.

If an interval collection g is aVitali covetingof F then, for each E C_ F,we candefine

Hs"(E

and dim

(E)

by

and dim

(E)

sup{a

Hs’(E

oo inf{a

[Hs(E

0}

where usual Hausdorff dimension

dim(E

results when g istakentobe the collection of all closed intervals.

We

automaticallyhave

dim(E )<dim

_s

(E)

for any Vitali coveting g sotodemonstrate equalityweneed only show dim

(E)<dim(E ). A

property of dimension which will be useful in this is the result dim

(UEt)= sup

dim_s

(EL)

for any countable collection

{EL

We

willcall g a bounded Vitalicoveting of F if, for each xeF,there existsa

sequence

of intervals from g

(which

wewilldenoteby {lj

(x)}j

such that

(i)

xdj

(x)

for each j

(ii)

d(l(x))0

d

(t

+(x ))

b

(x)>o.

(iii)

i

_d

(I

(x))

We

willsaythat aboundedVitali coveringof F is open if, for each x

eF,

the

sequence {lj

(x)}j

can be chosensothat x

d(x

for each j

(where

lj(x

denotes the interior of

I

(x)).

Theorem 2.1 deals withopenbounded Vitalicoverings.

THEOREM 2.1.

Let

g be an open bounded Vitali coveting of a closed interval F C_R. Then dim_s

(E)

dim(E

forall E C_F.

PROOF.

Let

E CF. Thenwecanwrite

E=U

uEi,

=2 =1

where E,m {x

eEI

inf

d

(I

+l(x ))

d

(I

(x))

>_

1/i

and d

(I t(x

))

>_

1/m

}. To provethe theorem it issufficientto

show

dims(Ei

_< dim(Ei).

Let

O<6<l/m

bearbitraryandsuppose {Ft}t isacountable coveting of

Ei

byclosed intervals with d

(Ft)--<6

for each k. Without loss ofgeneralitywe canassume

Fk

CF and

Ft

flEi

,

#

6. For

each k we will show that there exist four intervals from g whichcover

Ft

t’lE

,

and whose diameters do not exceed

d(Ft).

For

every x

eFtflEi,at

there exists

l(t)(x

such that d

(lj

(k

)(x

))>d (Ft)>d

(lj(t)+t(x)).

Writing

Ft

[at

,bt

then we must have

[at

,x

]_C/j(t)(x

or

Ix

,bt

]_C/j()(x

).

Without loss ofgeneralitywewill assume there exists at least one x

eFt

t’lE,.at forwhich

[a

,x

]_C./()(x

since theargumentisanalogous when beginningwith the assumption

[x

,bt

]_C/

()(x).

Let

Xo sup{x

eft

f’lEi,at

[at

,x

]_C./j

(t)(x)}.

Now

x

oFt

_CF

and so there exists k’ such that

d(lt,(Xo))<_i

d(F).

Let

x6-_<Xo be some point in F

tClEi,

such that

[a

t.xo-]_C/(t)(x0-)

and

I

()(x

6- )lql

,(x

o)

#

6.

Thisispossiblefrom the definition of x and theassumption x

odt (x o).

If x

>x

xF tqE, then it follows that

[x

,bt

ICIj

(t)(x

and wedefine x inf{x

Ft

NEi

x,bt

ICIj

(t)(x

)).

We

can then find x

+

_>x

with x+

Ft

NE,

and k" such that

[x

+

,bt

]_CIj

(t)(x

+

),

d

(It

,,(x 1))_<i

d

(Ft)

and

Ij

(t)(x

+_)rqlt

,,(x 1)

_#

-

Then

rt

fqei

,,.

c_

lj

(t)(x

6-

)ult

,(x 0)ult ,,(x

1)UI (t)(x

+

). Now

d

(b

()(x

o-

))

d

(6 ()(

o-

))

d

(,

()(x:

))

d

(F

--<

d

(lj(

t)+l(x0-

))

<-

andsimilarly d

(F

_<

i. Thus we obtain

[d

(I(t)(x0-

))0

+

d

(lt

,(x0))

+

d

(It,,(x 1))

+

d

(I(t)(x

+

))] _<

4i

o

d

(F

)0.

--1 =1

Hence

weconclude

inI d

(I)o

<

4i inf

’

d

(F)0.

,

_e, _1

e,

,

-!

g

F

d teal

d_(t) d_(F

ng

0+ _weobtain

Hs(Ei

4i

H (E,

,

).

s

sho dim

(Ei

dim(E,

,

asrequiredd complettheprfof the theorem.

moregeneral u is the follongtheorem in wch the

n

ruirement

isreled.

