proceedings of the american mathematical society Volume 106, Number 2, June 1989
DIMENSIONS AND MEASURES OF QUASI SELF-SIMILAR SETS
K. J. FALCONER
(Communicated by Kenneth R. Meyer)
Abstract. We show that sets with certain quasi self-similar properties have equal Hausdorff and box-packing dimensions and also have positive and finite Hausdorff measure at the dimensional value. A number of applications of these results to particular examples are given.
1. INTRODUCTION AND DEFINITIONS The idea of dimension is fundamental in the study of fractals, see Mandelbrot [5]. Various definitions of dimension have been proposed, such as the Haus- dorff dimension, which is mathematically convenient being based on Hausdorff measures, and the box counting dimensions, which are computationally conve- nient. It is often desirable to know whether such differing definitions give the same dimensional value for a particular fractal. Moreover, it is very useful to know whether a set of Hausdorff dimension s is an 5-set, that is whether the 5-dimensional Hausdorff measure is positive and finite. Much of the fundamen- tal work done by Besicovitch and others (see Falconer [3]) relates specifically to sets that satisfy this rather strong condition. Rigorous calculations of Hausdorff dimension normally involve calculating the 5-dimensional Hausdorff measures of a set, and then finding the value of 5 at which the measure jumps from infinity to zero. Calculation of such mea- sures often requires careful estimates involving admissible coverings of the set. Similarly, careful estimates are required to find box counting dimensions before these dimensional values may be compared. This paper discusses the opposite approach. We give conditions on a set of Hausdorff dimension s that guarantee that the 5-dimensional Hausdorff measure is positive and/or finite without any need to calculate the actual value of 5. From this information, s can often be found easily. (For example, given that the middle third Cantor set has positive finite measure at the critical value, it is a trivial exercise using the scaling property of Hausdorff measures to show that the Hausdorff dimension is log 2/ log 3.) Similarly, we give conditions that
Received by the editors May 10, 1988 and, in revised form September 5, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 58F12, 28A75. Key words and phrases. Dimension, fractal, self-similar set, repeller.
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ensure that the box-counting and Hausdorff dimensions are equal without the need to evaluate them. We call these theorems implicit in that they enable us to draw conclusions about dimensions that are not explicitly calculated. The first result of this na- ture, which we quote as Theorem 1, was due to McLaughlin [6]. Our results are applicable to fractals that are renormalisable in some sense or that have a quasi self-similar structure. We give a number of such applications in §3. Let (F ,d) be a metric space. For 0 < 5 < oo the s-dimensional Hausdorff outer measure of F is given in the usual way by
^s(F) = hminf\j2\Ui[s:Fç\Jur,0<[Ux[<ô\, I j i J where | | denotes the diameter of a set. This leads to the definition of the Hausdorff dimension of F : dimHF = inf{5: ßf(F) < oo} = sup{5: ?t(F) > 0}. Although the usual situation is of a fractal F embedded in some larger metric space, usually R" , for many purposes, such as in the above definitions, we may work entirely with the restriction of the metric to F . Many other definitions of dimension are encountered in the literature, some more satisfactory and useful than others. Let (F ,d) be a metric space, and, for e > 0, let «(e) be the smallest number of sets of diameter at most e that cover F . We define the lower and upper box counting dimensions by (1.1) dimB7" = lim-log «(e)/loge
(1.2) dim^F = lim-log«(e)/loge.
