HATHIC/92/262

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

NUMERICAL INVARIANTS OF CANTOR SETS

Sergio Plaza

MIRAMARETRIESTE

T

IC/92/262

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

NUMERICAL INVARIANTS OF CANTOR SETS

Sergio Plaza *

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

In this note we give a review of basic properties of limit capacity and Hausdorff dimen-

sion and some applications oi these concepts to dynamical systems and numerical sets.

MIRAMARE - TRIESTE

September 1992

Proyectos Fondecyt 0499/91 and Dicyt 9133 P.S.

Permanent address: Departamento de Matematicas, racultad de Ciencias, Universidad de Santiago,

Casilla 5659, Correo 2, Santiago, Chile.

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I.- LIMIT CAPACITY

Lei(X,p) be a metric space. Letv4 c X be a compact set, for each e > 0,n(e) will be denote the smallest number of e-balls (i.e., balls of radius e) needed tocover A the limit capacity, d{ A), of A is defined as

Remark Note that s > d( A) implies lim£_o n(e)£J = 0

Example: The middle-a-Cantor set, K{a) C [0 ,1 ] , 0 < a < j , is defined asfollows

where ATo ( a ) = [ 0, 1 ] and for each x > 1, Jf,> i ( a ) is obtained from A"j( a) deletingfrom each connected component (interval) of /f ,•( a ) , which have length, say i, a centralinterval of length al. See figure 1.

oi

Figure 1

it is clear that for each t > 1, K,(a) D K(ct) and iC,(a) consist of 2 ' intervals oflength ( !LYL)'. Therefore

d(K(a)) = lim • l n ( 2 " J

^S)n) l n ( 2 ) - l n ( l - a )

For a = } we obtain the classical ternary Cantor set, C, and d(O = j |

Remark. From this example we immediately see that the limit capacity is not invariantby homeomoiyhisms, since central Cantor sets in the line are all homeomorphic, forinstance, to the classical ternary Cantor set, but it is clear that taking different values ofat's we obtain central Cantor sets with different limit capacities.

II.- BASIC PROPERTIES OF THE LIMIT CAPACITY

1.- Let A c Rm be a compact set (j 0) (R™ with the euclidean metric). Then0 <d((A) <m.

In fact, let C C Rm be a closed cube that contain A, then for each e > 0 ,viA.fi) < n(C,£). Setting 0 < e < 1,0 < ' " ^ u " ^ ! n ^ $ I , and takinglimit (upper) as e —» 0, we let 0 < d(A) < d(C). Now it is easily show thatd(C) = TO.

2.- Let v4 c X be a compact set. Let r e ]0 ,1[ and c > 0 be constants. Set

3.- From the definition of d(A) it follows immediately that

d(A) - inf {d > 0 : for some to > 0 and any e E]O,to[, 4 admit a coverby e — balls with no more than t~d elements}

4.- Let (X, p) and (X', p') be metric spaces. Let A c X and A' c X' be compactsets. Then d(A x A') <d(A) + d(A').

5.- A m a p / : (X,p) -» (X' ,p ' ) such that for all x,y G X, <Hf{x) ,f<v)) <c • d{x,y)^ for some constants c > 0 and -7 > 0 is called a (c,7)-H0ldermap. When 7 = 1, / is called a Lipschitz map. We say that / is Lipschitzhomeomorphism if / and f~l are Lipschitz, this is equivalent to the existence oftwo constants c,c > 0 such that for all 1, y € X,cp{x,y) < p'(f(x),f(y)) <cp{x,y).

Now if / : X —> X' is a (c, 7) -Holder map then for any compact set E C X,d(f(E)) < ~t-xdiE). In particular, if / i s a Lipschitz map, then d(f(E)) < d(E)and consequenUy, if / is a Lipschitz homeomorphism, d{ f(E)) = d(E).

In fact, let e > 0 be given. If n( E, e) = n. Let B a ball of radius e, then / ( B ) is aball ofradiusc-e*\ so with nballsofradiusce7 we cover f(E). Thusn(/(fi) ,cE'T) <n = n( E, e), and the proof follows.

Example (R. Mane") There exists a countable compact set Km c Rm with d( /Cm) = m.Construction: let ai > a2 > • • • > 0 be a sequence of positive real numbers such thatY^Zi On < 1 • Choose a sequence of closed cubes C C Rm with edge of length an

and such that

(1) \x - y\ > on for each 1 G Cn and y e Ck,ni k,n> 1,

(2) lim,^,,, sup i e G t |x| = 0.

I T

In each Cn choose points z, (n),l < i < [ a i - " ] m ([] denote theentire part function)whose coordinates are separated by intervals of length o j .

Set Km = { zj"* : n > 1, 1 < t < [ a i ~ n ] m }. It is clear that Km is a countablebounded set. Now in order to cover Km by balls of radius EB = ^ we need at least oneball centered at each of one point x\n). Thus n( Km, £ ) > [ ai""]™. And then

= m.

III.- HAUSDORFF MEASURES

Let (X,p) be a metric space. Let U C X the diameter of U, denoted \U\, issup{p(i, y) : I , J £ [/}. Let E c X and 6 > 0 be given, we say that a collection{Ui}i£i is 6-coverof E if, B c Ul6/Lr, andO < |C/,-| < 6.

