Abbas Edalat and Joseph Parry. Department of Computing,. Imperial College,. 180 Queen's Gate,. London SW7 2BZ, UK. Email: ae @ doc.ic.ac.uk. Abstract.
URL: http://www.elsevier.nl/locate/entcs/volume13.html 10 pages
An Algorithm to Estimate the Hausdor Dimension of Self-Ane Sets Abbas Edalat and Joseph Parry Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK. Email: ae @ doc.ic.ac.uk
Abstract
We present an algorithm, based on Falconer's results in 4,6], to eectively estimate the Hausdor dimension of self-a ne sets in Rn : For a given nite set of contracting non-singular linear maps T1 Tm , we obtain a decreasing sequence of computable real numbers converging to Falconer's dimension d. For almost all (a1 am ) Rmn , the number d is the Hausdor dimension of the unique nonS empty compact subset F satisfying F = mi=1 (Ti (F ) + ai ). Similarly, we obtain an increasing sequence of computable real numbers converging to Falconer's lower bound d which is indeed a lower bound for the Hausdor dimension of F if the sets Ti (F ) are disjoint for 1 i m.
2
1 Introduction
An iterated function system (IFS) is given by a nite collection of contracting maps S1 Sm on R n , i.e. with jSi(x) Si(y)j cijx yj for some 0 < ci < 1, all 1 i m and all x y 2 R n . A theorem of Hutchinson 9] shows that there exists a unique non-empty compact set F R n , called the invariant set or the attractor of S1 Sm, satisfying m F = S i (F ): i=1
In fact, F which is usually a fractal can be obtained as follows (see for example 5]). Let B be any large enough closed ball with Si(B ) B , for all 1 i m. Then F = \1 k=1 Fk with
Fk =
(1)
m
i1 ik =1
Si1 Si2
Sik (B ):
c 1998 Published by Elsevier Science B. V.
Edalat and Parry
Attractors of IFSs present a large and rich class of fractals with various applications in science and technology 5]. The main tool to study fractals is provided by the various notions of dimension. The Hausdor dimension is mathematically the most satisfactory notion of dimension whereas the box dimension is one which is most easily computed. The problem of estimating the Hausdor or the box dimension of the attractor of an IFS has been an active area of research 3,4,6,10]. We will recall the denitions of the Hausdor and box dimensions below. For any F R n , 0 s n and > 0, let s s jUi j : F U i 0 < jU i j < g H (F ) = inf f
X
i2I
i2I
where I is a nite or countable indexing set and jU j denotes the diameter of the set U R n . The number Hs(F ) increases as decreases and the s-dimensional Hausdor measure of F is dened as s s H (F ) = lim H (F ) !0 which can be innite. The Hausdor dimension of F is dimH F = inf fs : Hs(F ) = 0g = supfs : Hs(F ) = 1g: The Hausdor dimension has the desirable properties of dimension, but is in general very dicult to compute. The box dimension, on the other hand, has far less pleasing properties but is easier to calculate. It is dened as follows. A -mesh cube in R n is a cube of the form: m1 (m1 + 1)] m2 (m2 + 1)] mn (mn + 1)] where mi 2 Z for i = 1 n. Let N (F ) be the number of -mesh cubes which intersect a non-empty bounded set F R n , then the upper and lower box dimensions are, respectively, dimB (F ) = lim sup log N (F )= log !0
inf log N (F )= log : dimB (F ) = lim !0
In case the upper and lower box dimensions are equal then the common value is the box dimension, dimB (F ), of F . In general we have dimH (F ) dimB (F ) dimB (F ): The only general algorithmic technique for computing dimension is the box-counting algorithm for estimating the box dimension of a given fractal set F . See for example 1]. It consists of plotting log N (F ) against log for a range of values of and then taking the line of best t. The problem with this method is that we cannot say anything about the error in our estimate of dimB F . However, for fractals of iterated function systems, there are analytical results for computing the dimension. 2
