The current state of fractal billiards

1 MB·English

by  Michel L. Lapidus

THE CURRENT STATE OF FRACTAL BILLIARDS
MICHELL.LAPIDUSANDROBERTG.NIEMEYER
2
1
0 Abstract. If D is a rational polygon, then the associated rational billiard
2 table is given by Ω(D). Such a billiard table is well understood. If F is a
closed fractal curve approximated by a sequence of rational polygons, then
ct thecorresponding fractal billiardtableisdenoted byΩ(F). Inthispaper, we
O survey many of the results from [LapNie1-3] for the Koch snowflake fractal
billiardΩ(KS)and announce new results on two other fractal billiardtables,
1 namely, the T-fractal billiard table Ω(T) (see [LapNie6]) and a self-similar
Sierpinskicarpet billiardtableΩ(Sa)(see[CheNie]).

] We build a general framework within which to analyze what we call a
S sequenceofcompatibleorbits. Propertiesofparticularsequencesofcompatible
D orbits are discussed for each prefractal billiard Ω(KSn), Ω(Tn) and Ω(Sa,n),
forn=0,1,2···. Ineachcase,weareabletodetermineaparticularlimiting
.
h behaviorforanappropriatelyformulatedsequenceofcompatibleorbits. Such
t a limit either constitutes what we call a nontrivial path of a fractal billiard
a
tableΩ(F)orelseaperiodicorbitofΩ(F)withfiniteperiod. Inourexamples,
m
F willbe either KS, T or Sa. Several of the results and examples discussed
[ inthispaperarepresented forthefirsttime.

We then close with a brief discussion of open problems and directions for
1
further research intheemergingfieldoffractalbilliards.
v
2
8
2
0 Contents
.
0
1. Introduction 2
1
2. Rational billiards 3
2
1 2.1. Flat surfaces and properties of the flow 6
: 2.2. Unfolding a billiard orbit and equivalence of flows 7
v
i 3. The fractals of interest 9
X
3.1. Cantor sets 10
r 3.2. The Koch curve and Koch snowflake 12
a
3.3. The T-fractal 14
3.4. Self-similar Sierpinski carpets 15
4. Prefractal (rational) billiards 18
2010 Mathematics Subject Classification. Primary: 28A80, 37D40, 37D50, Secondary:
28A75,37C27,37E35,37F40,58J99.
Keywords and phrases. fractalbilliard,polygonalbilliard,rational(polygonal)billiard,law
of reflection, unfoldingprocess, flat surface, geodesic flow, billiardflow, iterated function system
andattractor,self-similarset,fractal,prefractalapproximations,Kochsnowflakebilliard,T-fractal
billiard,self-similarSierpinskicarpetbilliard,prefractalrationalbilliardapproximations,sequence
ofcompatibleorbits,hookorbits,(eventually)constantsequencesofcompatibleorbits,footprints,
Cantorpoints,smoothpoints,elusivepoints,periodicorbits,periodicvs. denseorbits.
The work of M. L. Lapidus was partially supported by the National Science Foundation
undertheresearchgrants DMS-0707524andDMS-1107750,aswellasbytheInstitutdesHautes
Etudes Scientifiques (IHES) inBures-sur-Yvette, France, where hewas avisitingprofessor while
thispaperwaswritten.
1
2 M.L.LAPIDUSANDR.G.NIEMEYER
4.1. A general structure 18
4.2. The prefractal Koch snowflake billiard 21
4.3. The T-fractalprefractal billiard 25
4.4. A prefractal self-similar Sierpinski carpet billiard 27
5. Fractal billiards 29
5.1. A general framework for Ω(KS), Ω(T) and Ω(S ) 29
a
5.2. The Koch snowflake fractal billiard 31
5.3. The T-fractalbilliard 31
5.4. A self-similar Sierpinski carpet billiard 33
6. Concluding remarks 33
References 35
1. Introduction
Thispaperconstitutesasurveyofacollectionofresultsfrom[LapNie1,LapNie2,
LapNie3]aswellasthe announcementofnew results onthe T-fractalbilliardtable
Ω(T) (see [LapNie6]) and a self-similar Sierpinski carpet billiard table Ω(S ) (see
a
[CheNie]).
