• No results found

The dimension of the boundary of the Lévy Dragon

N/A
N/A
Protected

Academic year: 2020

Share "The dimension of the boundary of the Lévy Dragon"

Copied!
6
0
0

Loading.... (view fulltext now)

Full text

(1)

VOL. 20 NO. 4 (1997) 627-632

627

THE DIMENSION

OF

THE

BOUNDARY OF

THE

LiVY

DRAGON

R DUVALL

Department

ofMathematical Sciences

University ofNorth CarolinaatGreensboro Greensboro,

NC

27412,

USA

and J. KEESLING

Department

ofMathematics

University ofFlorida

P.O Box

118105,358 LittleHall Gainesville,

FL

32611-8105,

USA

(Received

April 7,

1997)

ABSTRACT.

In

thispaper we describe the computations done by the authors in determining the dimensionoftheboundary of the

Lvy

Dragon. A

general theorywas

developed

for calculatingthe dimensionofa self-similar tileandthetheorywasappliedtothisparticularset. The computationswere

challenging

It

seemed thatamatrix which was 25 25would havetobe analyzed.

It

waspossibleto

reduce the analysistoa 752 x752 matrix.

At

lastitwasseen thatif

A

wasthe

largest

eigenvalue ofa certain734 734matrix,then

dimH(K)

Perron-Frobeniustheory playedanimportantrole

inanalyzingthis matrix.

KEY WORDS AND PHRASES:

Hausdorff dimension,iterated function systems, attractors, fractal geometry

1991

AMS SUBJECT

CISSIFICATION CODES:

Primary28A20, 54E40, 57Nxx.

1.

INTRODUCTION

Self-similarsetshave becomeanimportantareaof research. Theyare notonly interesting geometric figures, but

they

have also

proved

tobeuseful inimage compression and storage

],

[2],

and

[3].

It

is

possible that thoseself-similar setsthatare tilesmight

play

animportant

role

inimage compressioninthe

future.

So,

thesesetsdeservetobestudied insomedetail. The worksummarized here is anattemptto extend whatisknownaboutthesesets.
(2)

628 P DUVALL AND KEESLING

In

[4], P

L6vy studied thecomplexcurvewhich has cometobeknown as the

Lvy Dragon

He showedthat the

Dragon

isself-similar,andarguedthat theplanecanbe tiledby copies ofit L6vy’s paper

is atourde

force

onthissubject

It

is amazingthathewasabletodetermine somany propertms ofthis and other curves in

[4]

withouttheuseofmodemcomputers

Although L6vy found many properties ofthe

Levy Dragon,

he did not compute the Hausdorff

dimensionofitsboundary.

In [5,

page

236] G

Edgarasks whetherthedimensionoftheboundary ofthe

Dragon

is greater than one Inapreviouspaper

[6]

weanswered thatquestioninthe affirmative andin factdetermined aprecisevalue for this dimension.

In

theprocess of doingthiswedevelopedageneral theory for computingtheHausdorffdimensionoftheboundary ofself-similarfractaltiles

TheL6vy

Dragon

is theattractor for the

IFS

F

{I’1, f2},

where

1’1

and

f2

are similitudes of

Euclidean

space R

given by

fl(x,y)

(x-y

x

+y)

2 2

f2(x,y)=

(x+y+l y-x+l)

2 2

Much that we discoveredabout the

Dragon

holdsin abroadersetting

So,

it isappropriatetohave

fairly

general

notational conventions. TheHausdorffdimensionofa set

E

isdenoted by

dimn E,

andthe

’lower

andupper box-countingdimensionsof

E

aredenoted by

dimsE

and

d-sE.

Definitionsofthese dimensionfunctions can befoundin

[7],

[8]

and

[9].

Let

12denote the collectionof sequences

I

{il, i2,---}

with

i

{

1,2,---,

n}. 12

willdenote the

setof sequences of length kwith entriesfrom

{

1,2,.--,

n}.

12 has the naturalmetricgiven by

f

1,

if’il

:/:

jl

d(I,

J)

c,

c.--c,,,

if

i

j for k

<

mand

im+l

For

I

12,

welet

I

denotethe finitesequence

{il,i2,...,i},

andwilloftenusethe shorthand

fh

for

thecomposition

f,a f,---f,

andch fortheproduct

c, c--.c,

Wewillalso needthemapsa andg"f2

K

given by

a(il,

i,

...)

{i2,

i3,

...}

and

g(I)

lira

f,t f,’" "f, (x)

Itisnotdifficultto

k---*oo

seethat

g(I)

is independent ofthe choiceofx

X.

Themapa is thewell-known shiftmap on the

sequence

space

2.

SUB-SELF-SIMILAR SETS

In [7],

Falconer

gave

an

(abstract)

wayto determinethe dimension of sub-self-similarsets Given a set

F

fl, f2,"’,

fn

}

of contractingsimilitudeson

R

t,

theclosedset

E

issaid tobe

sub-self-similar

for

F

if

E

C

I,.J,lf,(E).

It follows that

E

C

K,

where

K

istheattractorfor

F.

