Self-similar and self-affine sets; measure of the intersection of two copies

Transcription

1 Self-smlar and self-affne sets; measure of the ntersecton of two copes Márton Elekes, Tamás Kelet and András Máthé Alfréd Rény Insttute of Mathematcs, Hungaran Academy of Scences, P.O. Box 127, H-1364, Budapest, Hungary Department of Analyss, Eötvös Loránd Unversty, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary e-mal: Receved 2008 Abstract. Let K R d be a self-smlar or self-affne set and let µ be a self-smlar or self-affne measure on t. Let G be the group of affne maps, smltudes, sometres or translatons of R d. Under varous assumptons such as separaton condtons or we assume that the transformatons are small perturbatons or that K s a so called Serpńsk sponge we prove theorems of the followng types, whch are closely related to each other; Non-stablty There exsts a constant c < 1 such that for every g G we have ether µ K gk < c µk or K gk. Measure and topology For every g G we have µ K gk > 0 nt K K gk where nt K s nteror relatve to K. Extenson The measure µ has a G-nvarant extenson to R d. Moreover, n many stuatons we characterze those g s for whch µ K gk > 0. We also get results about those g s for whch gk K or gk K. Supported by Hungaran Scentfc Foundaton grant no and Supported by Hungaran Scentfc Foundaton grant no Supported by Hungaran Scentfc Foundaton grant no

2 Self-smlar and self-affne sets 1 1. Introducton The study of the sze of the ntersecton of Cantor sets has been a central research area n geometrc measure theory and dynamcal systems lately, see e.g. the works of Igudesman [12], L and Xao [17], Morera [23], Morera and Yoccoz [24], Nekka and L [25], Peres and Solomyak [26]. For nstance J-C. Yoccoz and C. G. T. de Morera [24] proved that f the sum of the Hausdorff dmensons of two regular Cantor sets exceeds one then, n the typcal case, there are translatons of them stably havng ntersecton wth postve Hausdorff dmenson. The man purpose of ths paper s to study the measure of the ntersecton of two Cantor sets whch are affne, smlar, sometrc or translated copes of a selfsmlar or self-affne set n R d. By measure here we mean a self-smlar or self-affne measure on one of the two sets. We get nstablty results statng that the measure of the ntersecton s separated from the measure of one copy. Ths strong non-contnuty property s n sharp contrast wth the well known fact that for any Lebesgue measurable set H R d wth fnte measure the Lebesgue measure of H H + t s contnuous n t. We get results statng that the ntersecton s of postve measure f and only f t contans a relatve open set. Ths result resembles some recent deep results e.g. n [16], [24] statng that for certan classes of sets havng postve Lebesgue measure and nonempty nteror s equvalent. In the specal case when the self-smlar set s the classcal Cantor set our above mentoned results were obtaned by F. Nekka and Jun L [25]. For other related results see also the work of Falconer [5], Feng and Wang [8], Furstenberg [9], Hutchnson [11], Järvenpää [13] and Mattla [19], [20], [21]. As an applcaton we also get sometry or at least translaton nvarant measures of R d such that the measure of the gven self-smlar or self-affne set s 1. Feng and Wang [8] has proved recently The Logarthmc Commensurablty Theorem : they showed logarthmc commensurablty of the smlarty rato of a homogeneous self-smlar set n R wth the open set condton and of a smlarty map that maps the self-smlar set nto tself see more precsely after Theorem 4.9. They also posed the problem of generalzng ther result to hgher dmensons. For self-smlar sets wth the strong separaton condton we prove a hgher dmensonal generalzaton wthout assumng homogenety Self-affne sets. Let K R d be a self-affne set wth the strong separaton condton; that s, K = ϕ 1 K... ϕ r K s a compact set, where r 2 and ϕ 1,..., ϕ r are njectve and n some norm contractve R d R d affne maps and denotes dsjont unon. For any p 1,..., p r 0, 1 such that p p r = 1 let µ be the correspondng self-affne measure; that s, the mage of the nfnte product of the dscrete probablty measure p{} = p on {1,...,r} under the representaton map π : {1,...,r} N K, {π 1, 2,...} = n=1 ϕ 1... ϕ n K. In Secton 3 we show Theorem 3.2 that small affne perturbatons of K cannot ntersect a very large part of K; that s, there exsts a c < 1 and a neghborhood U