Let i5 c X, s > be given. We define the a-dimensional Hausdorff measure of Eas

en

K'( E) = lim inf { ^ \Ui\' : {C/i},>i is a countable 6 - cover of E}~~* i=i

Note that if we set W| = inf {^™i |t/i|* : {£/<}<>! is a countable fi-cover of S }then 6 < 6' implies H'S(E) > K J . Thus fixing £ cf J£ and 3 > 0, it follows that thefunctions —> K j ( £ ) is non-increasing, and therefore always exists lim «_o H\(E) andis equal to sup S>Q H't{E). HenceK'CE) is well defined for subsets of X. Moreover,the set function E —> H"{ E) is an outer measure in X,

IV.- BASIC PROPERTIES OF THE s-DIMENSIONAL HAUSDORFF MEA-SURES

L- Scaling Property. Let A c Rm and X > 0. Define \A = { Xx : i E A). ThenH'(XA) = V

In fact, if {Ui} is a countable £-cover of A then {A[/,} is a countable A6-covero f M . Hence, WXS(A) < £ ( >, |A[/,f = X ' ^ > t |C/a*. This inequality holdsfor any 6-coverofA then letting 6 -* 0 , we obtain H'(XA) < \"H'{A). Theoiher inequality is obtained replacing X by \~l and A by A A

2.- Let / : X -* Y be a (c, 7)-Holder map. Then, for any 3 > 0, JiH/(4)) <c*H'(A) for any subset A of Jf.

In fact, if {[/J is a countable 6-cover of A, then {f{A n Ut)}i>i is cfi -coveroff (A), since \f(A nUi)\ < c\Ui\\ ThusEJ /MnUi ) !* < c* £<>

In particular, if/ is a Lipschitz homeomorphims, weobtain c*K'( A) <H'(KA)) <cfH'(A). From this, follows that if / : FT1 —• R™ is an isometry, i.e., ||/ (z ) - / ( y ) ||=|| x-y ||, then forany subset^ of it™ and any 3 > 0, we havethat H'( f( A)) = H*{ A), consequently the Hausdorff measures are translationand rotation invariants, i.e., H\A + u) = H'{A) and K'(R(A)) = H'(A)where A + u = {x + u : x G 4} and R : Rm -* Rm is a rotation.

3.- For each m > 1 there is a positive constant km, which only depends on m, suchthat for every A c Rm,

where £m denote the m-dimensional Lebesgue measure in R™.The constant Ji:m is the m-dimensional volume of the unitary ball in Rm, and infact is equal to ^

V.- HAUSDORFF DIMENSION

Let A c X. Note that, as function of a,H"(A) is non-increasing. In fact, lett > s,andlet{EA} be a countable 6-cover of ,4, then £™, |E/i|( = £ " 1 \Ui\*~'\Ui\' <6'-" YXi IC/. I * - Thus taking the infima over all countable S-cover of A, we obtainH\(A) < £ (-'H|. Then, if t > s, and H'(A) < +00, letting S -> 0, in the aboveinequalities we obtain H ' ( J4 ) = 0. From this it fallows that in the graph of H*(A)againts 3 there is a critical value so at which H"{ A) jumps from +00 to 0. This criticalvalue is called the Hausdorff dimension of A and it is denoted by HD( A).

CD

Figure 2

Note that K*°(A) may be 0, +00 or satisfies 0 < Hn{A) < +00, where soHD(A).

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VI.- BASIC PROPERTIES OF THE HAUSDORFF DIMENSION

1.- Let A C Rm be an open set, then HD(A) = m.In fact, as A is an open set it contain a ball of positive m-dimcnsional volumeand since Hm( A) = ikII,£m(J4), we conclude that HD(A) > m. On the otherhand, it is easy show that HD(A) < m.

2.- Let 5 be a smooth ^dimensional submanifold of flm, then HD(S) = n.

3.- If A C B C X, then HD{A) < HD(B).

In fact, from the inequality H'(A) < H*(B) for any a > 0, the assertionfollows.

4.- Countable Stability. Let {Ei} be a countable sequence of subset of X. Then

In fact, as for each ; , E, c U£i Et it follows that HD( Ej) < HD{ \J£X Ed, andthen sup 1£j5oo HD(E,) < HDiu^Ei). On the other hand, if 3 > HD(Ei)for alii > 1, then «•(£,) = 0. AsW(u£i(£i) < YZ\ H'(Ei), it followsthatW(u^,(B,-) = 0,andso/f£>(U£,£,-) < s. Therefore HD(\J^iEi) <sup i&za, If D(Et).

5.- If E C X is a countable set, then HD{ E) = 0. It follows immediately frompro-perty 4.-

6.- Let / : .X -» Y be a (c,T)-H61der map, then HD(f(E)) < -)~lHD(E),for all subset BofX, In particular, if / is a Lipst;^ homeomorphism thenHD(f(E)) = HD{E). In fact, if s > HD(E), thenW(E) = 0 and soH^'ifiE)) < c^'W'^iE) = 0. Therefore, for all d > -y~lHD(E) wehaveWJ(/(B)) = 0, that is, HD(f(E)) <

7.- Let (X,p) and (X',p') be metric spaces. Let A c X and 5 c X'. ThenHD{A x B) > HD(A) + HD(B). In particular, if J4 ,B are compact andd(,4) = HD(A) and <i(5) = HD(B), then / T D ^ x B) = W D ( 4 )HD(B) = d(A) + d(B) = d{A x B).

Now we give some calculation Lemmas

1.- Let E C R and s > 0 be given. Suppose that for each e > 0 there is a sequenceof intervals, {In}*en, such that E C \J^sU, and |/n | < e for all n € JVand 5Z^i [Ail° < 1. Then HD{E) < s. The proof of this assertion followsimmediately from the definition of Hausdorff dimension.