Note

inparticularthat thetopological boundary

OK

of

K

iss.s.s.,since the interior int

K

of

K

ismappedimo itselfbyeachof

theopen mappings

f,.

Thefirst crucial observationabouts.s.s,sets is

PROPOSITION

2.1.

Let

E

beaclosedset. Then

E

iss.s.s,for

F

ifand

only

if

E

g(A)

for

somecompactset

A

C f2 such that

a(A)

C

A.

Furthermore, if

E

is s.s.s., such a set

A

isgiven by

A

{I

flg(a

(I))

E

for all k

_> 0}.

Theset

A

inProposition2.1 isfundamentalin the calculationof

dimnE.

For

k

>

0,let

Ak

be the

setoffinite

sequences

obtainedby truncating elements of

A

after kterms.

For

s

>

0 definite
(3)

629

U/,(u)

c

u,

ana

l,

(V)

l;(V)

,

if

#

j.

THEOREM

2.2.

Let

E

be aclosedset which iss.s.s,with respecttothefamily

{fl,

f2,

"-’,

f,}

of

contractingsimilitudes whichsatisfiesthe opensetcondition. Letsbethe numbersatisfying

-(s)

1 Then8

dimnE

dirnBE

dimBE.

3.

BOUNDARIES OF ATTRACTORS

One

obvious difficulty in applying Theorem 2.2 is that one has to sum over finite sequences

determinedbyaninfinitecondition Itwasimportanttoshowthatmembershipin

A

canbe determined from finite data

PROPOSITION

3.1.

Ak

{I

6_

fklfI

(K)

N

OK

:fi

0}

Once

wehad Proposition3.1 wehadastrategy forcomputing

dimnOK. Suppose

forconvenience

that the c, all havethe samevalue c, and

suppose

thatwecan determine cr lira

[Ak[

/’,

where denotesthe cardinality of theset

S.

Then

Thus dimncgK is the solutionto ac 1, or

dimnOK

In(a)

In(c)

4.

THE

LIVY

DRAGON

We now revert to letting

K

denotethe

IMvy

Dragon

with its

IFS{f],f2}

Lvy

[4]

showed,

among

other things, that

K

tilesthe

plane

in the sensethatthe

plane

canbewrittenas the union of congruent copiesof

K

thatmeet

only

intheirboundaries.

Let

S

bethe unit

square

with vertices

{(0,

0),

(0,1), (1,1), (1,

0)}

and let

To

C

S

be the triangle

with vertices

{

(0, 0), (1/2,1/2), (1,

0)}.

L6vy

viewed

K

as lira

F’(To).

k--oo

(4)

630 PDUVALL AND KEESLING

S

isthe union of fourtriangles in

S

withhypotenuseanedge of

S

and thirdvertex at

(1/2,

1/2)

Wedenote the triangulation of

R

consisting ofthesetrianglesand theirintegraltranslatesby

To

To

and itsneighborhoodin

To

areshown inFigure2 If

T

is aright triangleintheplane,werefertothevertex opposite the hypotenuse as the top vertex of

T,

and the other vertices as

left

and rght so that we encounterthe vertices in theorderleft, top, andfightas we traversetheboundaryinclockwise order The

left

(rght) edge of

T

is the edge determined by the left (fight) and top vertices of

T

The

triangulation

To

has a subdivision

T

obtained by subdividing each triangle

T

into the two triangles

determinedbythe lef (fight) edge of

T

andthemidpoint of the

hypotenuse

of

T

Continuingin the obvious way gives a sequence

To

-

T

-.--

-

Tk

--..

of subdivisions of

R

into fight isoceles

triangles. Eachtrianglein

T

has diameter

For

each

T

T,

let

T

r be thetrianglein

/

whosehypotenuseistheleft edge of

T

and whichhasavertexoutsideof

T.

Define

T

similarly.

One

can easily see that

F(T0)

T0

U

T0

.

In

fact,thefollowing propositionis immediate

PROPOSITION

4.1.

For

each

>

O,

F(To)

is a union of 2 triangles in

T.

F+(To)

is obtainedfrom

F (To)

by replacingeach

T

in

F

(To)

by

T

and

T

Weused theformula at the end ofSection 3 to compute the Hausdorffdimension of

OK

The difficulty thatwe encountered was that

K

andOKare very complicated objects, andwecan onlysee

finiteapproximationstothem,so it wasnotclearhowtocompute

A

What we were abletowork with was

Bk

(Ik

e ak If,

(To)

i)Fk

(To)

=fi

0

}.

Thenwewereabletodeterminetherelationshipbetween

Ak

and

Bk

tocompletethe calculation

\

/

Figure3 TheNdghborhood Ordering

One

ofthe difficulties indealingwith

K

is that it is noteasytolocate pointsinthe interiorof

K

by lookingat

F

k

(T0).