In§§2and3,wesurveythenecessarybackgroundmaterialforunderstandingthe
remainderofthearticle. Morespecifically,in§2,weintroducethenotionofaratio-
nal polygonal billiard, a flat surface determined from a rational polygonal billiard
and discuss the consequence of a dynamical equivalence between the billiard flow
1
andthe geodesicflow. This dynamicalequivalenceallowsus toexpressanorbitof
a rational billiard table as a geodesic on an associated flat surface, and vice-versa,
with the added benefit of being able to determine the reflection in certain types of
verticesofarationalbilliardtable. Furthermore,in§3,weprovideadditionalback-
ground material from the subject of fractal geometry necessary for understanding
the construction of the Koch snowflake KS, T-fractal T,2 and a Sierpinski carpet
Sa, as well as particular orbits and nontrivial paths.
We thencombinethebackgroundmaterialpresentedin§§2and3toanalyzethe
prefractalbilliard tables Ω(KS ), Ω(T ) and Ω(S ), for n=0,1,2 . We begin
n n a,n
···
by providing a general language for prefractal billiards and subsequently focus on
determining sufficient conditions for what we are calling a sequence of compatible
periodic orbits. While §§4.2–4.4 contain specific results and specialized definitions,
there is an over-archingtheme that is more fully developed in §5.
In addition to providing a general language within which to analyze a fractal
billiard,wediscussin§§5.2–5.4howonecandeterminewell-definedorbitsofΩ(KS),
Ω(T) andΩ(S ), as wellasnontrivialpaths of Ω(KS)andΩ(T) thatconnecttwo
a
elusive points of eachrespective billiard. Relying onthe main resultof [Du-CaTy],
the second author and Joe P. Chen have shown that it is possible to determine a
periodic orbit of a self-similar Sierpinski carpet billiard Ω(S ); additional results
a
andproofsareforthcomingin[CheNie],butasynopsisisprovidedin§§4.4and5.4.
1
Thereferences [GaStVo,Gut1,MasTa,Sm,Ta1,Ta2,Vo,Zo]provideanexcellentsurveyof
thevarioustopicsinthefieldofmathematicalbilliards,aswellasspecificresultspertinenttothe
theoryofrational polygonal billiardsandassociatedflatsurfaces.
2Toourknowledge, theT-fractalT wasfirstintroduced in[AcST].
FRACTAL BILLIARDS 3
Many of the results in §§4 and 5 are being announced for the first time. Specif-
ically, §§4.3 and 5.3 contain new results on the prefractal T-fractal billiard Ω(T )
n
and the T-fractal billiard Ω(T) (see [LapNie6]); §§4.4 and 5.4 contain new results
for a prefractal Sierpinski carpet billiard Ω(S ) and self-similar Sierpinski carpet
a,n
billiard Ω(S ), where a is the single underlying scaling ratio (see [CheNie]). As
a
thesesections constitute announcementsofnew resultsonthe respectiveprefractal
andfractal billiards,we will provide in future papers [LapNie4, LapNie5, LapNie6]
detailed statements and proofs of the results given therein. Given the nature of
the subjectoffractal billiards, wewillclosewith adiscussionofopenproblemsand
possible directions for future work, some of which are to appear in [CheNie] and
[LapNie4, LapNie5, LapNie6].
2. Rational billiards
In this section, we will survey the dynamical properties of a billiard ball as it
traversesa regionin the plane bounded by a closedand connected polygon. In the
latter part of this article, we will remove the stipulation that the boundary be a
polygonandfocusonbilliardtableshavingboundariesthatarefractalorcontaining
subsetsthatarefractal(whilestillbeingsimple,closedandconnectedcurvesinthe
plane).
Under ideal conditions, we know that a point mass making a perfectly elastic
collision with a C1 surface (or curve) will reflect at an angle which is equal to the
angle of incidence, this being referred to as the law of reflection.