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2 The dimension of self-ane sets A map S : R n ! R n is called a similarity if there exists c > 0 such that n jS (x) S (y )j = cjx y j for all x y 2 R . A similarity transformation is the composition of a rotation, a reection and a translation. A self-similar set is the invariant set of an IFS consisting of contracting similarity maps, i.e. maps S1 Sm with jSi(x) Si(y)j = cijx yj for all x y 2 R n , where 1 > ci > 0, for i = 1 m. It is well-known 11,9] that the Hausdor dimension and the box dimension of a self-similar set are equal and given by the unique solution of the equation, cs1 + cs2 + csm = 1 provided that S the open set condition holds, i.e. there exists an open set U such that mi=1 Si(U ) U with the union disjoint. The problem of calculating the Hausdor dimension of self-ane sets has been more challenging. Let T1 Tm be contracting linear maps on R n . Then, for ai 2 R n (i = 1 : : : m), the invariant set of the IFS consisting of the maps T1 + a1 T2 + a2 Tm + am is called a self-ane set. For these sets, the Hausdor and Box dimensions can be dierent 10,7] and the Hausdor dimension may not vary continuously even for a smoothly parametrized family of self-ane sets 4,15,7]. Several authors 10,3,2,14], have calculated the Hausdor dimension of particular classes of self-ane sets. Falconer in 4,6] developed a new technique to study the dimension of self-ane sets and showed a generic result which we will now explain without giving any proofs. For any non-singular, contracting, linear map T , the singular values 1 > 1 2 3 n > 0 of T are the positive square roots of the eigenvalues of T t T , where T t is the transpose of T equivalently they are the lengths of the principle semi-axes of the ellipsoid T (B ) where B is the unit ball centered at the origin. Falconer's singular value function is dened by 8 < 12 dssde se+1 s n s (T ) = : (1 2 n)s=n n < s where dse is the least integer greater than or equal to s. s(T ) is continuous and strictly decreasing in s and it satises the submultiplicative property s(TU ) s(T )s(U ). For each integer k 1, let X s k (s) = (Ti1 Ti2 : : : Ti ): (2) 1ij m
k
Then k : R + ! R is continuous and strictly decreasing in s. Furthermore, the sequence k (s) is submultiplicative: k+l(s) k (s)l (s) for all integers k l 1. It follows, from the properties of submultiplicative sequences, that 3
Edalat and Parry
the limit
(s) = klim ( (s))1=k !1 k exists and is continuous and strictly decreasing in s. There is therefore a unique number d for which (d) = 1. By constructing a Hausdor like measure on the product space f1 : : : mg! and using potential theory, Falconer proved the following theorem. Theorem 2.1 4] For all (a1 : : : am ) 2 R mn , we have: dimH (F ) d. For almost all (a1 : : : am ) 2 R mn with respect to the Lebesgue measure on R mn we have: dimH (F ) = dimB (F ) = min(d n): In 6], Falconer obtained a lower bound d for dimH (F ) when the sets Ti(F ) are disjoint for i = 1 : : : m. The technique used is similar to the one above for the upper bound d. The singular values of the inverse map T 1 are given in terms of those of T , i.e. i as above, by: n1 n1 1 : : : 11 > 1: Hence, for 0 s n, (sdse+1) 1 n s s(T 1) = n1 : : : nd se+2 ndse+1 = (T )=(1 : : : n ): And, for s > n, s(T 1) = (det T )n=s. Then, (s(T 1))1 is continuous and strictly decreasing in s, and it satises the supermultiplicative property (s((TU )1))1 (s(T 1))1(s(U 1 ))1 : For any integer k 1, let k (s) = (s(Ti1 Ti2 : : : Ti )1)1:
X
1ij m
k
The function k : R + ! R is continuous and strictly decreasing. Furthermore, k (s) is a supermultiplicative sequence: k+l (s) k (s)l (s) and, therefore, the limit (s) = klim ( (s))1=k !1 k exists and is continuous and strictly decreasing in s. It then follows that there exists a unique number d such that (d) = 1. Theorem 2.2 6] If the sets Ti(F ) are disjoint for i = 1 : : : m, then d dimH (F ). Falconer, in the same paper, provided more results on the upper bound d. He obtained sucient conditions, involving the projection of F into n 1 linear subspaces of R n , such that dimH (F ) = d. The dimension d is also known 4
Edalat and Parry
to coincide with the box dimension for a large class of connected self-ane sets 3,6] and also a class of totally disconnected self-ane maps 8]. Despite these analytic results, the problem of actually computing d or d is non-trivial. In fact, Falconer states in the concluding section of 4] that this computation is not easy as the rate of convergence of the submultiplicative sequence is not known.
3 Estimating d from above and d from below In this section, we show how one can obtain a computable decreasing sequence which tends to d and a computable increasing sequence which tends to d. Although we still cannot say anything about the rate of convergence of these two sequences, they will give successive upper and lower bounds converging to d and d respectively.