We

11 1a und Vitalicovetingof F completeif for each xF the

uence

{I

(x)}

n chensothat when-ever x isan

endint

of

ly

(x)

(if

at

all)

there

ests

Ig th d

(I)d

(I

(x))

and >0 such that

(i)

(x-

1 if x isthelefthand

endint

of

1

(x)

(t

n

the le nd

endint

of F

(ii)

(x

+)

I ifx isthe

fit

d

endnt

of

1i

(x) (but

notthefighthand

endint

of F

).

O2.2.

t

F acl inte of R and g ampleteund Vitalicovetingof F.

en

dim

s(E)=dim(E)

forl E

F.

PROF.

e

f

foHo

tt

of

corem

2.1 until diuion of x isrch.

d

endint

of

I

,(x 0)

thennomfifion is

rr.

However

if x is

c

lefthd

endint

of

I

,(x0)

we

n

to add in the inteal I profided by

(i)

in the deflation ofa complete coveting.

We

then sclt

x that

ly

()(x

)I

.

e

ao

mififionmust done in theseof x if x is

e

fit

d

endint

of

l(,)(x

).

uently

F

OEi,

n coveredbysix inteals g, none of wh

ete

ex

d

(F,).

m

e

hth

ol

eorem

2.2arefisbyadeclam

o

coveringswch includ the

r-ac

inter-valsbut is much more emeive. the next fion we idera

tque

lotaingthe dimeion ol r-raints.

3. COMPUTING DIMENSION.

In

[2], [3],

and

[4]

Billingsleydevelopedatechniqueforcomputingthe dimensions ofsetsdefined interms

og

(t,

())

of r-adicexpansions byconsideringlimitsof the form

n-.oolim

log

X(l.

(x))

where

-

isa(suitably

chosen)

We

begin by reviewing the nrydefinitionsandresultsfrom

[2]

and

[4].

Let

11,1_ be astochastic processonaprobability space

(X

,r,#)

taking values inacountable, possibly finite,set S.

A

subsetof X of the form

c(il,i2

in)= {xCX

ill(x)=

il,I2(x)=

i2

In(x)=

in}, where i1,i2

in

are members of S, is called an n-cylinder. Assume that each "oo-cylinder" has #-measure zero i.e.

#({xX

II1(x)=il,

I2(x)=i2

})=0

for each

sequence

il,i2

Let E

beasubset of X.

For

each 0<a<1 define

L(E)=

lira inf

la(ci)

._.o tlq_DE (c,)<

where each

c

isan n-cylinder

(for

some n

).

With methods similarto thoseused in the case of a-outer Hausdorff measure it can beshown that L isanouter measure on the subsets of X and for each E

_CX

there exists a unique point a0,

0!:_a0_<l,

suchthat

L

(E)

oo for

a<a0

while L

(E)

0 for

a>ao.

Thenumber a0 iscalledthe

/-dimension

of E anddenotedby

dim,(E

).

This valuegenerally dependson

#,

E,

and theunderlyingstochastic

process

It,

12

(since

thecylindersare determinedbythe

process). It

is

notdifficult to showthat

#(E

)>0

implies

dim,(E

1 andthis fact willprove veryuseful.

We

will alsoneed thefollowingresult.

THEOREM

3.1. (Billingsley):

Let

(X

,’,#)

and 11,12 beasdefined aboveand, for each x

X,

let

cn

(x)

denote the n-cylinder containing x. That is,

cn

(x)

{x

’X

[I

t(x

’)

1

l(x

In

(x ’)

In

(x)}.

Let

beanotherprobabilitydistribution on

(X

whichassignsmeasurezerotoeach oo-cylinder. If

{x

log-(cn(x))

}

E C_ CX

,,-lim

log/(c,,

(x))

0

(3.1)

then

dim,(E

0

din(E

).

[]

Billingsley’s idea wastocompute the /-dimensionof

E

by constructingsome measure

.

for which

/(E )>0

and

(3.1)

holds,thereby obtaining

dim(E

0.

He

showedhow thistechnique couldbeusedto cal-culatethe usualHausdorffdimension of certain subsets of

[0,1]

by identifyingeachpointwith its r-adic expan-sion

(r

chosen suitably), the

sequence

of r-adicdigits comprising the stochasticprocess and an n-cylinder correspondingto an r-adic intervaloflength

r-.

Since coveringsby r-adic intervalsproduceusualHausdorff dimension, Theorem3.1 can beappliedwith suitable

-

and Lebesguemeasure.

We

nowextendthis tech-niquetogeneralized expansions.