2.- (Eggleston [4])Let E C R, be a compact set and let s > 0 be given. Supposethat there is an e > 0 with the following property: given any finite collection ofclosed non-overlapping intervals I\, • • •, /n such that |/,-| < e, t = 1,2, • • •, n

.j | / ; | ' < 1. it follows that U^.,/, does not cover E. TtenHD(E) > a.

Example. For the middle-a- Cantor set if (a) C [0,1], taking s = ta(ft?!ff(i_^t wehave that given any e > 0 there is n e JV suh that (^y2)* < e, thus we may takethe 2* intervals forming Kn(a), each of one have length (^y2)", it is clear that itsunion cover K(a), and £ £ i |/,f - 2»< I f * ) - , now rw ^

e^2"") = 2 —. therefore £*_*, |/<|* = 2 » 2 - * 1and then HD(K(a)) < h (2 )*gi-..) = d(K(ot)).

Now recall that a mass distribution on a set F is a measure /j with support containedin F and such that 0 < /*( F) < + oo.

Let /i be a mass distribution on F and suppose that for some s > 0 there is a numberc > 0 such ttt«t /*([/) < c\U\* for all set !7 with \U\ < 6. Then H'(F) > ^ ands< HD(F).

In fact, if {C/,}iejir is any 6-cover of F then 0 < n(F) < /i(Uit/,) < Xc ^ i l^il"- T n u s " ^ < Z)i l^tl*- Then taking infima over all countable 6-covers of JF1,we have that H|( F) > ^ , so letting 2 -• 0, we conclud that H'( F) > ^f- > 0,and /TD(.F) > a.

As an application of this mass distribution principle, we will estimate a lowerbound for the Hausdorff dimension of the middle-a-cantor set K( a) c [ 0,1]. Let /ibe the mass distribution on K( a) obtained as follows: to each of the 2 n basic intervalsof Kn(a), each of one have length ( ^ y , we assign mass equal to 2 ~*. Let U be aset with \U\ < 1 and let fc be the integer such that (i-j2)**1 < \U\ < (^j2)*. ThenU can intersect at most one of the 2 k basic intervals of Kt( a), so

u(U) <C 2 ~ * = ( ( * ~ a ) * ) -w^f*> — ( 1—o\bt1^*) IjyIWn'Tj-t.il-e.i

Thus, taking c = (i^)ta<-T*> and s = ^^-JL^t-a)' w e ^ ^ ^tt/) < c|C/|* andhence H'(K(a)) > 0. Then from the distribution mass principle it follows thatHD(K(a)) > ^ I ' / i - e . ) = d(^(«)>-

Finally, note that in this example HD(K(a)) = diK{a)) =

A map / : Rn -> fln is called a similarity if / is the composition of a translation arotation and a magnification of ratio less than 1.

Let Ao, Ai, • • •, At, it > 2, be compact connected set in Rn, with Ao D A ; , ; =1, • •, k and Ai, • - -, A* being disjoints. Suppose that there exist similarities / ; : Ao —»A;. Define

I I

and set E = n™_i U* ...timml A*, . j_ . Let d be the unique solution of the equation

Then we have the following

Theorem (P.A.R Moran). Under the above conditions, 0 < Hd{E) < oo and thend = HD{E).

In general, we say that the Cantor set E constructed as above is derived from thepattern Ao, Ai, • • •, A*. For example, the classical ternary Cantor set, C, is defined inthis form whereAo = [0 ,1 ] , A! = [0 ,} ]andA 2 = [%,l].mAfi,h : [0 ,1] -* A*,t = 1,2 are given by / i ( i ) = f and fi{x) = ^ . Thus from Moran's Theorem, wehave that HD(C) = d where d is the unique solution of the equation 2( j)d = 1, i.e..

As an application of Moran's Theorem we show the following assertion: let 0 <a < 1 be given, then there is a Cantor set K, C [0,1] such that HD{K,) = a.

In fact, let k > 2 and chosen A such that kA' = 1, i.e., A such that a = r ^ y -

Then kA < 1, since k > 2 implies fc- > k, so kA < k^A = 1. Thus we canfind A: disjoint closed intervals Ai, • •, A of Ao = [0,1] each of one with length A.Regarding this as a pattern, in the sense of Moran's Theorem, which generates a Cantorset Kt we see that HD( Ka) = a.

Moreover, with the above notation let AJ1 be the Cartesian product of it copies ofAo and let AJ*, • • •, A ^ be the fcm distinct set of the form

Aj, x x A, , 1 < ii , - - •, im < k

Then regarding this as a pattern generating a Cantor set F we see that HD(F) =nHD( E). This can be used to prove the following assertion:

Let n > 2 and 0 < s < n, be given. Then there is a Cantor set Ka C Rn such thatHD(K.) = 3.

Examples:

Symmetric perfect sets

Let $ e ]0 ,1[ be given. We consider the trisection of the interval [0 ,1] in parts,respectively, equals to £, 1 -2(,£. The central interval will be called black interval andthe other two intervals (extreme intervals) will be called white intervals. This processof trisection is called a disection of type ( 2 , 0 -

Now let {&}«=# be a sequence of real numbers, 0 < & < j , for all t e N. Webegin with interval [0,1] and do a disection of type ( 2 , £ i ) , in the two remaining whiteintervals we do a disection of type ( 2 , £2) and so on. In the k\h step we have a set Kk

which consists of 2 * white intervals, all of them of length JI*. 1 & • The intersection set

is a perfect set determined by the sequence {£,}lGir- Any central Cantor set is obtainedin this way for a suitable choice of the sequence {£,}.