Considera compactregionΩ(D) in the plane with simple,closedandconnected
boundary D. Then, Ω(D) is called a planar billiard when D is smooth enough to
allow the law of reflection to hold, off of a set of measure zero (where the measure
is taken to be the Hausdorff measure or the arc length measure on D). Though
the law of reflection implicitly states that the angles of incidence and reflection be
determined with respect to the normal to the line tangent at the basepoint, we
adhere to the equivalent convention in the field of mathematical billiards that the
vectordescribingthepositionandvelocityofthebilliardball(whichamountstothe
positionandangle,sinceweareassumingunitspeed)bereflectedinthetangentto
thepointofincidence. Thatis,employingsuchalawinordertodeterminethepath
on which the billiard ball departs after impact essentially amounts to identifying
certain vectors. Such an equivalence relation is denoted by and, in the context
∼
of a polygonal billiard, is discussed below in more detail.
For the remainder ofthe article,unless otherwise indicated, whenD is a simple,
closed,connectedandpiecewisesmoothcurvesoasto allowthe lawofreflectionto
hold, we assume D is a closed and connected polygon. In such a case, we will refer
to Ω(D) as a polygonal billiard.
Wecanrigorouslyreformulatethelawofreflection,inthecontextofapolygonal
billiard,as follows: the vector describing the directionof motion is the reflection—
through the tangent at the point of collision—of the translation of the vector pre-
viouslydescribingthedirectionofmotion. Onemayexpressthe lawofreflectionin
terms of equivalence classes of vectors by identifying these two vectors to form an
equivalenceclassofvectorsintheunittangentbundlecorrespondingtothebilliard
table Ω(D); see Figure 1. (See [Sm] for a detailed discussion of this equivalence
relation on the unit tangent bundle Ω(D) S1.)
×
4 M.L.LAPIDUSANDR.G.NIEMEYER
Denote by S1 the unit circle, which we let represent all the possible directions
(or angles) in which a billiard ball may initially move. To clearly understand how
one forms equivalence classes from elements of Ω(D) S1, we let (x,θ),(y,γ)
× ∈
Ω(D) S1 andsay that (x,θ) (y,γ)if and only if x=y and one of the following
× ∼
is true:
(1) x=y is not a vertex of the boundary D and θ =γ;
(2) x=yisnotavertexoftheboundaryD,butx=yisapointonasegments
i
of the polygonD and θ =r (γ), where r denotes reflectionin the segment
i i
s ;
i
(3) If x = y is a vertex of D, then we identify (x,θ) with (y,g(γ)) for every g
inthe groupgeneratedby reflectionsinthe twoadjacentsideshavingx(or
y) as a common vertex.
For now, we shall denote by [x,θ] the equivalence class of (x,θ), relative to the
equivalence relation .
∼
The collection of vertices of Ω(D) forms a set of zero measure (when we take
ourmeasure to be the Hausdorffmeasureor simply, the arc-lengthmeasure on D),
since there are finitely many vertices.
Thephasespaceforthebilliarddynamicsisgivenbythequotientspace(Ω(D)
×
S1)/ . In practice, one restricts his or her attention to the space (D S1)/ .
∼ × ∼
Thebilliardflowon(D S1)/ isdeterminedfromthecontinuousflowonΩ(D)
× ∼ ×
S1)/ as follows. Let x0 be an initial basepoint, θ0 be an initial direction and
∼
ϕ (x0,θ0)beaflowlinecorrespondingtotheseinitialconditionsinthephasespace
t
(Ω(D) S1)/ . The values t for which ϕ (x0,θ0) (D S1)/ constitute the
× ∼ j tj ∈ × ∼
returntimes (i.e.,timesatwhichϕ (x0,θ0)returnstothesection,orintersectsitin
t
a non-tangential way). Then, the discrete map ftj(x0,θ0) constitutes the section
map. In terms of the configuration space, ftj(x0,θ0) constitutes the point and
angle of incidence in the boundary D. Since Ω(D) is the billiard table and we are
interested in determining the collision points, it is only fitting that such a map be
called the billiard map. More succinctly, we denote ftj by fj and, in general, such
a map is called the Poincaré map and the section is called the Poincaré section.