We get exactly the same values if we take «(e) to be the smallest number of closed balls of radii e with centres in F that cover 7". Moreover, by relating «(e) and «(2e), a simple exercise shows that we obtain the same values for dimg and dimÄ if, in (1.1) and (1.2), we let «(e) be the largest number of disjoint closed balls all of radii e with centres in F . There are other equivalent formulations of the box counting dimensions; different ones are convenient in different situations. The important difference between Hausdorff dimensions and the box count- ing dimensions is that in calculating Hausdorff dimensions different weights \U\S are attached to covering sets U, whereas in caculating box counting di- mensions, the same weight ss is attached to each covering set (notice that dimfl F — inf{s: lim«(e)es < oo} , etc.). From the definitions,
dimw F < dimfl F < dimB F. In a certain sense, the Hausdorff dimension and upper box counting dimension may be regarded as the smallest and greatest values of any reasonable definition
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of dimension. For example, the packing dimension of Tricot [7] always lies between these two values. 2. Implicit theorems For completeness we quote the result of McLaughlin [6], and then give a minor, but useful, generalisation. Theorem 1. Let (F ,d) be a nonempty compact metric space, with dim^ F = s. Suppose that there exist X0, a > 0 such that for any set N ç F with \N\ < X0 there is a mapping cp: N -* F such that (2.1) ad(x,y)<[N[d(cp(x),cp(y)) for x,yeN. Then as Theorem 2. Let (F ,d) be a nonempty compact metric space, with dim^ F = s. Suppose that there exist X0, a > 0 and an integer m such that for any set N ç F with \N\ < X0 there are sets Nj with N ç \J"=XN. and mappings cp.-.Nj^F ( 1 < j < m) such that ad(x,y)< |A|ú?((9;(x),^(v)) for x,ye Nj. Then asm~l < ¿FJ(F) • Proof. Suppose that 0 < ßfs(F) < asm~ . Then given 0 < 3 < min{A0, \a) we may find sets Ux, ... ,Uk with \UX\<â and F ç (J.=1 {/, such that k ¿^\U{\V~* ,TTlS < a S m —1 i=l and so with ;=1 for some t < s. By the hypotheses of the theorem we may find sets Uj and mappings cpij:U¡ . -» F (1 < j < m) such that U¡ C |T=i ty / an<1 ad(x,y)<[Ul[d(cpi](x),cpij(y)) for x, y e Ujj . Thus we have \ k m k ( k \ ( k \ ,=1 7=1 q=l \i=l J \q=l J But F C \J. pfj(U) ; hence F has a covering by sets of diameter less than jS with the same bound on the tth power of the diameters. Repeating this argument, we see that there are sets V. covering F , of diameters at most 2-" License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 546 K. J. FALCONER such that J2\V¡f < alm~x . It follows that SF\F) < a'm~x, and dim/ < s, a contradiction. D We now show that (2.1 ) implies that dim^ F = dimß F = dimB F . Theorem 3. Let (F ,d) be a nonempty compact metric space, and suppose that there exist X0, a > 0 such that for any set N ç F with [N\ < X0 there is a mapping cp: N —>F such that (2.2) ad(x,y)<[N[d(cp(x),cp(y)) for x, y € N. Then dimw F = djmB F = dimfi F. Proof. We always have dim^ F < dimg F < dimß F. Suppose that s > dim^ F, then (2.3) 0 = Jr*{F) < a . Choose an integer k and subsets Ux, ... ,Uk of F with 0 < \U¡\ < min{a/2, X0} for each i, such that F ç (J;=1 U¡ and k (2.4) Í2\Utf«f- i=i By hypothesis there exist cp.: U¡-* F such that (2.5) ad(x,y)<[Ul[d(cpi(x),cpi(y)) if x,y e Ü¡. Let J — {ix, ... ,i : \ < i: < k} be the set of q term sequences, and let J = U°l, J be the set of all such finite sequences. Let J^ = {ix, i2, ... : 1 < i; < k} be the corresponding set of infinite sequences. For each (ix, ... ,i ) e Jq let Vh„..,iq= d(x,y) Let 0 < b = minx<¡ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSIONS AND MEASURESOF QUASI SELF-SIMILARSETS 547 sequence in J. Repeated use of the inclusion F C U,.