d First we consider d. As we have seen the function : R + ! R is continuous and strictly decreasing in s. Furthermore, (0) = m > 1 and, for s n, (s) m 1, which implies that for large s, (s) < 1. Therefore, there exists a unique real number s with (s ) = 1. We have no analytical formula which directly computes s but it is easy to see from Equation 2 that s is computable for all k 0. In fact, the singular values of any n n matrix T can be computed from the eigenvalues of T T . This is particularly simple in R 2 where most applications occur. In fact, the singular values of any 2 2 matrix 0 1 @a bA c d are given by q p A + B ( A B )2 + C 2 where A = (a2 + c2)=2, B = (b2 + d2)=2 and C = ab + cd. Therefore, the singular values of T 1 T 2 : : : T k for each term in (s) in Equation (2) are computable. Using, for example, the bisection method, one can then compute the unique root s of (s) = 1. It is easy to see that this computation is exponential in k. We will now construct a monotonic subsequence of s . Lemma 3.1 For any sequence of positive integers n , we have: s 1 s 1 2 s 1 2 3 :::: Proof. From + (s) (s) (s) we get + (s) 1 for s max(s s ). Since is decreasing, it follows that s + max(s s ) 5 3.1
The upper bound
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Edalat and Parry
and more generally,
sk1+k2+:::k max(sk1 sk2 : : : sk ): Putting ki = q for 1 i r, we get srq sq for all integers r q 1. The result now follows immediately. In particular, we have sr s1 for all integers r 1. Proposition 3.2 For any sequence of positive integers ni 2 we have: lim sn1n2 :::n = d: i!1 r
r
i
For any integers q r 0, we get by the submultiplicative property: (qr (s))1=qr ((r (s))q )1=qr = (r (s))1=r : Hence, the sequence fi(s) = (n1 n2:::n (s))1=n1n2 :::n is decreasing and tends to 1=k (s) = klim ( : k (s)) !1 Let ti = sn1n2 :::n . Then, fi(ti ) = 1 = (d) fi(d). Thus, d ti for all i 0, i.e. d limi!1 ti . On the other hand, for any > 0, (d + ) < 1 since is continuous and strictly decreasing. Therefore, for large enough i we have: fi(d + ) < 1 = fi(ti ) which implies ti < d + for large i. It follows that limi!1 ti d. Corollary 3.3 inf sk = lim inf sk = d: k1 k!1 Proof.
i
i
i
In particular, for ni = q 2, the subsequence sq decreases to d. For computation, we can use q = 2 to obtain a decreasing sequence of upper bounds for d. However, as the computation of sk is exponential in k, it is clearly more ecient to use the sequence s0k dened inductively by s01 = s1 and s0k+1 = min(s0k sk+1). Then, s0k is decreasing and limk!1 s0k = d. We do not know if the sequence sk is in general decreasing and if in fact limk!1 sk = d. n
d There is a similar analysis for d. Recall that k : R ! R is continuous and strictly decreasing. Furthermore, k (0) = mk > 1 and for large s, k (s) < 1. Hence, there is a unique uk > 0 with k (uk ) = 1. From k+l (s) k (s)l(s) we get k+l(s) 1 for s max(sk sl ). It follows that sk+l max(sk sl) and more generally, sk1+k2+:::k max(sk1 sk2 : : : sk ): Similar to the previous case, we get u1 ur for any r 1, and for any sequence ni 2 we have: un1 un1n2 un1n2n3 : : : : 6
3.2
The lower bound
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Edalat and Parry
and lim !1 u i
n1 n2 :::ni
= d. We therefore get: sup u = lim sup u = d: 1 !1 k
k
k
k
In particular, for n = q 2, the subsequence u increases to d. For computation, we can use q = 2 to obtain a decreasing sequence of upper bounds for d. More eciently, we can use the sequence u0 dened inductively by u01 = u1 and u0 +1 = max(u0 u +1). Clearly, u0 is increasing and lim !1 u0 = d. Here again the elements of the sequence hu i are computable. As before we do not know if lim !1 u = d. But in the next section we will see an example for which the sequence hu i 0 is not increasing. qn
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T (F ) (1 i m) are disjoint, then the sei 1 of intervals is nested and shrinking, and we have d d] = hu0 s0 ]T
Corollary 3.4 If the sets
T u s ] = 1 k
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o
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quence
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0 0 1 u s ]. k
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4 Two examples A Miranda programme for computing s and u has been implemented in 12]. In this section we use the above results to estimate d and d for a self-similar set where an analytic formula for these quantities exist. Therefore, our estimates can be compared with the actual values. For a few classes of self-ane sets, there are computable analytic formulas to calculate the Hausdor dimension 2,14,13]. However, in nearly all of them the ane maps induce multiplicative sequences h i 0 and h i 0, with (s) = (1(s)) and (s) = (1(s)) . This implies that the sequences hs i 0 and hu i 0 are constant with values d and d respectively. In order to compare our results with actual values, we need to investigate a non-multiplicative set of maps. Paulson 13] has examined the following self-ane set called the \carpet". It is generated by an IFS with four ane 7 k
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Edalat and Parry
maps
S1 S2 S3 S4
given by (
S1 x
(
S2 x
(
S3 x
(
S4 x
0 )=@ 0 )=@
0
c
0 1
0 1
0 1 )=@
0 1 0 )+@ A
x
1 A(
0 1 )+@
0
c
c
1 A
1 A
c
0A
(
0 1 )+@ A c
x
0
c
c
x
1
c
0
c
0
c
0 1 )=@
1 A(
1 0A (
0 )+@
x
0
1 0 where 0 1 is a constant. He computed the functions k ( ) analytically for these matrices and obtained a simple formula for the limiting case: the lower bound satises the equation p 4( 2 )d = 1 In the case where = 1 3 he found the upper bound = 1 845 and the lower bound 4 log 2 = log(9 2) = 1 843 c
c
< c