Note that if,

fc-.oo

then K has zero Lebesgue measure.The 2 * end points of the intervals in Kk are given by the relation

e i ( l - O + £ z £ i ( l - 6 ) + ••• + e * £ i 6 • • 6 - i ( l -tk)

where £, are 0 or 1, t" = 1, • • -, k. The points of K are are given by the series

where e< G { 0 , 1 } for alt i > 1. Setting rk = £ i • • • £ k-\ ( I - £ *) it is easily to see thatthe condition 0 < d < j for all t > 1 is equivalent to rk > rk+1 + rk+2 + •••.

Homogeneous Cantor Sets

Let 171, • • •, tjjt be k positive real numbcis, such that 0 <TJI <TJ I < • • • < *J* < 1.Let £ be a positive real number, which satisfies the following inequalities

Consider the k disjoints white intervals [ tj;, rjjr + £], j = 1,2, • • •, k, we say that wehave maked a disection of type (k,rj\,- • ,Vk,0 of [0.1]- Making a disection of type(k, Tji, • • -, Tjt, 0 in the intervals [ TJ; , TJ; + £] we have Jfc2 white intervals each of onewith length £2 . Now we repeat this process at infiniium. In the nth step we nave kn

white intervals each of one of length £*. Let i d be the set of these ifcn white intervals.Then

is a Cantor set, cslled homogeneous Cantor set. Now as Jfcf < 1 —171 < 1 we have thatthe total length of Kn is equal to (k()n , thus the Lebesgue measure of K is m( K) =lim n^oo ()fcO n = 0. As before, we see that the points x of K are given by the series

T

= vUi) + (v(h) + • • • +en-1

where rj(;») are the TJ, and where;* range over ihe values 1, •• •, n. It is clear that eachpart of K contained in some of the intervals k[,i= 1, • • •, n is equivalent to K by themagnification of ratio £.

As a good exercise the reader can estimate or calculate the Hausdorff dimensionand the limit capacity of the Cantor sets defined above.

VII.- EQUIVALENT MANNERS TO CALCULATE HAUSDORFF DIMEN-SION

We note that in the definition of s-dimensionaJ Hausdorff measure we must takeall admissible countable S -cover of of the set E C X. Now we give an equivalentmanner to calculate the Hausdorff dimension, but this times the covers are reduced. LetE CX and a, 5 > 0 be given. SetBJ(£) = inf{£V|Bi | J : {Bi} is countable B -cover of by balls}. As before, letting { - » 0 w e obtain an outer measure B*(E) =hms^o B,\{E) and a dimension HDB(E) as the value at which B'( E) jumps from+ ootoO. We claim that, HD(E) = HDB(E),fotallE c X. In fact, for all a > 0 and6 > 0 it is clear that H\(E) < B\(E), since any countable 6-coverof£by6-balls is anadmissible cover in the definition of H\{ E). On the other hand, if {C/J is a :cuntablefi-cover of E, then for each t > 1, we take B, to beany ball of radius |[/j|(< 6) containingUi. Thus JV | & r < 2' E i \Ui\', then taking infimagives B^S(E) < HUE). Letting6 -+ 0 , it follows that H'(E) < B"(E) < l'H'(E). From these inequalities itfollows immediately that the value so at which H'( E) and B*( E) jumps from +oo to0 are equals. Therefore HD(E) = HDg(E).

Now, for X = Rm, with the euclidean metric we have some other equivalent formto define (he Hausdorff dimension.

Let B(Rm) denote the family of bounded subset of Rm. Denote T = { F :B( Rm) -» [0, +oo] } be the family of set functions. If Fi,F2 € T, put F\ < F2 ifand only if there is \ > 0 such that Fi(E) -< XF2(E) for all E € B( R"1). And weput Fi x F2 it and only if F\ -< Fi and F2 -4 F\.

Now let:

F * be the family of dyadic cubes in Rm, that is, C G F * if and only if C has sidelength 2 " " , n e N, and each of its projections pr^C) on the ith axis is a half-openintervaloftheform[£t,*£—[,fcj € Z .

F " be the family of semidyadic cubes in Rm, that is, C G F " if and only if Chas side length 2 " * , » e N, ai.d each of its p r ; ( O projections on the ith axis has thef ^ £ ) ^ [ * Z

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O the family of all open balls in Rm.

Let 11 be a covering of E c fl™, we call || TL j|= sup{ diam( R) : R e K } thediameter of the covering H. Then, K is a g-cover of E if || R j |< e.

Let .F denote one of the families of subset of R1" defined above.For any a > 0 , define

{?-m)%(E) = M{Y.R!^l{diam{E))a : H c J countable e-coverof E)

and let

(T - m)a(E) = lil

Whenever T = B( Rm), ( ^ - m)° is the Hausdorff outer measure denned before,and we still using the notation H" in this case. When J = O w e obtain spherical outerHausdorff measure, in this case (O - m)a = Ba and we have proved that

(O-m)a(E)=H"(E)

for all a > 0 . Further in this case we have that these outer HausdoifF measures jumpfrom +00 to 0 at the same value, which is the Hausdorff dimension of E.

When ? = r * it is easy to see that

for all a > 0 . And then they jump from +00 to 0 at the same value, which again is theHausdorff dimension of E

For E c Rm, denote by QE the family of open balls Br( 1) , r > 0 with centres inE. We now get

Ka(E) <(OE- m)a(E) < cKa(E)

for a suitable constant 0 0. In this case, we also see that the value at which (O f i -m)"(E) jump from + 00 to 0 is just the Hausdorff dimension of E.

