McMullen, Curtis T.

Nagoya Mathematical Journal

McMullen, Curtis

Harvard University - DASH

 Hausdorff Dimension of General Sierpinski Carpets  (Article begins on next page)The Harvard community has made this article openlyavailable.Please share how this access benefits you. Your storymatters.Citation McMullen, Curtis T. 1984. Hausdorff dimension ofgeneral Sierpinski carpets. Nagoya MathematicalJournal 96: 1–9.PublishedVersionhttp://projecteuclid.org/euclid.nmj/1118787639Accessed February 18, 2015 2:04:09 AM ESTCitable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:3354157Terms of Use This article was downloaded from Harvard University'sDASH repository, and is made available under theterms and conditions applicable to Other PostedMaterial, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAAC. McMullenNagoya Math. J.Vol. 96 (1984), 1-9THE HAUSDORFF DIMENSION OF GENERALSIERPINSKI CARPETSCURT McMULLEN§ l IntroductionIn this note we determine the Hausdorίf dimension of a family ofplanar sets which are generalizations of the classical Cantor set. Givenn> m and a set Jϊ consisting of pairs of integers (ί, j) with 0 < i < nand 0 < j < m, define the set R byR = {(έπWe refer to R as a general Sierpίήski carpet, after Mandelbrot [4], sinceSierpiήski's universal curve is a special case of this construction [6].It is clear that R = {J[fi(R)9 where r = \R\ and the ft are affine mapscontracting R by a factor of n horizontally and m vertically. Whenn = m these maps are actually similarity transformations, and a well-known argument shows the dimension of R is log r/log n (following e.g.Beardon [1]). If n > m, however, a different approach is required, essen-tially because squares are stretched into narrow rectangles under itera-tion of the maps /*.Our method relies on elementary probability theory to address thegeneral case, and we obtain the following result.THEOREM. The Hausdorff dimension of R is given bywhere t3 is the number of i such that (ί,j)eR.This settles a question of Hironaka's [2] concerning the dimensionof a certain continuous plane curve whose self-similarities entail theReceived December 27, 1982.2 CURT MCMULLENstretching described above. By the theorem, the dimension of Hironaka'scurve is Iog2(l + 2(log32)); his example is essentially the same as theSierpiήski carpet R for m = 2, n = 3 and R = {(0, 0), (1,1), (2, 0)}. (SeeFig. 1.)The theorem is proved in Section 2. In Section 3 we recall thenotion of metric dimension and discuss the following result.PROPOSITION. The metric dimension of R is given bym.dim(R) = logm s + logn (r/s)where s is the number of j such that (i,j)eR for some i, and r = \R\.The equality of the metric and Hausdorίf dimensions of R whenn = m sheds some light on the distinguished character of that case.Fig. 1 (Hironaka's Curve)SIERPINSKI CARPETS 3§ 2. The Hausdorff dimension of RLet X be a metric space. A collection of sets (Xt) is an ε-cover ofX if X = UΓ Xt and diam (X*) < ε for all i. The δ-dίmensional Hausdorffmeasure of X is given byμδ(X) — supinf {2]diam(Xi)δ: {Xt) is an ε-cover ofε>0We define the Hausdorff dimension of X by= sup {δ: μδ(X) = 00} .The determination of the Hausdorff dimension of R is organized into thefollowing steps.1. We reformulate the dimension of R in terms of coverings by aselected class of rectangles.2. The covering problem is lifted to a sequence space through amap ψ: Sr -» R.3. A probability measure is introduced on Sr, and we define asequence of functions fk which measure the difference betweenthis measure and the ^-dimensional Hausdorff measure on R.Here δ is the value of dim R claimed in the theorem.4. We prove a) Πϊn fk ;> 1 on all of Sr andb) lim fk = 1 almost everywhere.5. We use (a) to show dim R < δ.6. We use (b) to show dim R > δ.Each step corresponds to a lemma below.First, consider coverings of R by rectangles of the formriwhere I = [k logn m] is the integer such that n > mkjnι > 1. (In the sequelI will always be related to k in this way.) These rectangles are of coursetailored to the structure of -R. To a covering C = {Rk(p, q)} we associatethe integers Nk = the number of Rk{p, q)e C with V = k.LEMMA 1. The Hausdorff measure μδ(R) = 0 iff for any ε > 0 thereexists a covering C