Furthermore, the obvious benefit of having a visual representation of fj(x0,θ0)
in the configuration space is exactly why one restricts his or her attention to the
section (D S1)/ . Specifically, all one really cares about in the end, from the
× ∼
perspective of studying a planar billiard, are the collision points, which are clearly
determined by the billiard map.
Inordertounderstandhowonedeterminesthenextcollisionpointanddirection
of travel, we must further discuss the billiard map f . As previously discussed,
D
f : (D S1)/ (D S1)/ , where the equivalence relation is the one
D
× ∼→ × ∼ ∼
introduced above. More precisely, if θ0 is an inward pointing vector at a basepoint
x0, then (x0,θ0) is the representative element of the equivalence class [(x0,θ0)].
The billiard mapthen acts on (D S1)/ by mapping [(xk,θk)] to [(xk+1,θk+1)],
× ∼
where xk and xk+1 are collinear in the direction determined by θk and where θk+1
is the reflection of angle θk through the tangent at xk+1. In general, we have
fk[(x0,θ0)]=[(xk,θk)], for every k 0.
D ≥
Remark2.1. Inthesequel,wewillsimplyrefertoanelement[(xk,θk)] (Ω(D)
∈ ×
S1) by (xk,θk), since the vector corresponding to θk is inward pointing at the
∼
basepoint xk. So as not to introduce unnecessary notation, when we discuss the
FRACTAL BILLIARDS 5
Figure 1. A billiard ball traverses the interior of a billiard and
collideswiththeboundary. Thevelocityvectorispointedoutward
atthepointofcollision. Theresultingdirectionofflowisfoundby
eitherreflectingthevectorthroughthetangentorbyreflectingthe
incidence vectorthroughthe normalandreversingthe directionof
the vector. We use the former method in this paper.
billiard map f corresponding to the nth prefractal billiard Ω(F ) approximating
Fn n
afractalbilliardΩ(F),wewillsimplywritef asf . Whendiscussingthediscrete
Fn n
billiardflowon(Ω(F ) S1)/ ,thekthpointinanorbit(xk,θk) (Ω(F ) S1)/
n n
× ∼ ∈ × ∼
will instead be denoted by (xkn,θkn), in order to keep track of the space such a
n n
point belongs to (namely, with our present convention, Ω(F ) S1). Specifically,
n
×
k refers to the number of iterates of the billiard map f necessary to produce the
n n
pair (xknn,θnkn). An initial condition of an orbit of Ω(Fn) will always be referred to
as (x0,θ0).
n n
An orbit with a finite period is called a closed orbit. If there exists m 1 such
≥
that fm(x0,θ0) = (x0,θ0), then the resulting orbit is called periodic; the smallest
D
positiveintegermsuchthatfm(x0,θ0)=(x0,θ0)iscalledtheperiod oftheperiodic
D
orbit. In the event that a basepoint xj of fj(x0,θ0) is a corner of Ω(D) (that is,
D
a vertex of the polygonal boundary D), then the resulting closed orbit is said to
be singular. Inaddition, if there exists a positive integer k suchthat the basepoint
x−k offD−k(x0,θ0)is acornerofΩ(D) (here,fD−k denotes the kth inverseiterate of
f ), then the path traced out by the billiard ball connecting xj and x k is called
D −
a saddle connection.
We say that an orbit O(x0,θ0) is dense in a rational billiard table Ω(D) if the
pathtraversedbythe billiardballinΩ(D)isdenseinΩ(D). Thatis,the closureof
the setofpoints comprisingthe pathtraversedby the billiardballisexactly Ω(D).
Likewise, the points of incidence (i.e., the footprint) of a dense orbit will be dense
in the boundary D.
Definition 2.2 (Footprint of an orbit). Let O (x0,θ0) be an orbit of a polygonal
D
billiard Ω(D) with an initial condition (x0,θ0) D S1. Then
∈ ×
(1) (x0,θ0):=O (x0,θ0) D,
D D
F ∩
the trace of the orbit on the boundary D, is called the footprint of the orbit
O (x0,θ0). When we are only interested in a prefractal billiard Ω(F ), we de-
D n
note the footprint of an orbit by (x0,θ0).