3 2 M. Elekes, T. Kelet, A. Máthé of the dentty map n the space of affne maps such that for any g U \ {dentty} we have µ K gk < c. We also prove Theorem 3.5 that no sometrc but nondentcal copy of K can ntersect a very large part of K; that s, there exsts a constant c < 1 such that for any sometry g ether µ K gk < c or gk = K Self-smlar sets. Now let K R d be a self-smlar set wth the strong separaton condton and µ a self-smlar measure on t; that s, K and µ are defned as above wth the extra assumpton that ϕ 1,..., ϕ r are smltudes. In Secton 4 we prove Theorem 4.1 that for any gven self-smlar set K R d wth the strong separaton condton and self-smlar measure µ on K there exsts a c < 1 such that for any smltude g ether µ K gk < c µk = c or K gk. In other words, the ntersecton of a self-smlar set wth the strong separaton condton and ts smlar copy cannot have a really bg non-trval ntersecton. Let K, µ and g be as above. An obvous way of gettng µ K gk > 0 s when gk contans a nonempty relatve open set n K. The man result Theorem 4.5 of Secton 4, whch wll follow from the above mentoned Theorem 4.1, shows that ths s the only way. That s, for any self-smlar set K R d wth the strong separaton condton and self-smlar measure µ on K a smlar copy of K has postve µ measure n K f and only f t has nonempty relatve nteror n K. An mmedate consequence Corollary 4.6 of the above result s that for any fxed self-smlar set wth the strong separaton condton and for any two selfsmlar measures µ 1 and µ 2 we have µ 1 gk K > 0 µ2 gk K > 0 for any smltude g. As another corollary Corollary 4.7 we get that for any gven selfsmlar set K R d wth the strong separaton condton and self-smlar measure µ on K there exst only countably many n fact exactly countably nfntely many smltudes g : A K R d where A K s the affne span of K such that gk K has postve µ-measure. Let K R d be a self-smlar set wth the strong separaton condton and let s be ts Hausdorff dmenson, whch n ths case equals ts smlarty and box-countng dmenson. Then the s-dmensonal Hausdorff measure s a constant multple of a self-smlar measure one has to choose p = a s, where a s the smlarty rato of ϕ. Therefore all the above results hold when µ s s-dmensonal Hausdorff measure. In Secton 4 we also need and get results Proposton 4.3, Lemma 4.8, Theorem 4.9 and Corollary 4.10 statng that only very specal smlarty maps can map a self-smlar set wth the strong separaton condton nto tself. Theorem 4.9 and Corollary 4.10 are the already mentoned generalzatons of The Logarthmc Commensurablty Theorem of Feng and Wang [8]. In Secton 5 we apply the man result Theorem 4.5 and some of the above mentoned results Lemma 4.8 and Theorem 4.9 of Secton 4 to characterze those self-smlar measures on a self-smlar set wth the strong separaton condton that can be extended to R d as an sometry nvarant Borel measure. It turns out that, unless there s a clear obstacle, any self-smlar measure can be extended to R d as an sometry nvarant measure. Thus, for a gven self-smlar set wth the

4 Self-smlar and self-affne sets 3 strong separaton condton, there are usually many dstnct sometry nvarant Borel measures for whch the set s of measure 1. Let us smply call a measure defned on K sometry nvarant f t can be extended to an sometry nvarant measure on R d. Many dfferent collectons of smltudes can defne the same self-smlar set. We call {ϕ 1, ϕ 2,...,ϕ r } a presentaton of K f K = ϕ 1 K... ϕ r K holds; n other words, K s the attractor of the terated functon system {ϕ 1, ϕ 2,...,ϕ r } wth the extra condton of dsjontness. The noton of a self-smlar measure on K depends on the partcular presentaton. However, we show that the noton of sometry nvarant self-smlar measure on K s ndependent of the presentatons Theorem 5.5. By ths theorem we can defne a natural number for each self-smlar set satsfyng the strong separaton property, an nvarant, whch does not depend on the presentaton Theorem 5.7. Ths nvarant s equal to the dmenson of the space of sometry nvarant self-smlar measures, and s related to the algebrac dependence of the smltudes of some any presentaton of K. In Secton 6 we show that the connecton between dfferent presentatons of a self-smlar set can be very complcated. Ths sheds some lght on why results and ther proofs n Secton 5 are complcated. The structure of dfferent presentatons of a self-smlar set n R has been also studed recently and ndependently by Feng and Wang n [8], where a smlar example s presented Self-affne sponges. Take the unt cube [0, 1] n n R n and subdvde t nto m 1... m n boxes of the same sze m 1,..., m n 2 and cut out some of them. Then do the same wth the remanng boxes usng the same pattern as n the frst step and so on. What remans after nfntely many steps s a self-affne set, whch s called self-affne Serpńsk sponge. A more precse defnton wll be gven n Defnton For n = 2 these sets were studed n several papers n whch they were called self-affne carpets or self-affne carpets of Bedford and McMullen. Bedford [2] and McMullen [22] determned the Hausdorff and Mnkowsk dmensons of these selfaffne carpets. The Hausdorff and Mnkowsk dmenson of self-affne Serpńsk sponges was determned by Kenyon and Peres [15]. Gatzouras and Lalley [10] proved that except n some relatvely smple cases such a set has zero or nfnty Hausdorff measure n ts dmenson and so n any dmenson. Peres extended ther results by provng that except n the same rare smple cases for any gauge functon nether the Hausdorff [28] nor the packng [27] measure of a self-affne carpet can be postve and fnte n fact, the packng measure cannot be σ-fnte ether, and remarked that these results extend to self-affne Serpńsk sponges of hgher dmensons. Recently the frst and the second lsted authors of the present paper showed [4] that some nce sets among others the set of Louvlle numbers have zero or non-σ-fnte Hausdorff and packng measure for any gauge functon by provng that these sets have zero or non-σ-fnte measure for any translaton nvarant Borel measure. Much earler Daves [3] constructed a compact subset of R wth ths