=1Ut = |J,=1 cpi l(F) gives that (2.8) F c M V] : . /,.í,es Further, (2.7) together with repeated application of (2.4), that ¿Z,i=i(a~l\Ui\Y < 1, gives (2.9) (be)scardS< £ (uT'lt/,-,!)'■: ■(«r^r^JD'WTI- < tël? ■ /./',€S Hence F may be covered by at most \F[s(be)~s sets of diameter at most e . It follows that lim«(e)ei < oo, so that dim^F < s, and hence dim^F < dim^ F. D Note that it would be possible to extend Theorem 3 in the way that Theorem 1 has been extended to Theorem 2. Our next result concerns transformations in the opposite direction, from a set onto small neighbourhoods of itself. We give conditions for the Hausdorff and box counting dimensions to be equal, as well as for the critical Hausdorff measure to be finite. Theorem 4. Let (F,d) be a compact metric space with dim^F = 5 < oo. Suppose that there exists a,rQ > 0 such that for any ball B in F of radius r < r0 there is a mapping y/ : F —yB satisfying (2.10) ard(x,y) (2.11) n(E)e>a~5. Choose t > s such that (2.12) n(e)e'>a~'. Let Bx, ... , B., be closed disjoint balls of radius e with centres in F and write ô = mini,jd(Bj,Bj) > 0. By hypothesis there exist mappings y/¡,: F —y Bi for 1 < / < «(e) such that (2.13) aed(x,y) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 548 K..J. FALCONER We may define a measure p on the class of finite unions of sets y/i o ■• -o y/¡ (F) by letting p(y/j o • •• o y/. (F)) — «(e)-9 . This extends to a measure p on the rj-algebra generated by these sets, which in turn give rise to an outer measure p on the subsets of F. Let U be any subset of F with [U[ < a (2.15) (ae)qô>[U[>(ae)9+xô. Then by (2.14) U intersects at most one set y/¡ o • • • o y/. (F), so p(U) < n(efq < (ae)"' < (aeô)~'\U\' by (2.12) and (2.15). Now suppose that F c (j°lx Ui where \U¡\ < aeâ for each i. Then 1 = p(F) < £~,/*(£/,) < («*<*>"'£*.](//. Hence %\F) > 0, implying that dimw F > t > s, contrary to hypothesis. We conclude that «(e)e* < a~s for all sufficiently small e, giving dimßF < s. Finally, given a maximal set of disjoint balls of radii e with centres in F, the set of concentric balls of radii 2e cover F . It follows that %?S(F) < oo. 3. Examples and applications Example 1. Dynamical repellers. Let M be a Riemann manifold, and / an expanding map of class C +n on M (that is the tangent map of / satisfies a Holder condition with exponent n > 0). We assume that / is a compact set satisfying the following properties: (i) / is expanding on J, that is there are constants c > 0 and a > 1 such that (3.1) [[(Txf")(u)[[>ca"[\u[\ for x e J and « > 1, when Txf is the tangent map off at jc. (ii) / is invariant under / and f~ , that is /(/) c / and f~ (/) c /. (iii) / is topologically mixing on J, so that for every open set U intersect- ing J there is an « > 0 such that J c f"(U). The set / is called a (mixing) repeller for /, and often may be identified with the Julia set of /. We use our results to show that if / is conformai, that is if T f is always a scalar multiple of an isometry, then J has equal box counting and Hausdorff dimensions. Moreover, / has positive finite Hausdorff measure at the critical value, a result obtained by Ruelle [9] and Bowen [2] using Markov partitions and Gibbs measures, who also give the Hausdorff dimension of J in terms of the pressure of certain functions. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSIONS AND MEASURESOF QUASI SELF-SIMILARSETS 549 Theorem. Let f be a Cl+>1 conformai mapping on a Riemann manifold M with mixing repeller J. If s = dimHJ, then dimBJ = dimBJ = 5 and 0 whenever w e J and x,y e Br (w) with x ^ y. If f'(x),f'(y) e Bro(f'(w)) for 0 d(f-\x),f-\y)) > y"-1-id(fi(x),fi(y)) so that T7(l - c(2r )V'*) <_d(f"(x),f"(y))_ i=0 ( o)y )-\\Twf\\\[Tf{w)f\\.