A

generalized expansion of a number in

[0,1]

will be defined as follows.

For

each n 1,2 let

kn

>_2 be anintegerand choose values

0<an,l<

<an ._1<1,

setting an,0 0 and an,k, 1. The initial proportions a_1, a_1.

-

determine a division of

[0,1]

into the disjoint intervals

[a

,

,a 0,1

kl-2,

and

[311_1,1

].

We

will indicate that a point x in

[0,1]

falls into the interval

(i

0,1 k

1-1)

bythe notation 1

l(x

i. 1

l(x

will be the first term in theexpansionof x

(with

respecttothechoices

an

).

At the second stage eachinterval {x

[I

l(x )=i

isdivided into k disjoint subin-tervals determinedbythegiven proportions 32,1 32,k2_I. Thissplits

[0,1]

into k

lk

disjointintervals

which are most conveniently expressed in the form {x

ll(X )=i,

12(x)=j} for some choice of =0,1

k1-1

and j =0,1

k2-1.

Letting

dn,,

=an,+l-an, for each n and i, wecan alternatelywrite {x I

(x

)=i,

I2(x

)=j {x a_1.i

+

a_2,d

,

<_

x

<

a,i

+

a=,y

+ld

1,,

(but

including the

{x I

l(x

)=i1,

12(x

)=i

In

(X)=i,,

{X a1,, -F a2,td1,, --F a3,t d2,,d1,a -I" d- _{tin ’n}

dn

_--l,t, d1,,

<_

x

<

a,i

+

a2.,d1. -F d-

an

_,i.

+dn

_-t.,. d.i

}.

e

uen

1

l(x ),

12(x

is the generalized

exmion

of x,

tng

valu in the ctable t S

{0,1

k.-l[n

1,2 }. If

r2

isa

ifive

integerand

k.

r, a..i =i/r for each n,then the result is the usual r-adic

exmion

of x.

(If

x has more than one r-adic

exmion

ts

methpr duc theteinating

one.)

We

ll using coverings

com

of inteals lonng to the lltion g of nylindem c

(i

i.

)=

{x

[I

(x

)=i

I.

(x)=i.

generat

bythe generalized

exnsiom.

(Note

that me nylindem may

emp;

ts

curs ifsome

i

S {0,1

k-1}.)

orderthat g a

compete

und Vitalicovetingof

[0,1]

wendtomake the resmicfion:

e

diameters of the

(nonemp)

nylindem

s

at acontrledrate. If

c. (x)

is the nylinder containg the

int

x then

d

(q

+(.

))

i.

d

(c. (x))

=i.

d.

+d. +,(x) b

(x )>0.

(3.2)

It

isosyto

tt

(3.2)

impliesthe diameter of ch nylinder

s

to zero n and forc S to

afitet.

We

now

ed

tothe

mn

result.

OM3.2.

t

I

t(x

),

12(x

reprintthegeneriz

exmion

of x

[0,1]

th

rt

to achoi

ofo

a.,i, 1 k.-1, n 1,2 d

su

the rdtinginte collation g of

n-cyHndefisfi

(3.2).

t

definoverthe nyHnde bythe relafiom:

(c

(i

,i

i. ))

p.

(i

,i

i.

(3.3)

where

.

(i

i,

)1,

p.

(i

i.

0 ifone or more

i

k,

p.

(i

i.

_,i p.

_(i

i._)

(comistency

condition),

p

(i

1,and

lim p.(i

i.)

0.

en

een

uquelyto adiff

bili

sfionthe rel

m

Bof

[0,1],

d if

E C

/x,[0,1]

Hm

logp.(ll(X)J2(x)

l.(x))

=0

(3.4)

then

m(E

0

dim(E

).

(E )>0

then

dim(E

0.

PROF.

s

H

follow from

eorem

3.1 and

eorem

2.2.

It

isclr

om

thecucfi of the n-cylinde

(d

rcfion

(3.2))

that the nylinde

(n

1,2 generate the reitsof

[0,1]

d that

(3.3)

definadiffu

bili

msure that

een

uniquelyto

e

relsets. Regardingthegenerzed

Theorem 3.2 canfrequentlybeappliedtoCantorsetsbuiltfromgeneralized expmions.

We

say C is a generalized Cantorset if itcanbeexpressedin the form C {x

(1 t(x ),/2(x

S"

where

S"

issome subset of the countable

product

,,Xt{0,1.

k,,-1}.