5 4 M. Elekes, T. Kelet, A. Máthé property. So t was natural to ask whether the self-affne carpets of Bedford and McMullen have ths stronger property. In Secton 7 we prove Corollary 7.7 that for any self-affne Serpńsk sponge K R n wth the natural Borel probablty measure µ see n Defnton 2.15 on K and t R n, the set K K + t has postve µ measure f and only f t has non-empty nteror relatve to K. For ths we prove Theorem 7.4 that for any self-affne Serpńsk sponge K R n and translaton vector t R n we have µ K K +t = 0 unless K or t are of very specal form. We also characterze Theorem 7.9 those Serpńsk sponges for whch we do not have nstablty result for translatons and the natural probablty measure µ. In fact, we get that µ K K +t can be close to 1 only for the same specal sponges that appear n the above mentoned result. In Secton 8 we show Theorem 8.1 that for any self-affne Serpńsk sponge K R n the natural probablty measure µ on K can be extended as a translaton nvarant Borel measure ν on R n. We also extend ths result Theorem 8.2, Corollary 8.3 to slghtly larger classes of self-affne sets. 2. Notaton, basc facts and some lemmas In ths secton we collect several notons and well known or farly easy statements that we wll need n the sequel. Some of these mght be nterestng n ther own rght. Of course, only a few of them are needed for each specfc secton. Though some of these statements may be well known, for the sake of completeness we ncluded the proofs. Notaton 2.1. We shall denote by the dsjont unon and by dst the Eucldean dstance Affne maps, smltudes, sometres. Defnton 2.2. A mappng g : R d R d s called a smltude f there s a constant r > 0, called smlarty rato, such that dstga, gb = r dsta, b for any a, b R d. The affne maps of R d are of the form x Ax + b, where A s a d d matrx and b R d s a translaton vector. Thus the set of all affne maps of R d can be consdered as R d2 +d and so t can be consdered as a metrc space. It s easy to check that a sequence g n n ths metrc space converges to an affne map g f and only f g n converges to g unformly on any compact subset of R d. Defnton 2.3. For a gven set K R d wth affne span A K let A K, S K and I K denote the metrc space wth the above metrc of the njectve affne maps, smltudes and sometres of A K nto tself, respectvely. Note also that all these three metrc spaces wth the composton can be also consdered as topologcal groups.

6 Self-smlar and self-affne sets Self-smlar and self-affne sets and measures. Defnton 2.4. A K R d compact set s self-smlar f K = ϕ 1 K... ϕ r K, where r 2 and ϕ 1,..., ϕ r are contractve smltudes. A K R d compact set s self-affne f K = ϕ 1 K... ϕ r K, where r 2 and ϕ 1,..., ϕ r are njectve affne maps that are contractons n a norm whch gves the standard Eucldean topology. By the n-th generaton elementary peces of K we mean the sets of the form ϕ 1... ϕ n K, where n = 0, 1, 2,.... We shall use mult-ndces. By a mult-ndex we mean a fnte sequence of ndces; for I = 1, 2,..., n let ϕ I = ϕ 1... ϕ n and p I = p 1 p 2... p n. We shall consder I = as a mult-ndex as well: ϕ s the dentty map and p = 1. Note that the elementary peces of K are the sets of the form ϕ I K. These sets are also self-smlar/self-affne; and f h s a smltude / njectve affne map then hk s also self-smlar/self-affne and ts elementary peces are the sets of the form hϕ I K. Note also that every pont of K s contaned n an arbtrarly small elementary pece. Defnton 2.5. Let K = ϕ 1 K... ϕ r K be a self-smlar/self-affne set, and let p p r = 1, p > 0 for all. Consder the symbol space Ω = {1,..., r} N equpped wth the product topology and let ν be the Borel measure on Ω whch s the countable nfnte product of the dscrete probablty measure p{} = p on {1,..., r}. Let π : Ω K, {π 1, 2,...} = n=1 ϕ 1... ϕ n K be the contnuous addressng map of K. Let µ be the mage measure of ν under the projecton π; that s, µh = ν π 1 H for every Borel set H K. 1 Such a µ s called a self-smlar/self-affne measure on K. One can also defne see e.g. n [7] self-smlar or self-affne measures as the unque probablty measure µ on K such that µh = r =1 p µ ϕ 1 H holds for every Borel set H K. It was already proved by Hutchnson [11] that the two defntons agree. Lemma 2.6. Let K = ϕ 1 K... ϕ r K be a self-affne set, p p r = 1, p > 0 for all, and let µ be the self-affne measure on K correspondng to the weghts p. Then for every affne subspace A ether µa K = 0 or A K. Proof. Let {x 1, x 2,..., x k } be a maxmal collecton of affne ndependent ponts n K. Choose U 1,...,U k convex open sets such that x j U j j = 1,...,k and whenever we choose one pont from each U j they are affne ndependent. Snce