One

needsto findobjects of positivearea whichpersist fromiteration to the next, but

assoon as atriangle

T

appearsatonelevel,its interior is discardedatthe nextleveland isreplaced by twotrianglesexteriorto

T. It

is not untilthe iterationshave become sufficiently intertwinedto create covered triangles thatthe discarded mass is filledback in by adjacent triangles Through computer experimentationwediscoveredthatcoveredtrianglesdo not exist until the 14thiteration, andonly8of

the

214

trianglesthatcomprise

FI4(To)

arecovered!

However,

iratriangle

T

E

Tk

iscovered,then

T

is

theunion oftwotriangles in

Tk+l

which are also covered.

It

follows that intTC

Fk+"’(To)

for all
(5)

6 31

Figure4.

Ns

and

FSTo

PROPOSITION

4.2.

Bk

C

Itfollows that

[Bk[ <_

[Ak[

We

alsousedthefact thatateach stage

F

(I)

isa tilingof

R

So,

in

using

B

twoways, wewereabletodetermine lower andupper bounds for

dimHoaK

whichwerethe same number thusgivingus acompletedetermination of this dimension

Let/3

lira

IBm[

1/

TIIEOREM

4.3.

n(/)

5.

COMPUTATIONAL ISSUES

In

thelightof Theorem4.3,ourgoalwastounderstandtheasymptoticbehavior of the number

[Bk[

oftriangles in

Fk(To)

which are notcovered.

To

this end, we needed to study the neighborhood

structuresintroduced in Section 4.

In

ordertokeeptrackof thenumberof neighborhood typesateach stage in

F

k

(To)

weneededamatrix which was

215

x

215

However,

by experimentingwith this matrix, wefound thatwecould reduceit to a smaller 734 x 734 matrix withnonnegative integerentries which was irreducible Perron-Frobenius

theory

was used to determine that there was a unique largest eigenvalue

A

tothis matrix. Thiseigenvaluedeterminedthecardinalityof

[Bk[

tobe

A

kasymptotically

This allowed usto compute the dimension oftheboundary of the Lrwj

Dragon

We

were ableto

determineprecise bounds for

A

usingintegerarithmetic and also determine as estimate for

A

tohigh precision usingthe

power

method.

By

Section4,

In(A)

dimn,

OK

ln(v/)

Floating point computations using the power method give the estimate

A

1.934007183 The

rigorous estimatethat we obtainedusing integerarithmetic was 1.824190

<

A

<

1.974189. Thislast

estimate

proved

that

A

wasgreater than

V/,

and that the

Hausdorff

dimension ofoakisgreater than1

Thus, the

boundary of

the

Dragon

has

Hausdorff

dimension greater than 1, and

Edgar’s

questionwas
(6)

632 P DUVALLAND KEESLING

ACKNOWLEDGEMENT.

Thisresearch was done while the first authorwasaVisiting Professorat

the University of Florida He gratefully acknowledges UF’s hospitality, as well as research support from the University of NorthCarolina, Greensboro

REFERENCES

BARNSLEY, M,

FractalsEverywhere

(2nd

ed

),

Academic

Press,

1993

[2] BARNSLEY, M

and

HURD,

L.,

Fractal

Image

Compression,

A

K.

Peters,

1993

[3] FISHER,

Y.

(ed.),

Fractal

Image

Compression:

Theory

and

Apphcatton

to

Dtgaal

Images,

Springer-Verlag, 1995

[4]

LIVY,

P,

L,es

courbesplanesougauchesetlessurfaces compos6edeparties sembladesautout, Journal de l’EcolePolytechmque

(1938),

227-247,249-291

[5]

LIVY,

P,

Plane orspacecurvesandsurfaces consisting ofparts similartothewhole, Classtcson

Fractals

(G

Edgar,ed),Addison-Wesley,pp 181-239.

[6] DUVALL,

P and

KEESLING, J,

TheHausdorff dimension of theboundaryof theL6vy

Dragon,

submitted.

[7] FALCONER, K

J,

Sub-self-similar sets,

Trans.

Amer.

Math.

Soc.

347,8

(1995),

3121-3129

[8] FALCONER, K J.,

Fractal

Geometry

MathemattcalFoundations and

Apphcattons,

JohnWiley

and

Sons,

1990

[9] EDGAR, G A., Measure, Topology,

and Fractal

Geometry,

Springer-Verlag, 1990

References

Related documents

Figure 2 indicates the representative chromatographic peaks of FFAs in the sediment samples in both seasons while the free fatty acid levels of the river sediments

Field experiments were conducted at Ebonyi State University Research Farm during 2009 and 2010 farming seasons to evaluate the effect of intercropping maize with

Surgical Site Infection Reduction Through Nasal Decolonization Prior to Surgery..

It was decided that with the presence of such significant red flag signs that she should undergo advanced imaging, in this case an MRI, that revealed an underlying malignancy, which

Also, both diabetic groups there were a positive immunoreactivity of the photoreceptor inner segment, and this was also seen among control ani- mals treated with a

penicillin, skin testing for immediate hyper-.. sensitivity to the agent should

After successfully supporting the development of the wind power technology, an approach is needed to include the owners of wind turbines in the task of realizing other ways, other

19% serve a county. Fourteen per cent of the centers provide service for adjoining states in addition to the states in which they are located; usually these adjoining states have