--[[Tf„_l{w)f\[d(x,y) n-l Thus (35) bu <^d(f"(x),f"(y)) «\J w,y \y>) < b \TJ"[[d(x,y) for constants 0 < bx < b2 < 00 for x ¿ y, provided that f'(x), f'(y) e Br (f'(w)) for 0 < i < n - 1. Let w e J, 0 < r < min{r0,rQb21}. Choose the least non-negative integer « such that r0/ß (3.6) VJ-rd{x'y) - d{f"{x)' f"iy)) - rjd(X'y) ■ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 550 K. J. FALCONER Taking cp as the restriction of f" to Br(w)nJ in Theorems 1 and 3, it follows that ßf*{J) > 0 and dimfl / = dimß = 5 . Since / is topologically mixing on the compact set J, we may fix a positive integer m such that fm(Br¡(w)) D J for every w e J, where r, = bxr0/b2ß . There is a number v such that (3.7) d(f"(x),fm(y)) that /": B(w)r —»Br r0 (f"(w)) is an injection, satisfying r-±d(x,y) By invariance of dimension in the Riemann manifold M, f"(Br(w)) D Bri(f"(w)). Restricting f": Br(w) nJ - f"(Br(w)) n J, let «: /?,_(/"») n J -* Br(w) n J be the restriction of the inverse of f" to Br¡ (f"(w)) n /. Thus d(x,y) Let g: J ^y Br (f"(w))nJ be any (set theoretic) right inverse of the surjection fm:Bri(f"(w))nJ-+J. From (3.7) d(x,y) It follows from Theorem 4 that %?S(J) < oo, as required, a Example 2. Self-similar sets. For i = \, ... ,k let ^: R" —►R" be contracting similarity transformations, i.e. (3.8) d(y/i(x),y/l(y)) = cid(x,y) where 0 < c, < 1 and <7 is the Euclidean metric. It is well known (see Hutchin- son [4]) that there exists a unique nonempty compact F c R" that is self-similar in the sense that k F = \J V,(F). i=l In general F will be a fractal. It is easy to see that if z e F then there is a (not necessarily unique) infinite sequence (ix,i2, ...) such that the sequence of compact sets y/i o • ■■ o y/. (F) decreases to z as q —yoo. Hence given r < \F\ there exists a least integer q such that ar —< C: 'I c. '2 ■■ci V [F\ ' —< r where a = min.^.<.. c;. Writing y/ = w. o • ■• o y/¡ (3.9) ard(x,y)/\F[ < d(y/(x), y/(y)) < rd(x,y)/\F[ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSIONS AND MEASURESOF QUASI SELF-SIMILARSETS 551 by (3.8). Thus taking x = y/~\z) we see that y/ maps F into the ball of centre z and radius r. It follows by Theorem 4 that, if 5 = dim^ F, then %fs(F) < oo and dim^ F = dimfi F = dimfi F. Suppose now that the y/x satisfy the open set condition of Hutchinson [4], namely that there exists an open set U such that k (3.10) UDÏJy/^U) i=i with this union disjoint. Let X < 1 be given. Then for every infinite se- quence (ix,i2, ...) with 1 < /. < k, there is a least integer q such that aX < c. c ■■-c < X. Let S be the set of curtailed sequences (/. , ... ,i) obtained in this way. It follows from iterating (3.10) that U?\Jy/ho...oy/.q(U) s with this union disjoint. Moreover, it is easy to show that F ç \Js y/.qo ■■■o y/iq(V). Let N c F have \N\ = X and let Ux, ... , Um de- note the sets y/¡ o • •• o y/. (U) with (ix, ... ,i ) e S that have closures which intersect N. Then aX\U[ < \U¡\ < X\U\. The U¡ are disjoint and are contained in a ball of radius A(l + \U\) with center in N. Comparing vol- umes it follows that \m[ < m0 = (1 + [U\)"a~n, which is independent of X. Let N-NnU(l License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 552 K. J. FALCONER (positive inverse), where g(x) = 1 - 1.528x2 + 0.105x4 H- is the fixed point function of the Feigenbaum renormalization equation and a = 2.503... (see Rand [8] or Falconer [3] §8.7). It follows that the closed invariant Cantor- like attractor of g has positive finite 5-dimensional Hausdorff measure, where 5 = 0.538... . Example 3. Recurrent sets. This is essentially a generalisation of the previous example, which is discussed in Bedford [1]. For i = 1,... ,k let y/^. R" —> R" be contracting similarity transformations of ratios 0 < c¡ < 1. Let T be a subset of {1,2, ... ,k} x {1,2, ... ,k}, and assume that T satisfies a transitivity condition that if 1 < i,j < k then there is a sequence /' = ix,i2,...,iq = j with (im,im+{) e T for 1 < m < q - 1. Let Jq = {(1 = ixJ2,...,iq):(im,im+l) eT, 1 < m < q- 1} and J = {(1 = ix,i2, ...): (im ,im+x) e T, 1 < m < oo} . Let F be the decreasing sequence of sets given by Jq where E is some compact set with y/¡(E) c E for each i. We are interested in the dimension and measure of the set >-fK.Il q ' 1=1 which is independent of the initial E chosen. Let p = max(/ .. min{i : i - ix, ... ,it= j for some sequence ix, ... , it with t > 1}, and let 0 < a = min, c < 1. If z e F there is a (not necessarily unique) infinite sequence (\ = ix,i2, ...) with (im, im+x) eT for 1 < m < oo and with the sequence of compact sets ^ o çr. o • • • o y/. (F) decreasing to z . Hence given r < [F\ there is a least integer q such that ar —< c,C; 1 ¡2 ■■ c lq' \F\ ' —< r. with (im , im+x) e T. There is a sequence i ,iq+x, ... , i t = 1 where 1 < t < p - 1 and (im, im+x) eT for q < m c,'I c,■ '2 ■ ■ ■ c, 'q <— a . Let S c— M°°^«=1 7 Q be the set of curtailed sequences^ obtained License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIMENSIONSAND MEASURESOF QUASI SELF-SIMILARSETS 553 in this way. Let N ç F have \N\ = X. Just as in the case of self-similar sets, there is an integer mQ such that at most m0 of the sets y/i o • • • o y/. (U) with (ix, ... ,i ) e S have closures intersecting N. We work with each of these sets U0 = y/i o • • • o y/. (U) in turn. For some 2 < t < p we can find a sequence 1 = j\ , ... ,jt = iq with (jm ,jm+i) € F for 1 < m < t - 1. If y/ = (y/ho---o y/.) o (^ o • • • o y/.)~x then d(x,y) 9=1 As above 0 < %? (Fu)7(f) < oo for each i ; by the transitivity of T, s = dim^ F (•) is independent of i. By assumption FU)=\J{¥j(F{i)):(j,i)eT} is a disjoint union, so that (ß?s(F(l))\ í^s(F(x)y = MC \ß?s(F{k))) \Jfs(F(k)) where M denotes the matrix with (i,j)th entry of 1 if (i,j) e T and 0 otherwise, and (c\ 0 Cs = Vo ck. Since MCS is an irreducible matrix with non-negative entries, the Perron- Frobenius theory tells us that '^S(FW)\ »s(F{k))j is, to within a scalar, the unique eigenvector of MCS with positive entries, with corresponding eigenvalue the unique eigenvalue of largest absolute value, equal to the spectral radius of MCS. This eigenvalue is strictly decreasing with s, so that dim^ FK is the value of s making the largest eigenvalue of MC equal to 1. □ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 554 k. j. falconer References 1. T. Bedford, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. (2) 33 (1986), 89-100. 2. R. Bowen, Hausdorff dimension of quasi-circles, Pub. Math. I.H.E.S. 50 (1979), 11-25. 3. K. Falconer, The geometry of fractal sets, Cambridge Univ. Press, Cambridge, England, 1985. 4. J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747. 5. B. B. Mandelbrot, The fractal geometry of nature, W. H. Freeman, San Francisco, Ca., 1982. 6. J. McLaughlin, A note on Hausdorff measures of quasi-self-similar sets, Proc. Amer. Math. Soc. 100(1987), 183-186. 7. C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. 8. D. Rand, University and renormalisation in dynamical systems in New directions in dynamical systems, London Math. Soc. Lecture Notes 127 (1988), 1-56. 9. D. Ruelle, Bowerís formula for the Hausdorff dimension of selfsimilar sets, in Scaling and self-similarity in physics-renormalization in statistical mechanics and dynamics, Progr. Phys. 7 (1983). School of Mathematics, University Walk., Bristol BS8 1TW, England License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 0 we require the open set condition (3.10) to hold. Let X < 1. For every sequence (/',, i2, ... ) e J there is a least q such that aX <