7 6 M. Elekes, T. Kelet, A. Máthé K U s a nonempty relatve open subset of K, we may choose an elementary pece ϕ Ij K n U j for each j. Let ε = mn 1 j k p Ij > 0. We shall use the notaton we ntroduced n Defnton 2.5. For 1 r and ω = 0, 1,... Ω, let σ ω =, 0, 1,.... Thus ν σ H = p νh for all Borel subset H of Ω. Suppose that A s an affne subspace such that µa K > 0. Thus ν π 1 A > 0. It s easy to prove see a possble argument later n the proof of Lemma 2.12 that ths mples that there exsts an elementary pece σ J Ω such that ν π 1 A σ J Ω > 1 εν σ J Ω = 1 εp J. Snce ν σ J σ Ij Ω = p J p Ij p J ε j = 1,...,k, the set π 1 A must ntersect the sets σ J σ Ij Ω. Therefore the set A must ntersect the sets πσ J σ Ij Ω = ϕ J ϕ Ij K j = 1,...,k. By pckng one pont from each A ϕ J ϕ Ij K, we get a maxmal collecton of affne ndependent ponts n K snce ϕ J s an nvertble affne mappng. As ths collecton s contaned n the affne subspace A, we get that K s also contaned n A. Remark 2.7. In ths paper one of our man goals s to study µ K gk, where g s an affne map of R d. By the above lemma f the affne map g does not map the affne span A K of K onto tself then µ gk K = 0 snce K ga K. The other property of affne maps we are nterested n s K gk, whch also mples that g maps A K onto tself. Thus t s enough to consder those affne maps g of R d that map the affne span A K of K onto tself. Snce then both K and gk are n A K, only the restrcton of g to A K matters. Ths s why n the next secton we shall study A K, S K and I K the njectve affne maps, smltudes and sometres of A K nto tself nstead of all affne maps, smltudes and sometres of R d. Therefore f we state somethng about µ gk K or about the property K gk for every affne map, smltude or sometry g, t wll be enough to prove them for g A K, g S K or g I K, respectvely. Note also that self-smlar sets and measures are self-affne as well, so results about self-affne sets and measures also apply for self-smlar sets and measures Separaton propertes. Defnton 2.8. A self-smlar/self-affne set K = ϕ 1 K... ϕ r K or more precsely, the collecton ϕ 1,..., ϕ r of the representng maps satsfes the strong separaton condton SSC f the unon ϕ 1 K... ϕ r K s dsjont; open set condton OSC f there exsts a nonempty bounded open set U R d such that ϕ 1 U... ϕ r U U; strong open set condton SOSC f there exsts a nonempty bounded open set U R d such that U K and ϕ 1 U... ϕ r U U;

8 Self-smlar and self-affne sets 7 convex open set condton COSC f there exsts a nonempty bounded open convex set U R d such that ϕ 1 U... ϕ r U U; measure separaton condton MSC f for any self-smlar/self-affne measure µ on K we have µ ϕ K ϕ j K = 0 for any 1 < j r. We note that the frst three defntons are standard but we have not seen any name for the last two n the lterature. It s easy to check the well known fact that we must have K U where E denotes the closure of a set E for the open set U n the defnton of OSC and SOSC, COSC. It s easy to see U can be chosen as a small ε-neghborhood of K for the frst mplcaton that for any self-affne set SSC = SOSC = OSC. Usng the methods of C. Bandt and S. Graf [1], A. Schef proved n [30] that, n fact, SOSC OSC holds for self-smlar sets. In [30] for self-smlar sets SOSC = MSC s also proved. Snce the proof works for self-affne sets as well we get that for any self-affne set SOSC = MSC. It seems to be also true that COSC = SOSC and so COSC = MSC but we do not prove ths, snce we do not need the frst mplcaton and the followng lemma s stronger than the second mplcaton. Lemma 2.9. Let K = ϕ 1 K... ϕ r K be a self-affne set n R d wth the convex open set condton and let µ be a self-affne measure on t. Then for any affne map Ψ : R d R d we have µ Ψ ϕ K ϕ j K = 0 1 < j r. Proof. Let 1 < j r and U be the convex open set gven n the defnton of COSC. Let A K be the affne span of K. Snce ϕ U A K and ϕ j U A K are dsjont convex open sets n A K, ϕ U A K ϕ j U A K must be contaned n a proper affne subspace A of A K. Snce K U A K, ths mples that ϕ K ϕ j K A, and so Ψ ϕ K ϕ j K ΨA. 2 Snce ΨA s an affne subspace, whch s smaller dmensonal than the affne span A K of K, we cannot have K ΨA, so by Lemma 2.6 we must have µ K ΨA = 0. By 2 ths mples that µ Ψϕ K ϕ j K = 0. We also note that one can fnd a self-smlar set n R that satsfes even the SSC but does not satsfy the COSC [8, Example 5.1], so SSC and COSC are ndependent even for self-smlar sets of R.