Fn n n
Fortheremainderofthearticle,whendiscussingpolygonalbilliards,wewillfocus
our attention on what are called rational polygonal billiards, or, more succinctly,
rational billiards.
6 M.L.LAPIDUSANDR.G.NIEMEYER
Definition 2.3 (Rational polygon and rational billiard). If D is a nontrivial con-
nected polygon such that for each interior angle θ of D there are relatively prime
j
integers p 1 and q 1 such that θ = pjπ, then we call D a rational polygon
j ≥ j ≥ j qj
and Ω(D) a rational billiard.
2.1. Flat surfaces and properties of the flow. In this subsection, we will dis-
cuss what constitutes a flat surface and how to construct a flat surface from a
rational billiard. Then, in §2.2, we will see how to relate the continuous billiard
flow on (Ω(D) S1)/ with the geodesic flow on the associated flat surface.
× ∼
Definition 2.4 (Flat structure andflat surface). Let M be a compact, connected,
orientable surface. A flat structure on M is an atlas ω, consisting of charts of the
form(Uα,ϕα)α A, where Uα is a domain(i.e., a connectedopen set) in M and ϕα
isahomeomorp∈hismfromU toadomaininR2,suchthatthefollowingconditions
α
hold:
(1) The collection Uα α A covers the whole surface M except for finitely
many points z ,{z ,.}..,∈z , called singular points;
1 2 k
(2) all coordinate changing functions are translations in R2;
(3) the atlas ω is maximal with respect to properties (1) and (2);
(4) foreachsingularpointz ,thereisapositiveintegerm ,apuncturedneigh-
j j
borhood U˙ of z not containing other singular points, and a map ψ from
j j j
this neighborhoodto a punctured neighborhood V˙ of a point in R2 that is
j
a shift in the local coordinates from ω, and is such that each point in V˙
j
has exactly m preimages under ψ .
j j
We say that a connected, compact surface equipped with a flat structure is a flat
surface.
Remark 2.5. Note that in the literature on billiards and dynamical systems, the
terminologyanddefinitionspertainingtothistopicarenotcompletelyuniform;see,
for example, [GaStVo, Gut1, GutJu1, GutJu2, HuSc, Mas, MasTa, Ve1, Ve2, Vo,
Zo]. We haveadoptedthe abovedefinitionforclarityandthe reader’sconvenience.
Wenowdiscusshowtoconstructaflatsurfacefromarationalbilliard. Consider
a rational polygonal billiard Ω(D) with k sides and interior angles pjπ at each
qj
vertex z , for 1 j k, where the positive integers p and q are relatively prime.
j j j
≤ ≤
The linearportions ofthe planar symmetriesgeneratedby reflectioninthe sides of
thepolygonalbilliardΩ(D)generateadihedralgroupD ,whereN :=lcm q k
N { j}j=1
(the least common multiple of the q ’s). Next, we consider Ω(D) D (equipped
j N
with the product topology). We want to glue ‘sides’ of Ω(D) D× together and
N
×
constructanaturalatlasontheresultingsurfaceM sothatM becomesatranslate
surface.
As a result of the identification, the points of M that correspondto the vertices
of Ω(D) constitute (removable or nonremovable) conic singularities of the surface.
Heuristically, Ω(D) D can be represented as r Ω(D) 2N , in which case it is
× N { j }j=1
easy to see what sides are made equivalent under the action of . That is,
∼ ∼
identifies opposite and parallel sides in a manner which preserves the orientation.
SeeExample 2.6andFigure2foranexampleofaflatsurfaceconstructedfromthe
equilateral triangle billiard Ω(∆).
FRACTAL BILLIARDS 7
Figure 2. The equilateral triangle billiard Ω(∆) can be acted on
by a particular groupof symmetries to produce a flat surface that
is topologically equivalent to the flat torus. In this figure, we see
that opposite and parallel sides are identified in such a way that
theorientationispreserved. Thisallowsustoexaminethegeodesic
flow on the surface. We will see in §2.2 that the geodesic flow on
theflatsurfaceisdynamicallyequivalenttothecontinuousbilliard
flow.
Figure 3. Partially unfolding an orbit of the square billiard Ω(Q).