of R with J] Nkm~δ1i < ε.4 CURT MCMULLΈNThe proof is straightforward upon observing that (a) any boundedset can be covered by N rectangles Rk(p, q) of smaller diameter, where Nis a fixed constant; and (b) m~k ~ diam Rk(p, q).Now let (xu y 1)1=0 enumerate the elements of R> and let Sr =ΠΓ {0,1, 2, , r — 1}. Then the map ψ: Sr -> R given byis a surjection. (It fails to be injective because some coordinates havemore than one "decimal" representation.) We will construct sets Ak(p, q)X Bk in Sr which correspond roughly to the sets Rk(p, q) in R under themap ψ. WritingP = Σί Xjnι-J 0 < Xj < n, I = [k logn m]andj 0<yj<mwe defineAk(p, q) = {(h, k, - ,ίk): Xtj = Xj for j = 1, ."'.., /,and yiό = ys for j = 1,andTo a covering C = {Afc(p, q) X jBfc} we also associate the integers Nk = thenumber of Ak(p, q) X Bk, e C with h! — k.LEMMA 2. The Hausdorff measure μδ(R) = 0 iff for any ε > 0, ίΛereexists a covering C of Sr with Σ Nkm~δlc < ε.Proo/. Note thatAfc(p, q)xBka Ψ~\Rk{p, q)) C U Ak(p + ί, q + j) X Bki=-l,0,ly = -1,0,1so we can pass back and forth between covers of R and covers of Sr.Furthermore we need only change Nk by a bounded factor in doing so.The result then follows from Lemma 1.The point of lifting the problem to Sr is that coverings of a productspace by cylindrical sets are more easily analyzed than coverings of Rby rectangles.SIERPI&SKI CARPETS 5We now study the size of the cylinders Ak(p,q) X Bk. For i —0,1, , r — 1, let at = the number of j such that yt = y^, where (xi9 ydlzl enumerates R as before. Then the cardinality of Ak(p, q) is given by theproduct au+1aiι+2 aik, for any (il9 i2, - ,ik)e Ak(p, q). The constants at will be used to define a probability measure μ onSr. First, let δ be the value of dimi? claimed in the statement of thetheorem. Note thatLetting bt = a^^^jm8, we have Σ &< = 1. We take μ to be the uniqueprobability measure on the Borel subsets of Sr (in the product topology)which satisfies, for any (iu i2, , ίk),μ((i» h, ' -', h) X Bk) = δ^δ^&i, bik .The existence of such a measure is a special case of the Kolmogorovconsistency theorem (see [5]).Define the functions fk on Sr by[(n n . . . n V°%n, m "1 l(aitai2 au) Jwhere / = [k logn m] as usual.LEMMA 3. If ze Ak(p, q) X Bk, thenμ(Ak(p, q) X Bk) =Proof. Pick (ιΊ, i2, , ίfc) e Ak(p, q) and define the sequence(sh s29 - , sk) = (αίl? σίa, , αi4) . Note that this sequence is independent of the choice of (iu i2, ,ιfc). Inparticular,to e c ^logΛ w((iu h, , ifc) X Bfc) = ^f2 ' gfc)gfcjS2 Sk)for each (il9 i2, , jfc) e Afc(p, g). But the cardinality of Afc(p, q) is Sk) SOX Bk) = ig((SiS2 Sz)as claimed.CURT MCMULLENIt remains only to show that the functions fk -> 1 in a suitable sense,so that for a covering C of Sr we haveand we can apply Lemma 2.LEMMA 4. a) lim fk(z) > 1 for all z e Sr.b) fk -> 1 almost everywhere (μ).Proof. Define gk9hk on Sr byσ(i j . . Λ (fh&*^(atιau ait)1/ιh (j 7* . . A — \n a a l(VO(iogn m-ι/k)Then fk(z) = hk(z)'gk(z)loSnm. We claim hk(z)-+l for all z. Indeed, 1 <at < n for all i, soandlognm - 1 = logwm - J^M^I —>0 as k—> oo .Therefore it suffices to show a) lim gfc(2) > 1 for all ze Sr and b) gk(z) —• 1a.e.(a) Fix 2τ = (ij, ί2, •) and let ^ = (αiχαia α<y)1/J. Then 1 < s^, soit is clear thatsince this holds for any positive sequence Sj bounded away from zero.(b) From the definition of μ it is almost tautological to assert thatthe functions(ii,i» •••) >bik β= 1,2,3, ...are independent, identically distributed random variables with respect toμ. Hence the sequence(bhbh . bίky<k k = 1, 2, 3,converges for almost every (iu i2, ) e Sr9 by Kolmogorov's strong lawof large numbers [3]. Recalling the definition of bt we haveSIERPINSKI CARPETSand hence gk->l almost everywhere. (In case n = m we have I = k sogk is identically 1).LEMMA 5. dim R < δ.Proof. We will construct an efficient cover for Sr.Fix ε > 0.Let Ck consist of those nonempty sets Ak(p, q) x Bk for which fk(z) > m~ε for ze Ak(p, q) X Bk. These sets are disjoint and satisfyby Lemma 3. Hence Mk = cardinality of Ck < m(δ+ε)k as the measure ofSr is one.Note that any ze Sr is covered by Ck for infinitely many k, sinceΠrn/fcOa:) > 1 > m~\ Hence C ={Jk>κCk is a covering of Sr for any choice of K. Choose K large enough that