9 8 M. Elekes, T. Kelet, A. Máthé Notaton Gven a fxed measure µ, we shall say that two sets are almost dsjont f ther ntersecton has µ-measure 0. The almost dsjont unon wll be denoted by. It s very easy to prove one by one each of the followng facts. Facts Let K = ϕ 1 K... ϕ r K be a self-affne/self-smlar set wth the measure separaton condton and let µ be a self-affne/self-smlar measure on t, whch corresponds to the weghts p 1,..., p r. Then the followng statements hold. 1. Any two elementary peces of K are ether almost dsjont or one contans the other. 2. Any unon of elementary peces can be replaced by an almost dsjont countable unon. 3. For any mult-ndex I we have µ ϕ I = p I µ; that s, µ ϕ I B = p I µb for any Borel set B K. 4. We have µ ϕ I K = p I for any mult-ndex I. 5. For any Borel set B K we have { µb = nf p I : B =1 =1 } ϕ I K. Snce SOSC and COSC are both stronger than MSC and one of them wll be always assumed n ths paper, the statements of ths lemma wll often be tactly used. Sometmes, for example, we shall even handle the above almost dsjont sets as dsjont sets and often consder Fact 5 as the defnton of self-affne/self-smlar measures. Lemma Let K = ϕ 1 K... ϕ r K be a self-affne set wth the measure separaton property or n partcular wth the SSC or SOSC or COSC and let µ be a self-affne measure on t. Then for every ε > 0 and for every Borel set B K wth postve µ-measure there exsts an elementary pece ak of K of arbtrarly large generaton such that µ B ak > 1 εµ ak. Proof. Snce µb > 0, usng Fact 5, B can be covered by countably many elementary peces ϕ I K N such that 1 + εµb > µ ϕ I K. By subdvdng the elementary peces f necessary, we can suppose that each s of large generaton. If there exsts an N such that 1 + εµ B ϕ I K > µ ϕ I K then we can choose ϕ I as a.

10 Self-smlar and self-affne sets 9 Otherwse we have 1 + εµ B ϕ I K µ ϕ I K for each N, hence 1+εµB = 1+εµ B ϕ I K 1+εµ B ϕ I K µ ϕ K, contradctng the above nequalty. Lemma Let K = ϕ 1 K... ϕ r K be a self-affne set wth the measure separaton property or n partcular wth the SSC or SOSC or COSC and let µ be a self-affne measure on t. Then for any Borel set B K and ε > 0 there exst countably many parwse almost dsjont elementary peces a K such that µ B a K > 1 εµ a K and µ B \ a K = 0. Proof. The elementary peces a K wll be chosen by greedy algorthm. In the n th step n = 0, 1, 2,... we choose the largest elementary pece a n K such that µ a n K a K = 0 0 < n and µ B a n K > 1 εµ a n K. If there s no such a n K then the procedure termnates. We clam that µ B \ a K = 0. Suppose that µ B \ a K > 0. Then by Lemma 2.12 there exsts an elementary pece ak such that µ B \ a K ak > 1 εµ ak. Then µ B ak > 1 εµ ak but ak was not chosen n the procedure. Ths could happen only f ak ntersects a chosen elementary pece a K n a set of postve measure. But then ether a K ak or a K ak, whch are both mpossble Self-affne Serpńsk sponges. Defnton By self-affne Serpńsk sponge we mean self-affne sets of the followng type. Let n, r N, m 1, m 2,..., m n 2 ntegers, M be the lnear transformaton gven by the dagonal n n matrx m 1 0 M =..., and let 0 m n D = {d 1,..., d r } {0, 1,..., m 1 1}... {0, 1,..., m n 1} be gven. Let ϕ j x = M 1 x + d j j = 1,...,r. Then the self-affne set KM, D = K = ϕ 1 K... ϕ r K s a Serpńsk sponge. We can also defne the self-affne Serpńsk sponge as { } K = KM, D = M k α k : α 1, α 2,... D, k=1