Example 2.6. Consider the equilateral triangle ∆. The corresponding billiard is
denoted by Ω(∆). The interior angles are π,π,π . Hence, the group acting on
Ω(∆)toproducetheflatsurfaceisthedihed{ra3lg3rou3p}D . Theresultingflatsurface
3
is topologically equivalent to the flat torus. We will make use of this fact in the
sequel.
2.2. Unfolding a billiard orbit and equivalence of flows. Considerarational
polygonal billiard Ω(D) and an orbit O(x0,θ0). Reflecting the billiard Ω(D) and
theorbitinthesideofthebilliardtablecontainingthebasepointx1 oftheorbit(or
an element of the footprint of the orbit) partially unfolds the orbit O(x0,θ0); see
Figure 3 for the case of the square billiard. Continuing this process until the orbit
isastraightlineproducesasmanycopiesofthe billiardtable asthereareelements
of the footprint; see Figure 4. That is, if the period of an orbit O(x0,θ0) is some
positive integer p, then the number of copies of the billiard table in the unfolding
is also p. We refer to such a straight line as the unfolding of the billiard orbit.
Given that a rational billiard Ω(D) can be acted on by a dihedral group D to
N
produce a flat surface in a way that is similar to unfolding the billiard table, we
canquicklyseehowthe billiardflowis dynamicallyequivalentto the geodesicflow;
see Figure 5 and the corresponding caption.
One may modify the notion of “reflecting” so as to determine orbits of billiard
tables tiled by a rational polygon D. As an example, we consider the unit-square
billiard table. An appropriately scaled copy of the unit-square billiard table can
be tiled by the unit-square billiard table by making successive reflections in the
8 M.L.LAPIDUSANDR.G.NIEMEYER
Figure 4. Unfolding an orbit of the square billiard Ω(Q).
Figure5. Rearrangingtheunfoldedcopiesoftheunitsquarefrom
Figure 4 and correctly identifying sides so as to recover the flat
torus, we see that the unfolded orbit corresponds to a closed geo-
desic of the flat surface.
sides of the unit square. One may then unfold an orbit of the unit-square billiard
table into a larger square billiard table. When the unfolded orbit of the original
unit-squarebilliardintersectstheboundaryoftheappropriatelyscaled(andlarger)
square, then one continues unfolding the billiard orbit in the direction determined
bythelawofreflection.3 Wewillrefertosuchanunfoldingasareflected-unfolding.
We may continue this process in order to forman orbitof a largerscaledsquare
billiard table. Suppose that an orbit O(x0,θ0) has period p. The footprint of the
orbit is then FB(x0,θ0) = {fBi(x0,θ0)}pi=−01. If s is a positive integer (i.e., s ∈ N),
then the footprint FBs(x0,θ0) := {fBi(x0,θ0)}is=(p0−1) of an orbit constitutes the
footprint of an orbit that traverses the same path s-many times. For sufficiently
larges N,anorbitthattraversesthesamepathasanorbitO(x0,θ0)s-manytimes
∈
canbereflected-unfoldedinanappropriatelyscaledsquarebilliardtabletoforman
orbitofthelargerbilliardtable;seeFigure6. Suchatoolisusefulinunderstanding
therelationshipbetweenthebilliardflowonarationalpolygonalbilliardΩ(D)and
a billiard table tiled by D, and will be particularly useful in understanding the
nature of particular orbits of a self-similar Sierpinski carpet billiard in §5.4.
Asonemayexpect,ifDisarationalpolygontilingabilliardtableΩ(R),thenan
orbit of Ω(R) may be folded up to form an orbit of Ω(D). This is done by making
successive reflections in D, the result being an orbit of Ω(D); see Figure 7 for the
case of a square billiard table.
3
Thatis,assumingtheunfoldedorbitislongenoughtoreachasideofthelargersquare.
FRACTAL BILLIARDS 9
Figure 6. Unfolding the orbit of the unit-square billiard in a
(larger)scaledcopyoftheunit-squarebilliard. Thisconstitutesan
example of a reflected-unfolding. The edges of the original unit-
square billiard table and the segments comprising the orbit have
been thickened to provide the reader with a frame of reference.