Finally, for (T ** - m)"( •) it is easily to see that

=Ha(E)

and so, the value at which (V " - m) °(E) jump from+00 to 0 is again the Hausdorffdimension of E.

Thus, all the outer measures denned above give us the Hausdorf dimension of E.

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VIII.- ARITHMETIC DIFFERENCE OF SETS

Let A, B C R, the arithmetic difference of A and B,A-B, is denned as

A- B= {t£ R : 3a G A, 3b € B such that t= a-b}.

It is easily seen that

wheren+ B = {x + p : x € R}-

Theorem ( see [1]). Let A,B C R, be compact sets. If d(A) + d(B) < 1, thenm( J4 - S) = 0, (m Lebesgue measure in R). In particular, if A c R is a compact setwith d(A) < 1, then m(A) = 0.

Theorem (see [1]). Let A, B C R. If J / D M ) + HD(B) > 1, then for m-almostevery X e R, m(>l - AS) ^ 0.

Now let M be a C°° compact boundaryless 2-dimensional manifold and <p : M —*M be a C7, r > 1 diffeomorphism. We say that <p is hyperbolic or Axiom A, if (a)the nonwandering set of <p, i i (<p), is a hyperbolic set, and (b) closure( Per( <p)) =fi ( yp), here Per( <p) denote the set of periodic points of yj. It is well known that if <p ishyperbolic then there exist a decomposition Q (tp) = A\ U •• • U A, in a disjoint union,each A, is a closed invariant sets and p/A< , i = 1, - -, n is topologically transitive.Each set A, in the above decomposition is called a basic set of<p. Now let A be a basicset of <p, we define the stable and unstable set of x € A as

W(x) = {y£M : d(<fin(x),<p*(y)) -+ 0 , O J H - O O }

and

These set are C injectively immersed submanifolds (if ys is CT) of A/. Their union,respectively, denoted by JF'( A) and J^C A) are called the stable and unstable foliationof A, If ip is C2 then these foliations can be extended to a C1 foliations (in fact, CUt,i.e., Cx with derivative e-Holder if p is C3) to a neighborhood of A.

Let / be a small line segment transversal to ?*( A) then we define the stable Haus-dorff dimension of A as HD"(\) = / / D ( J F ' ( A ) n / ) , i n a similar way we define theunstable Hausdorff dimension and the stable and unstable limit capacity of A. If ^ isC3 the definition of Hiy( A) and d"( A) , a = u or s does not depend on the line seg-ment / transversal to T* (A) (^u( A)). In general may occur that H Da{ A) ^ d"( A ) ,a = a or u.

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Now suppose that a 1-parameter family of C3 diffeomorphisms {¥>„}„£* have abasic set A^ and a fixed or periodic point p^ e A^ for which W(po) and W ( p o )have a homoclinic orbit 9, generic (i.e., quadratic) and unfolds generically, and for/i < 0, <Pn ^ hyperbolic. See figure 3.

Let / be small line segment through x e W(po) and tmsversal to F'(Ao) and^ ( A o ) . As a first approximation to the bifurcations values of ipu for /* > 0, thetangency bifurcations occur along the line / and correspond to the difference of the sets(Cantor sets) f( Ao) n / and T"{ Ao) n /. For a detailed description of this topic see[1].

Figure 3

IX.- NUMERICAL SETS

Let 6 > 1 be a fixed integer. Given x 6 [ 0 , 1 ] , then x can be expressed in base bas

where 0 < e n ( i ) < 6 - l . The functions cn( x) are called the digit functions of x inbase 6. If we impose that for each x, 0 < en( x) < 6 - 1 for infinitely many n's, thenall the functions en( x) , is uniquely determined. The lack of uniqueness is an issue onlyfor countable may x'a.

Now, let 0 < c < r be fixed integers. We define.

13

T T

" =

this set consists of all i e [ 0 ,1 ] , for which the sum of r consecutive base b digits havesum greatest or equal than c. In order to do most clear the exposition of the ideas, weset b = 2 , for the general case, the arguments are similar.

Let, £i, e.%, • • •, es be a finite sequence of O'a and Vs. Let

/ ( e i , • • - , £ * ) = clo3ure({xe [ 0 , 1 ] : ei(x) = eit- •• ,eN(x) = eN } )

this set is called a cylinder of rank N. We note that, each cylinder of rank N is a closedinterval of length 2 ~n and that two cylinders of the same rank do not overlap.

Let N > r, we say that a cylinder 7(gi, • • •, eN) of rank TV is admissible if

^ c> for n = 0 , 1 , • • •, JV — r.

It is clear that if JV < r, then any cylinder of rank N is admissible.Define T{N) to be the collection of all admissible cylinder of rank N, and let

F( N) be the union of these cylinders. Finally, set F = rw>i F( N), this set differs of7i( c, r) by a most a countable set of points, since Ti( c, r) is the intersection of the setsG(N), which consist of the union of admissible cylinders of rank /V, whose right endpoints was removed. Thus,

HD{Ti{c,r)) ~- HD(F).