Σ*>κm~'k < ε> ^i1611 ^e ^associated to C satisfy-v+2εϊk = 2] Mfcm-(δ+2ε) < 2 m"εfc < ε .Using Lemma 2 it then follows easily that dim R < δ.LEMMA 6. dim R > δ.Proof. Fix β < δ. We will show there exists an ε > 0 such thatJ^Nkm~βk > ε for any covering C of Sr; the dimension estimate is thenimmediate from Lemma 2.Let Eκ = {2eSr:/^) < mδ~β for all fc > K). We know /& tends to1 almost everywhere, and mδ~β > 1, so we can pick K such that μEκ > 0.Set ε = minίμίJ^, m~βK}.Now let C be any covering of S>. Ίί Nk Φ 0 for some k < K, thenΣNyϊn'** > ™>~βκ > e. So assume that iVfc = 0 for β < iί.Then for the elements of C such that Ak(p, q) X Bk f] Eπ Φ φ, wehaveμ(Ak(p, q) X Bk) = [fk(z)m-ψ < [mf'^m-γ = w^*where we have chosen z lying the intersection of the two sets, and usedthe assumption that k>K. Since C covers Eκ, we have J]Nkm~β1c >8 CURT MCMULLENμEκ > ε. By Lemma 2, μβ(R) φ 0; since β < δ was arbitrary, it followsthat άim(R)>δ.The preceding two lemmas complete the proof of the theorem.§ 3. The metric dimension of RLet X be a bounded finite dimensional metric space, and let N(ε)denote the number of elements in a minimal covering of X by ε-balls.The metric dimension of X is given bylog 1/εThe metric dimension can be thought of as the value obtained for theHausdorff dimension if we restrict ourselves to coverings all of whoseelements have the same size. Consequently the inequality m.dim (X) >dim(X) always holds.Now let Nk denote the number of (p9 q) such that Rk(p, q) Γϊ R Φ φ. It is easy to see there exist Cu C2 > 0 such that CίNk > N(m~k) > C2Nk, and therefore m.dim (R) = ϊίΐn log Nklllog mk. But Nk is precisely thenumber of ways to choose (x^Li &n(l 00*=i such thata) (xi9 yd e R for ί = 1, 2, , Z, andb) (xt> yt) e R for i = Z + 1, Z + 2, , A and some choice of xt. Hence Nk = rιsk~ι = (rls)ιsk where r is the cardinality of R and 8 is thenumber of y such that (x,y)eR for some x; and the metric dimensionof i? is given bym.dim (R) = US M(?M?1 = l0gm s + logm (Jl) Πm (1) ft logmfc \s / \k/= logm s + logm ί — j logw m = logm s + logn ί — jas stated in the introduction.There are two conditions under which the Hausdorff and metricdimensions of R agree. The first is when n = m: in this case the setR is geometrically self-similar and both dimensions are given by logn ras noted in the introduction. The second condition occurs when theconstants tό take on only one value other than zero. (Recall t5 is thenumber of i such that {i, j) e R).As an example of the latter case, suppose R = T X U. ThenSIERPINSKI CARPETS 9dim (R) = logm (ufilo*nm)) = logm u + log. t = m.dim (5)where u = |C7| and t = \T\. Indeed, JR=Tχϊ7 where T and Π aregeneral Cantor sets in R, and we are just asserting thatdim(S) = m.dim (J?) = dim(f) + dim(C7) .This also follows from a general theorem due to Wegmann [7] whichstates that dim {Ex X E2) = dim (EΊ) + dim (E2) whenever m.dim (Et) = dimCE*) for i = 1,2.We remark that the functions fk introduced in the preceding sectionare identically equal to one if the Hausdorff and metric dimensions of Ragree, and the proof can be simplified in this case. Furthermore oneeasily obtains the stronger statement that the Hausdorff measure of Rsatisfies 0 < μδR < oo. It would be interesting to know if this last as-sertion holds in general.EEFERENCES[ 1 ] A. F. Beardon, On the Hausdorff dimension of general Cantor sets, Proc. Camb.Phil. Soc, 61 (1965), 679-694.[ 2 ] H. Hironaka, Hironaka Heisuke no Suugaku Kyositsu, Tokyo 1980.[ 3 ] Michael Loeve, Probability Theory, New York 1963.[ 4 ] Benoit B. Mandelbrot, Fractals, San Francisco, 1977.[ 5 ] K. R. Parthasarathy, Probability Measures on Metric Spaces, New York, 1967.[ 6 ] W. Sierpiήski, "Sur une courbe cantorienne qui contient une image biunivoque etcontinue de toute courbe donne", Comptes Rendus, 162 (1916), 629-642.[ 7 ] H. Wegmann, "Die HausdorfF-Dimension von kartesischen Produkten metrischerRaume", J. reine angew. Math., 246 (1971), 46-75.Department of MathematicsHarvard UniversityCambridge Massachusetts 01238USA

Hausdorff Dimension of General Sierpinski Carpets

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Hausdorff Dimension of General Sierpinski Carpets

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