Figure 7. Illustrated in this figure is the process of folding up
an orbit of a square billiard table, as discussed at the end of §2.2.
In the first image, we see an orbit of unit-square billiard table.
Partitioningtheunitsquareintonine equallysizedsquares,wesee
that we can fold up the orbit by making successive reflections in
thesidesofthesquarescomprisingthepartition. Usingsufficiently
many reflections results in an orbit of one of the squares of the
partition.
3. The fractals of interest
We are primarily interested in fractals with boundaries either partially or com-
pletelycomprisedofself-similarsetsandfractalsthatareself-similar. Soastomake
the material discussed in §4–5 more accessible, we provide a few basic definitions
from the subject of fractal geometry.
Definition 3.1. Let (X,d) be a metric space and φ:X X.
→
(i) (Contraction). If there exists 0<c<1 such that
d(φ(x),φ(y)) cd(x,y)
≤
for everyx,y X,then φis calleda contraction (orcontraction mapping).
∈
(ii) (Similarity contraction). If there exists 0<c<1 such that
d(φ(x),φ(y)) =cd(x,y),
for every x,y X, then φ is called a similarity contraction. This unique
∈
value c (0,1) is called the scaling ratio of φ.
∈
Definition 3.2. Let (X,d) be a complete metric space.
10 M.L.LAPIDUSANDR.G.NIEMEYER
(i) (Iterated function system and attractor). Let φ k be a family of con-
{ i}i=1
tractions defined on X. Then φ k is called an iterated function system
{ i}i=1
(IFS).
An iterated function system is so named because the map Φ : K
K, given by Φ() := k φ () and defined on the space K of nonemp→ty
· i=1 i ·
compactsubsetsofX,Scanbecomposedwithitself. Indeed,foreachm N,
∈
we have
k k
(2) Φm()= ... φ φ ().
· i1 ◦···◦ im ·
i1[=1 im[=1
Furthermore, there exists a unique nonempty compact set F X (i.e.,
F K), called the attractor of the IFS, such that ⊂
∈
k
(3) F =Φ(F):= φ (F).
i
i[=1
(ii) (Self-similar system and self-similar set). In the special case where each
φ isacontractionsimilarity,fori=1,...,k,thenthe IFS φ k issaidto
i { i}i=1
be a self-similar system and its attractor F is called a self-similar set (or
a self-similar subset of X).
If X is complete, then so is K (equipped with the Hausdorff metric) and hence,
since it can be shown that Φ : K K is a contraction, it follows from the con-
→
traction mapping theorem that Φ has a unique fixed point (thereby justifying the
definition of the attractor F above) and that for any E K, Φm(E) F, as
∈ →
m (where, as in Equation (2), Φm is the mth iterate of Φ). (See [Hut].)
→∞
We state the next property in the special case that will be of interest to us,
namely, that of an IFS in a Euclidean space.
Theorem 3.3 ([Hut]; see also [Fa, Thm. 9.1]). Consider an iterated function
system given by contractions φ k , each defined on a compact set D Rn, such
{ i}i=1 ⊆
that φ (D) D for each i k, and with attractor F. Then F D and in fact,
i
⊆ ≤ ⊆
∞
(4) F = Φm(E)
m\=0
for every set E K such that φ (E) E for all i k. Here, the transformation
i
Φ:K K is gi∈ven as in part (i) of D⊆efinition 3.2.≤
→
Not every fractal is self-similar or embedded in Euclidean space. However, such
sets represent an important collection of examples of fractal sets. In the next
subsection, we will discuss the fractal subsets (self-similar or not) of R or of R2 of
direct interest to us in this paper.
3.1. Cantor sets. ACantorset is a setwith veryrichandcounter-intuitiveprop-
erties; topologically, it is a compact and totally disconnected (i.e., perfect) space.
In order to illustrate some of the properties that make a Cantor set so interesting,
werefertothecanonicalexampleofaCantorset: theternaryCantorset. Wefocus
on three methods for constructing the ternary Cantor set: 1) by tremas, 2) as the
unique fixed point attractor of an iterated function system, and 3) in terms of an
alphabet.