From this equality, we need only calculate the HausdortT dimension of the set F .L e t / = /<£!,-•-,£,.) be a fixed cylinder of rank r, if N > r, and / = /(171, • • •, rjjv)is an admissible cylinder of rank N, then we say that / is the type J if VN-J = £r->,j = 0 , 1 , ••• ,r - 1, that is, the last r digits of I coincide with the digits of J . Lets be the number of admissible cylinder of rank r, clearly a depends on r and c, andin the general case on the base b. We fix and arbitrary order for these cylinders, say,J\, • • •, J,. Then any admissible cylinder of rank N > r is of one of the type J\, • •, Jt.Now we define a xs matrix M = (my)ij.i. . . , , as follows: Fix 1 < i < s and letJ% = / ( e i , - • < ,e r) be the ith admissible cylinder of rank r. Consider the cylinders/ ' = /( £2 , - • • ,£ , , 1) and / " = I(E2 , - • - , £„ 0). It is clear that / ' is admissible, since£2 + • • • + er + 1 > £i + • • • + e r , so there is some j \ such that / ' = / , , , and we setmy, = 1. On the other hand /" may or may not be admissible, depending whetherei + • • • + eT + 0 is > c or < c. If /" is admissible, there is some h such that /" = J, , ,and we set m ^ = 1. Note that there is no problem in the above definition, since;'i j J2, because the last digits of / ' and /" are different . For the other entries notdetermined by the above procedure we set Tui; = 0. In this way we have defined a

14

Perron matrix, M, which has a (unique, say) O-eigenvalue, say XQ, such that for any othereigenvalue TJ of M, |rj| < \Q , and the corresponding eigenvector v = {in, • • •, v,) naseach v, > 0. Finally we have the following.

Theorem (Drobot & Turner [2]) With the above notation,

Remark. In the general case, that is, base 6 > 2 ,

where \o is the corresponding eigenvalue of the Perron matrix M obtained in this case.

Now in order to calculate the limit capacity of r t ( c , r ) , we use Lemma 4 of [2],which give us the following inequality

where 0 < c\ < C2 are constants and AN denote the number of admissible cylinder ofrank /v\

Now, as each admissible cylinder of rank N is an interval of length 6"^, we have

From this and the above inequality, we sec that

Thus /nXTi ( c, r)) = d( 7\( c, r ) ) .

X.- MARKOV PARTITIONS

In this section we consider a class of Cantor sets in the real line defined by MarkovPartitions whose intervals may intersect in boundary points. For each of these sets theHausdorff dimension and the limit capacity coincide and are calculated as for dynam-ically defined Cantor sets (see [1]). For a special subclass we apply Perron-FrObeniusTheory to obtain a simpler expression for this quantity. We apply this result to the classof Cantor sets described in [2],

15

I T

We say that a Cantor set K in R is defined by a Markov partition p if there is acollection of pairs {(Kui>\),{Kt,1n),-- ,(Kk,^k) h * > 1 where each Kt isanon-trivial closed interval in R and each fa : K< -* R is an affine map verifying:

(a) If x G Ki and j / e /fin then i < y.

(b) |V>i(*)| = A j > l i = l , 2 , . . . / f c .

(c) if «>jt(iM/f\)) n in t ( / f ; ) y 0 then if, c V « W -

(d) Denote K° = L i ? . ^ and define 0 : /C° -> R by V(*) = iM*) if z €/fj (for x€ KiHKi+i define V(x) arbitrarily), then for all i = 1,..., k, thereexis tsn=n(i) such that ^n(int(Ki)) C\irt{Kj) i t, ; = l,...,Jfc.

(e) JTVutiW^).

(f) /f = rV>o#m where AT™ = clousure(^-m(ira(K0))).

The collection p = {(K i , ^ i ) , • • •, (Kt, V>t)} ' s called a Markov partition asso-ciated to K (or which defines K).

For this class of Cantor sets the Hausdorff dimension and the limit capacity coincideand are calculated by successive approximations as for dynamically defined Cantor set,see [1]. Although the proof of this fact is almost the same as for dynamically definedCantor sets', we include it here for completeness.

From this 'esult and Perron-Frobenius Theory it is obtained that when A, = * fori = 1, • •, k, the Hausdorff dimension of K is given by g^£ where ij is the largestpositive eigenvalue of an associated transition matrix. We apply this last result to theCantor sets TUc, r) considered in [2].

Statement of the results.

Let K be the Cantor set defined by a Markov partition p = {(K\, V>i), • • -,(Kk , V>*)}. and Km be defined as in (0- Then Km = K|" U - • - U K^ where each/f™ is a nontrivial closed interval such that ^m( tnt( /(•")) = int(Jf,-) for somet = 1, - - •, Jfc. Redefining ^ " = ^m/int( K™) in the boundary of Kf a new Markovpartition p™ = {(if |", ^J") , - • • , ( i f £ , ^ ) } is obtained which defines the same Can-tor set A\

Denote \f - \(rf>p'(x)\ and define c^ by E*ri(^™)~*" = 1. Then:

Theorem (Bam6n & Plaza [3]).

HD(K) = c(K)= lim d™n—too

where HD and c denote respectively Hausdorff dimension and limit capacity.

The transition matrix T = {ty) given by a Markov partition p = {(Ki, ipi) : i =1, • • •, k } is the it x k matrix where tti = 1 if fai if,) covers Kj and ti}- = 0 if

16

not. By Perron-FrObenius theory, T has a positive real eigenvaluefor every other eigenvalue £ of T.

Theorem (Bamon & Plaza [3]). If A< = X for all i; = 1, • • •, k,

such that |£| < tj

= Xm for everyIn order to calcu-

Proof of theorem 2 and applications.First notice that if A, = A for every i = 1, • • • Jt, then

i = 1, • • •, km . Also ife™*-™*- = 1, and hence cU £ £late h'mm_KSO<2m >t is necessary to estimate Jfcm.