11 10 M. Elekes, T. Kelet, A. Máthé or equvalently K s the unque compact set n R n n fact, n [0, 1] n such that r MK = K + D = K + d j ; j=1 that s, K = M 1 K + M 1 D. By teratng the last equaton we get K = M k K + M k D + M k+1 D M 1 D = M k K + M k α k M 1 α 1. α 1,...,α k D Note that the k-th generaton elementary peces of K are the sets of the form M k K+M k α k +...+M 1 α 1 α 1,..., α k D and the only 0-th generaton elementary pece of K s K tself. Defnton By the natural probablty measure on a self-affne sponge K = KM, D we shall mean the self-affne measure on K obtaned by usng equal weghts p j = 1 r j = 1,..., r. Snce the frst generaton elementary peces of K are translates of each other n fact, so are the k-th generaton elementary peces, ths s ndeed the most natural self-affne measure on K. Usng 5 of Facts 2.11 we get that { } µb = nf µs : B =1S, S s an elementary pece of K N =1 for every Borel set B K. Let µ be the Z n -nvarant extenson of µ to R n ; that s, for any Borel set B R n let µb = B + t K. One can check that t Z n µ µ M l H + v = r l µh for any H K Borel set, v Z n, l = 0, 1, 2, Lemma Let m 1,..., m n 2 and M be lke n Defnton Let t R n be such that M k t > 0 for every k = 0, 1, 2,..., where. denotes the dstance from Z n. Then there exsts nfntely many k N such that M k t > 1 2maxm 1,...,m n. Proof. Ths lemma mmedately follows from the followng clear fact: u 1 2 maxm 1,..., m n = Mu mnm 1,..., m n u 2 u.

12 Self-smlar and self-affne sets Invarant extenson of measures to larger sets. Lemma Suppose that the group G acts on a set X, M s a G-nvarant σ- algebra on X, A M, M A = {B M : B A} and µ s a measure on A, M A. Then the followng two statements are equvalent: µ gb = µb whenever g G and B, gb M A. There exsts a G-nvarant measure µ on X, M such that µb = µb for every B M A. Proof. The mplcaton s obvous. For provng the other mplcaton we construct µ as follows. If H s a set of the form then let H = =1 B, where g 1, g 2,... G and g 1 B 1, g 2 B 2,... M A 4 µh = µ g B =1 and let µh = f H M cannot be wrtten n the above form. Frst we check that µ s well defned; that s, f we have 4 and H = j=1 C j, h 1, h 2,... G and h 1 C 1, h 2 C 2,... M A then µ g B = µ h j C j. 5 =1 Usng that B H = j=1 C j we get that g B = g j=1 B C j = j=1 g B C j and so =1 and smlarly =1 j=1 µ g B = µ j=1 g B C j = j=1 j=1 =1 =1 j=1 µ h j C j = µ h j B C j. µ g B C j, Thus, usng condton for B = g B C j and g = h j g 1, we get 5. Usng the freedom n 4 and that whenever H M can be wrtten n the form 4 then the same s true for any H H M, t s easy to check that µ s a G-nvarant measure on X, M such that µb = µb for every B M A. We wll need only the followng specal case of ths lemma. Lemma Let µ be a Borel measure on a Borel set A R n and let G be a group of affne transformatons of R n. Suppose that µ gb = µb whenever g G, B, gb A and B s a Borel set. 6 Then there exsts a G-nvarant Borel measure µ on R n such that µb = µb for any B A Borel set.

13 12 M. Elekes, T. Kelet, A. Máthé Remark The extenson we get n the above proof do not always gve the measure we expect t may be nfnty for too many sets. For example, f A R s a Borel set of frst category wth postve Lebesgue measure, G s the group of translatons and µ s the restrcton of the Lebesgue measure to A then the Lebesgue measure tself would be the natural translaton nvarant extenson of µ, however the extenson µ as defned n the proof s clearly nfnty for every Borel set of second category. Defnton Let µ be a Borel measure on a compact set K. We say that µ s sometry nvarant f gven any sometry g and a Borel set B K such that gb K, then µb = µ gb. Ths defnton makes sense snce by Lemma 2.18 exactly the sometry nvarant measures on K can be extended to be sometry nvarant measures on R n n the usual sense. As an llustraton of Lemma 2.18 we menton the followng specal case wth a pecular consequence. Lemma Let A R n n N be a Borel set such that A A + t s at most countable for any t R n \ {0}. Then any contnuous Borel measure µ on A contnuous here means that the measure of any sngleton s zero can be extended to a translaton nvarant Borel measure on R n. Note that although the condton that A A + t s at most countable for any t R n \ {0} seems to mply that A s very small, such a set can be stll farly large. For example there exsts a compact set C R wth Hausdorff dmenson 1 such that C C + t contans at most one pont for any t R \ {0} [14]. Combnng ths wth Lemma 2.21 we get the followng. Corollary There exsts a compact set C R wth Hausdorff dmenson 1 such that any contnuous Borel measure µ on C can be extended to a translaton nvarant Borel measure on R Some more lemmas. The followng smple lemmas mght be known but for completeness and because t s easer to prove them than to fnd them we present ther proof. Recall that the support of a measure s the smallest closed set wth measure zero complement. Lemma Let µ be a fnte Borel measure on R n wth compact support K. Then for every ε > 0 there exsts a δ > 0 such that u ε = µ K K + u 1 δµk. Proof. We prove by contradcton. Assume that there exsts an ε > 0 and a sequence u 1, u 2,... R n such that u n ε for every n N and µ K K+u µk > 0 n. By omttng some at most fntely many zero terms we can guarantee that every u n s n the compact annulus {x : ε x damk} where dam

