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 employed^toobtaina 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 )).

In

actually computingthe dimension ofa set it isrcquently useful tobeabletoconsideronly coverings fromarestrictedfamilyofintervals.

In [1]

Besicovitch established thatcoveringsby dyadic^intervals

(i.e.

^inter- vals of the form [j 2 (j

+ 1)2 ))

producethe samedimension forsets. Billingsley

[2]

extended this resultto r-adic intervalswhere r_>2 issomepositive integer.

However

in

[3]

Billingsleyremarked that he knew ofno

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 equality^weneed only show dim

(E)<dim(E ). A

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

(UEt)= sup

^dims

^(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 covering^of 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}_d

^(I _(I ^{+l(x ))} _(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

^{bearbitraryand}suppose {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 ofgenerality^wewill assume there exists at least one x

eFt

t’lE,.at forwhich

[a

,x

]_C./()(x

^{since theargumentis}analogous 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.

^{Thisispossible}from the definition of x and theassumption x

odt (x o).

^{If x}

>x

implies x

Ft

f3E,: then

I ^(t)(x 0- )ult ,(x0)

^covers

Fi NE,

,at. If however there exists x

>x

such that

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,g _1}

^F e,

^{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

2.2arefisby^adeclam

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)

^dif-

fuseprobabilitydistributionontheBorelsetsof

[0,1],

X represents Lebesgue measure, and

I (x)

^{isthe r-} adicinterval oflength r containing ^x.Thistechniquefor "r-adic

sets"

arose out ofresults ofBillingsley

[2,

4]

concerningthe dimensions of certainsetsobtainedfromdiscrete-timestochasticprocesses.

We

willshowthat these same results, along with Theorem 2.2, can be used todevelop the analogous technique for computing dimensionsofsetsdeterminedbymoregeneralized expansions.

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 value}generally dependson

#,

E,

and theunderlyingstochastic

process

It,

12 ^(since

^thecylindersare determinedbythe

process). It

is notdifficult to showthat

#(E )>0

implies

dim,(E

^{1 and}this fact willprove veryuseful.

We

will alsoneed thefollowing^result.

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 /-dimension^of

E

by constructingsome measure

.

^{for which}

/(E )>0

and

(3.1)

holds,thereby obtaining

dim(E

^0.

^He

^{showedhow this}technique could^beusedtocal- culatethe usualHausdorffdimension of certain subsets of

[0,1]

by identifyingeachpointwith its r-adicexpan- sion

(r

^chosen suitably), ^the

sequence

^{of r-adic}digits 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

^{nowextendthistech-}

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 aninteger}and 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 disjointsubin- tervals determinedbythegiven proportions 32,1 32,k2_^{I. This}splits

[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 ^a1.i

+

^a2,d

, <_

^x

<

^a,i

+

^a=,y

+ld

^1,,

^(but

including the fight hand endpointif k

1-1

^{and j k}

2-1). 12(x

will be the second term in the expansionof x.

Each interval {x I

l(x

)=i

,/2(x

)=j is then divided according^tothe proportions _{a a.1} ^a_a,k-l- Con- tinuingthis subdivision

process,

the n^th stageproducesasplitting^of

[0,1]

^{into k}

lk kn

^{disjointintervals}

{x I

l(x

)=i1,

12(x

)=i

In ^(X)=i,,

{X a_1,, -F a2,td_1,, --F a3,t d_2,,d_{1,a -I"} d- tin ’n

dn

--l,t, d_1,,

<_

x

<

^a,i

+

^a2.,d_{1. -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 n}ylinder 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 n}yHnde bythe relafiom:

(c (i

,i

i. ))

^p.

(i

,i

i. (3.3)

where

. ⁽ⁱ

^i,

^)1,

^p.

^{(i i.}

^{0 if}one 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 n}ylinde

(n

1,2 generate the reitsof

[0,1]

^{d that}

(3.3)

definadiffu

bili

^{msure that}

een

uniquely^to

e

relsets. Regardingthegenerzed

exion

^{I ,12 as asthasfic the}

bili s ([0,1],B,X), (X sgue meure)

^and noting

tt (c. (x))

p.

(ll(X)2(x I. (x))

^wle

x(. (x)) a,t=)ac= a..c=),

^itfollo

bom

rem

3.1 that if E fisfies the

hths

^of

eorem

3.2 then

dimx(E )= dim(E ^).

^Now

dimx(E

d

(E

^{are definby vedngsbom the nylinde}

^generat

^{bythepr I}

^d

^d

rcfi

(3.2)

^{emurthe n}yHndefo a

:ete

^{und Vitalivefingof}

^{[0,1]. From eorem}

^2.2

wecclude

dimx(E dim(E

and the resultfollo, o

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 n^th 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

^{7(c,, (i i. ))}

in the role of^{(ss2 0}E

^s)-t

and

-

^definedif

^iS

^{otherwise.by for all j}

7 correspondstochoosing uniformlyandindependently

among

the allowabledigits^ateach 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 n^th 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

and hence

dim(C

log

1/3 /

log a# asclaimed.

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

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Advances in Difference Equations

Special Issue on

Boundary Value Problems on Time Scales

Call for Papers

The study of dynamic equations on a time scale goes back to its founder Stefan Hilger (1988), and is a new area of still fairly theoretical exploration in mathematics. Motivating the subject is the notion that dynamic equations on time scales can build bridges between continuous and discrete mathematics; moreover, it often revels the reasons for the discrepancies between two theories.

In recent years, the study of dynamic equations has led to several important applications, for example, in the study of insect population models, neural network, heat transfer, and epidemic models. This special issue will contain new researches and survey articles on Boundary Value Problems on Time Scales. In particular, it will focus on the following topics:

•

Existence, uniqueness, and multiplicity of solutions

•

Comparison principles

•

Variational methods

•

Mathematical models

•

Biological and medical applications

•

Numerical and simulation applications

Before submission authors should carefully read over the journal’s Author Guidelines, which are located at

http://www .hindawi.com/journals/ade/guidelines.html. Authors should

follow the Advances in Difference Equations manuscript format described at the journal site

http://www.hindawi .com/journals/ade/. Articles published in this Special Issue

shall be subject to a reduced Article Processing Charge of C200 per article. Prospective authors should submit an elec- tronic copy of their complete manuscript through the journal Manuscript Tracking System at

http://mts.hindawi.com/

according to the following timetable:

Manuscript Due April 1, 2009 First Round of Reviews July 1, 2009 Publication Date October 1, 2009

Lead Guest Editor

Alberto Cabada,

Departamento de Análise Matemática, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain;

[email protected]

Guest Editor

Victoria Otero-Espinar,

Departamento de Análise Matemática, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com