Take V = ( 1 , • • •, 1) e Rk and directly obtain that km = £*_ , «;" where

T""<~u*) = «"' - (v™,- •• , v " ) . Let Ui*, • • • , « ? be the eigenvectors of T associatedto the eigenvalues v\,m •• ,Vk, a n d suppose that 17 = TJI is real and TJ > \r)i\ for t" =2 , • • •, k. Then ~v* = a i vf + • • • + ak v£ for some complex coefficients a,- and

It follows that

Proving theorem 2.

mln(A)

As an application we check that the Cantor sets Tt( c, r) considered in [2] are de-fined by Markov partitions. We first recall the definition of Tt(c, r) :

Tt,(c,r) = {.0102 - G [0,1] : 0 .1 + • •• + % , > c V n > 0 }.

where b, c and r are positive integers and .0102 • • • denotes the expansion in base b ofthe numbers in [0,1].

A Markov partition defining 7\(c, r) has the form p = { (K tj , VO t 0 < t <6 - 1 , I < j < Mi) where the intervals Kij are describe below and &( x) = bx-i.For each t = 0 , 1 , • • •, 6 - 1 the sets Ki = { x € [ *y-, j] : m + • - - + aT > c }are unions of finite number of intervals Kij•,, j = 1, • • •, Mi. It can happen that forsome Kij , ^,( Kij) does not cover completely all the k?j it intersects: the collection{(Kij , ^ i )} may not be a Markov partition. It is necessary to subdivide the intervalsK^ . If k^ = [aij, &j] then limlN>atf V(x) = 0 and l i m l / f l , V r ( i ) = 1 • In fact,

r r

ctij and Pij are of the form a,, = .ai a% • • • ar00 with 01 + 02 + •• • + ar = c and0ij = .Si • -Q.-iCft- 1) with* < rando] + • •• + o,_i + (r — s+ 1 ) ( 6 - 1) > cand51 + • - • + o,_ 1 + 1 < c . I t follows that the positive orbits of the boundary points of eachKij intersect UijKij in a finite number of points. Subdividing Kij by the points inevery one of these orbits, we obtain a new collection of intervals Kij, i = 0 , 1 , • • •, 6—1, ; = 1 , - Mi such that for every» = 0 , 1 , - • ,b - I, Kt = ufm\Kij = vfm\kijand the collection p = { (K,-j , VO : i = 0 , 1 , • • •, 6 - 1 , ; = 1, • •, Af,} is aMarkov partition associated to T\,(c, r) .

Example:For ^ ( 2 , 3 ) we have:

tfo = [ & . jl u [ fT - \ 1 u [ I - T! = K0.1.U ifo.2 U

That is, p = {(ATij , Vi) : > = 0 ,1 ,2 , 1 < ; < M,} with Mo = 3, Afi =2, and Mi = 1 is a Markov partition for ^ ( 2 , 3 ) . The transition matrix for thisexample is 6 x 6-dimensional. Following [2] we would obtain a 23 x 23 transitionmatrix.

Proof of theorem 1.

Step 1: c(K) < cU foreach m > 0.The proof that c( K) < do involves the fact that p is a Markov partition which

determines K. Tb prove c( K) < d^, m> 0 rewrite the same proof using the Markovpartition pm.

Let us prove c(K) < do- It is known that c(K) = inf(L) where L = { d >: there ezi.it £0 > 0 such that for every 0 < e < eo, K admits an e —covering with 110 more than £~d elements}.

If d > do then YA-\ K* < 1 Let ^ = mai{Ai, -, A*}. Then for d > do,there existcrfd) > 0 such that X-'<d) = Y%-\ K*-

Assertion 1: do < d e L =>• fd' > d- a(d) => d£ £ L].In fact, suppose do < d e L and let £0 > 0 be such that foreach 0 < e < e0 , K

admits an e -covering with no more than e~d elements. Then for each 0 < e <eo, KDKi admits a \ ~ l e -covering with no more than e~* elements. Hence for every0 < e < V 1 eo, KnKi admits an e -covering widi no more than Af' e"^ elements.Since X~l < A"1 for i = 1, ••-,*, it follows that foreach 0 < e < \~XBQ, Kadmits an e— covering with no more than J^f-i ^t~'£ d elements.

By induction, for 0 < £ < X~meo, K. admits an e -covering with no more than(Ef-t V*)m £~d elements. Since *-•«> = ^*_, \r*, (J^*ml X;d)n = \-™<.<t> forall m e N. Then if e e [X~<m+1)eo, A~m£o[, K admits an e - covering with nomore than:

18

elements.Hence, for every e e]0, £o [, K admits an e - covering with no more than

elements where C= ^ p . Assertion 1 follows.

Assertion 2: Let a(d) = d - a{d) defined for d > do. Then a(do) = do and0 < a'(d) < 1 for all d> do. In particular an(d) -+do as n—» oo.

In fact A-ff<<0 = Y,Kd implies that cr(d) = -[_V(A> } • f r o m t"'8- Assertion 2 iseasily proved.

Assertion 3: do < d£ L => a(d) - d- o{d) € L.In fact, by assertion 1 d! e L for every d! < a(d). By assertion 2 a(d') < a(d)

for every a(d) < d1 < d proving that a(d) € L.These three assertions prove that an(d) e L for every n. and hence c(K) <

do = li

Step 2: There exists a sequence dm such that

(i) dn < HD(K) forall m > 0.