14 Self-smlar and self-affne sets 13 denotes the dameter. Hence, by takng a subsequence, we can suppose that u n converges, say to u. Snce K K +u s a proper compact subset of K snce K s compact and u 0, K + u K s mpossble and K s the support of µ, we must have µk > µ K K + u = µk + u. It s well known see e.g. [29], Theorem that any fnte Borel measure s outer regular n the sense that the measure of any Borel set s the nfmum of the measures of the open sets that contan the Borel set. Thus µk+u < µk mples that there exsts an open set G K + u such that µg < µk. Then whenever u n u s less than the postve dstance between K and the complement of G, G contans K +u n and so µk > µg µk +u n. Ths s a contradcton snce u n u and µk + u n = µ K K + u n µk. Lemma Let µ be a probablty Borel measure on a compact set K R d such that any nonempty relatve open subset of K has postve µ measure. Then f the sequence g n of affne maps converges to an affne map g and µ g n K K 1 then µ gk K = 1. Moreover, K gk. Proof. Suppose that µ gk K = q < 1. Let gk ε denote the ε-neghborhood of gk. Snce n=1 gk 1/n K = gk K and µ s a fnte measure we have µ gk 1/n K µ gk K = q. Thus there exsts an ε > 0 for whch µ gk ε K 1+q 2 < 1. Snce g n converges unformly on K, for n large enough we have g n K gk ε and so µ g n K K 1+q 2, contradctng µ g n K K 1. Therefore we proved that µ gk K = 1. Then K \ gk s relatve open n K and has µ measure zero, so t must be empty, therefore K gk. 3. Self-affne sets wth the strong separaton condton Proposton 3.1. For any self-affne set K R d wth the strong separaton condton there exsts an open neghborhood U A K of the dentty map such that for any g U, gk K g = dentty. Proof. Let n denote the dmenson of the affne span of K. We shall prove that there exsts a small open neghborhood V A K of the dentty map such that for any g V we have gk K g = dentty. Ths would be enough snce then for any g V we get K g 1 K g = dentty, therefore U = V 1 = {g 1 : g V } has all the requred propertes. Smlarly as n the proof of Lemma 2.6, choose n + 1 elementary peces ϕ I1 K,..., ϕ In+1 K of K so that f we pck one pont from the convex hull of each of them then we get a maxmal collecton of affne ndependent ponts n the affne span of K. Let d = mn 1 n+1 dstϕ I K, K \ ϕ I K, then d > 0. Let V be a so small neghborhood of the dentty map that dstx, gx < d for any g V and x K. Let g V and gk K. Then, by the defnton of d and V we have gϕ I K ϕ I K for every 1 n + 1. Then the convex hulls of these

15 14 M. Elekes, T. Kelet, A. Máthé elementary peces are also mapped nto themselves. Snce each of these convex hulls s homeomorphc to a ball, by Brouwer s fxed pont theorem we get a fxed pont of g n each of these elementary peces. So we obtaned n + 1 fxed ponts of g such that ther affne span s exactly the affne span of K. Snce g s an affne map, the set of ts fxed ponts form an affne subspace, thus the set of fxed ponts of g contans the affne span of K. Snce g A K, g s defned exactly on the affne span of K, therefore g must be the dentty map. Theorem 3.2. Let K = ϕ 1 K... ϕ r K be a self-affne set satsfyng the strong separaton condton and let µ be a self-affne measure on K. Then there exsts a c < 1 and an open neghborhood U A K of the dentty map such that g U \ {dentty} = µ K gk < c. Proof. Accordng to our defnton of self-affne set see Defnton 2.4 there exsts a norm n whch every ϕ s contractve. Let dst ϕ denote the metrc determned by ths norm. Note that dst ϕ s equvalent to the Eucldean dstance; that s, there exst 0 < k 1 < k 2 such that k 1 dstx, y dst ϕ x, y k 2 dstx, y holds for all x, y R d. Usng Proposton 3.1 we can choose a small open neghborhood U A K of the dentty map such that even n the closure of U the only affne map g for whch gk contans K s the dentty map and so that dst ϕ x, gx < 1 for any g U and x K. 7 Snce A K s locally compact, we may also assume that the closure of U s compact. We clam that we can choose an even smaller open neghborhood V U of the dentty map such that ϕ 1 V ϕ U for = 1,...,r and that gϕ K ϕ j K = for any j and g V. Indeed, the frst property can be satsfed snce A K s a topologcal group and those g s for whch the second property do not hold are far from the dentty map. Now we clam that there exsts a c < 1 such that g U \V = µ gk K < c. Suppose that there exsts a sequence g n U \ V such that µ K g n K 1. Snce U \ V s compact there exsts a subsequence g n such that g n h U \ V. By Lemma 2.24 ths mples that hk K but n U \ V there s no such affne map h. We prove that ths U and ths c have the requred propertes; that s, g U \ {dentty} = µ K gk < c. If g U \ V then we are already done, so suppose that g V \ {dentty}. Let F denote the set of fxed ponts of g. The heurstcs of the remanng part of the proof s the followng. The affne map g moves K too slghtly. We zoom n on small elementary peces ak of K so that each gak ntersects only ak n K, but g moves ak far enough compared to ts sze. Techncally ths second requrement means that a 1 g a U \ V, so we can use the g U \ V case for the elementary pece ak. We fnd such an elementary pece around each pont of K that s not a fxed pont of g, and so we