The simplest case occurs when

S"

,,X-tS"

where

S,,

_C

{0,1

k,,-1}

is the set of "allowable" digits at the nth _{stage. The resulting}

Cantor

_{set is called} "independent" andcan be written as C {x

II,,

(x)S,,

for all n}. Theusual

Cantor

set

(minus

acountable collection of"endpoints" corresponding to somenumbers with morethanonetriadicexpansion)isanexample, resulting when k,, 3, a,,.i

i/3,

and

S,

(0,2}

for all n.

We

havethefollowingcorollary.

COROLLARY

3.2.

Let

C be a generalized independent

Cantor

set built fromgeneralized expansions whose n-cylinders satisfy

(3.2).

Let

s, denotethe size of the set ofallowable digits

S,,

atthe

nt

stage.

Suppose

there exists

d,

suchthat

d,,.i

d,, for each

eS,,.

Ifthelimit

log

(s

s

s

)-t

lira

log

dtd

d.

existsandequals 0

(3.5)

then

dim(C

0.

PROOF. We

applyTheorem 3.2 with C in the role of E and

-

definedby

(ss2

s)-t

if

iS

for all j

7(c,,

(i

i. ))

_{0 otherwise.}

7 correspondstochoosing uniformlyandindependently

among

the allowabledigitsateach stage, t

We

applyTheorem 3.2 tocomputethe Hausdorff dimension of a certaingeneralized "Markov"

Cantor

set

(i.e.

a

Cantor

setin which the allowable digitsat the nth stage depend onthedigit chosenat the

(n-1)

’

stage).

Whiletechniquesexist intheliteratureforcalculatingthe dimemion of self-similarsets

(see

[5], [6],

[7])

by obtainingtheso-called "similarity dimension", thefollowingsetisonlyself-similar in alimitingsense.

(It

can bepartitioned as a countablyinfinite union of

similitudes.) For

each n take

k

5,

and for n even set

d,o=dn=dn,=ct

and

dn,t=dn,a=a

(where a>O, a>O

satisfy

3a+2a

1)

whilefor n oddset

d,,,t=d,3=#

and

d,0=d,2=d,4=b

(where

>0, b>O satisfy

2+3b

1.)

We

set

St={1,3},

allowing only "1"or"3"tobeselectedatthe firststage. Letting

S, (i)

denote the allowabledigitsatthe nm stage given that "i" is selected at the

(n-l)

th stage, we use the rules

S(O)=

{1),

S(1)

{0,2},

S,, (2)

{1,3},

S

(3)

{2,4},

and

Sn

(4)

{3}. (These

rulescorrespondtothepermissiblemovesin a random walk on

{0,1,2,3,4}

with reflecting barriers at 0 and

4.)

We

will show the resulting

Cantor

set C {x

II

(x )S

and

1 (x)Sn

(In_t(x))

for each n has dimension log

1/3

/

log

aB.

Construct

a

Mar-kovprobabilityrule onthe n-cylinders accordingtothe initial distribution p

t(1)

p

t(3)

1/2

and transition probabilities

p(ll0)=l,

p(011)=l/3,

p(211)=2/3,

p(l12)=p(312)=l/2,

p(213)=2/3,

p

(413)

1/3,

p

(314)

1,and p

(i

J

0 otherwise.

(This

gives p

2(i ,J

P

t(i

)p (j and, induc-tively, p

(i

i

p,,_(i i,_Dp

(4,

i,,-D

.)

Theresultingdistribution ./ isclearly supportedon

the set C. Furthermoreit issufficienttocomputethe limit in

(3.4)

over oddintegersandweobtain,forany xC,

lim logp2

+(I

t(x

I2 +(x

))

lim

1o

2-t3

log

dldl(X)

d2n+t,z,+t(x

loga/3+t

ACKNOWLEDGEMENI. This research was supported by the Natural Sciences and Engineering Research Council ofCanada.

[]

BESICOVITCH,

A.S. Onexistenceof subsets of finitemeasureofsetsofinfinitemeasure. Indag.Math. 14

(1952)

339-344.

[2]

BILLINGSLEY, P.

Hausdorff dimension inprobability theory, Illinois

J.

Math. 4

(1960)

187-209.

[3]

BILLINGSLEY, P.

ErgodicTheoryandInformation, Krieger

(1978).

[4]

BILLINGSLEY,P. Hausdorff dimension inprobability theory

II,

IllinoisJ.Math.5

(1961)

291-298.

[5]

HUTCHINSON, J.E.

Fractalsandself-similarity, IndianaUniv. Math.

J.

30

(1981)

713-747.

[6]

FALCONER, K.J.

TheHausdorff dimension ofsomefractalsandattractorsofoverlapping construction,

J.

Statist. Phys.47

(1987)

123-132.