(ii) ii

In order to define dm first note that the sequence {d^,} is bounded. In fact km <km and \? > Xg where \Q - min{Xi, - • -, Xt}. Hence 1 = £*ri(>r>~'£~

< fc-V"4- ^ < SLet 0 < d and N 6 JV be such that d™ < d for all m and $N( K{C\K) = K i =

1,••-,*.Let C = aup{|(^w)'(i)|< ' : i G domotnt^)} and define dn.by

Let us now prove that dn < HD(K) for every m. In fact, suppose that HD(K) <d < dm for some m € JV. Let eo > 0 be small and let 0 be a finite covering of Ksuch that £ y e O \U\d < eo - Suppose eo > 0 be small enough such that

(i) U e 13 => U intersects at most two intervals KJ*.

(ii) Kf C\K]ly=<b^> for all U G U, [/ does not intersect both Kf and if

F O T ; G { 1 , ..,*„,} let 13; = {£/e U : C/n/T

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Assertion 4: With the assumption above, there exist ; e { 1, •• •, km } such that

In fact, let J = {; = 1, • • • ,Jfcm-i : Kf n Kfr i 0} and for each ; eJ let 0> * {U e U : U n Kf j $ and U n K^+l i «}. Assuming assertion4 false

> C~x Yfci 1X71^ Eueu, l^+ n7*; n(C/) | ' ' - £0

> 2 eo — £0 = eo •

Hence with the assumption HD(K) < d < d,,, assertion 4 is true. But this isimpossible because

is a covering of K, with less elements than the original covering U Repeating theargument successively we finally find a covering of K with just one element veryfying\U\d < eo. From this d^ < HD{K) for every m > 0.

It only remains to prove Step 2 (ii).

Assertions: 4 , - <L < l n ( 2l n ?^" ( ' ? = M B for ^ = " " " { * ! " : »= 1 • " . * » ) •

In fact, denote /(d) = D(Af)-<£- Then /'(d) = £<- l na™»<*r>~* <- lnCA™)^^) . Hence ^ < - i n tAm) and integrating between d^, and ^ theassertion follows.

Finally since Km —* 00 as m - » o o , step 2 (ii) is proved.

XL- AN APPLICATION TO DYNAMICAL SYSTEMS

Finally, we give an application of Hausdorff dimension to dynamical systems.Let / : R —* R be a map such that

20

* am m- w tt

(i) / is CV"J, for some a > 0;

(ii) | / ' O ) | > 1 ifzanf/f i) are in [0,1];

(iii) [0,1] C int(/([0,1])) and /^UO, ]]) C [0,1];

(iv) / has two fixed attracting points, one attracting each point in ] 1,+oo[ and theother attracting each point in ] - oo, 0[.

Figure 4

This map has two attractors A and B and there exists an invariant set A c [0,1].Now suppose i and y and chosen random from [0,1] subject only to the condition | x—y\ < £. What is the probability pE that x and y tend to different attractors? If/describesa physical system this question can be rephrased as follows: suppose that there is auniform error in determining the initial conditions of the system. What is the probabilitythat our prediction of the long-term behavior of the system will be incorrect? In fact,the probability pe tends to zero like e1 ~d as e goes to zero, where d is the Hausdorffdimension of A = n^o /"*{ [ 0,1]).

Theorem (Pelikan [6]). L e t / : [ 0 , l ]ing on / - ' ( [0 ,1 ] ) . Let A =

i lbea C1+a map, for some a > 0, expand-]). Theni/D(A)

The proof of this Theorem follows immediately, since under these conditions the Ais a dynamically defined Cantor set (see [1]).

Recall that the basin of an attractor g e R of / is the set B(q) = { x e R :fn(x) ~» q, n—* +oo}. Letnowe > 0 be given. Define Ie(x) = [ i - £ , x + £],andset

21

i r

= m ( { y €

where m denote the Lebesgue measure in R, and then set p« = /0 ' p,( x) d i .

Theorem (Pelikan [6]) Under the above conditions,

where d= HD(A) = d(A).

ACKNOWLEDGMENTS

The author would like thank to Professor A. Verjovsky for the suggestion and theencouragement to write these notes and, especially, for fruitful conversations aboutGeometry, and for the stimulating working atmosphere in the Mathematical Sectionof ICTP. He would also like to thank Professor Abdus Salam, the International AtomicEnergy Agency and UNESCO for hospitality at the International Centre for TheoreticalPhysics, Trieste, where part of UIIM work was done.

References

[1] J. Palis - F. Takens, Homoclinic bifurcations: hyperbolicity, fractional dimensionsand infinitely many attractors, Cambridge U. Pre^i, «o appear.

[2] V. Drobol - J. Turner, Hausdorff dimension and Perron Frobenius Theory, IllinoisJ. of Math., 33,1(1989), 1-9.

[3] Bamon, R. - Plaza, S., Cantor sets, numerical invariants and Perron-Frbbenius the-ory. To appear in Pitman Research Notes in Mathematics, 1992.

[4] H. G. Eggleston, Set of fractional dimension which occurs in some problems ofnumber Theory., Proc. London Math. Soc. 54(2) (1952), 42-93.

[5] P. A. P. Mi-ran, Additive functions of intervals and Hausdorf measures, Proc. Cam-bridge Phys. Soc. 42 (1946), 15-23.

[6] S. Pelikan, A dynamical meaning of fractal dimensions. Trans. Am. Math. Soc. 292(1985). 695-703.

[7] K. Falconer, Fractal Geometry, John Wiley & Son 1990.

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* # *

[8] K. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, 1985.

[9] S. J. Taylor & C. Tricot, Packing measuie and its evaluation for a Brownian path,Trans. Amer. Math. Soc. 288 (1985), 679-699.

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. • • • • • ; } * •