16 Self-smlar and self-affne sets 15 get a partton of K \ F nto elementary peces wth the above property. Fnally, by addng up the estmates for these elementary peces we derve µ gk K < c. Clam 3.3. For any x K \ F there exsts a largest elementary pece ϕ Ix K of K that contans x and for whch ϕ 1 I x g ϕ Ix U \ V. Proof. Let 1, 2,... be the sequence of ndces for whch {x} = ϕ 1 ϕ 2... ϕ n K, n=1 and let I n = 1,..., n. Snce g V, we have ϕ 1 1 g ϕ 1 U by the defnton of V. If for some n we have ϕ 1 I n g ϕ In V then by the defnton of V we have ϕ 1 I n+1 g ϕ In+1 = ϕ 1 n+1 ϕ 1 I n g ϕ In ϕ n+1 U. Therefore t s enough to fnd an n such that ϕ 1 I n g ϕ In V snce then takng the smallest such n, I x = I n has the desred property. Lettng y n = ϕ 1 I n x we have y n K snce {x} = n=1 ϕ I n K and ϕ 1 I n g ϕ In y n = ϕ 1 I n gx. Snce x s not a fxed pont of g, for n large enough we have dst ϕ gx, ϕin K > dst ϕgx, x 2 def = t > 0. Recall that dst ϕ was defned as a metrc n whch every ϕ s contractve. Hence for each there exsts an α < 1 such that dst ϕ ϕ a, ϕ b α dst ϕ a, b for any a, b. Then, usng the mult-ndex notaton α In = α 1... α n, we clearly have dst ϕ ϕ In a, ϕ In b α In dst ϕ a, b for any a, b. Then dst ϕ ϕ 1 I n gx, K > t/α In, hence dst ϕ ϕ 1 I n large enough. Thus for n large enough, ϕ 1 I n even n U by 7. g ϕ In y n, K > t/α In, whch s bgger than 1 f n s g ϕ In s not n V, snce t s not Clam 3.4. For any x K \ F we have gϕ Ix K K ϕ Ix K, where I x = I n = 1,..., n s the mult-ndex we got n Clam 3.3. Proof. Let k {0, 1,..., n 1} be arbtrary and let I k = 1,..., k. Then ϕ 1 I k g ϕ Ik V, hence for any l k+1 we have ϕ 1 I k g ϕ Ik ϕ k+1 K ϕ l K =, whch s the same as g ϕ Ik+1 K ϕ Ik ϕ l K = l k+1. Snce g ϕ In K g ϕ Ik+1 K, ths mples that g ϕ In K ϕ Ik ϕ l K = k {0, 1,..., n 1}, l k+1. Snce K \ ϕ In K = n 1 k=0 l k+1 ϕ Ik ϕ l K, ths mples that gϕ In K K ϕ In K. The elementary peces {ϕ Ix K : x K \ F } clearly cover K \ F. Snce for any x y we have ϕ Ix K ϕ Iy K = or ϕ Ix K ϕ Iy K or ϕ Ix K ϕ Iy K, one can choose a K \ F ϕ J K 8 =1

17 16 M. Elekes, T. Kelet, A. Máthé countable dsjont subcover. By Clam 3.4 we have gϕ J K K ϕ J K. 9 Snce g s not the dentty map of the affne span of K and F s the set of fxed ponts of the affne map g, the dmenson of the affne subspace F s smaller than the dmenson of the affne span of K, and so we cannot have gf K. By Lemma 2.6 ths mples that µ gf K = 0. Usng ths last equaton, 8, 9, and fnally the defnton of a self-affne measure we get that µ gk K µ gf K + µ gk \ F K = µ gk \ F K = =1 µ g =1 ϕ J K K = µ g ϕ J K ϕ J K = =1 µ g ϕ J K K µ ϕ J ϕ 1 J g ϕ J K K =1 = =1 p J µ ϕ 1 J g ϕ J K K. Snce ϕ 1 J g ϕ J U \ V, the measures n the last expresson are less than c. Thus µ gk K < p J c = µ ϕ J K c = µ ϕ J K c = c, whch completes the proof. Theorem 3.5. Let K R d be a self-affne set wth the strong separaton condton and let µ be a self-affne measure on K. Then there exsts a constant c < 1 such that for any sometry g we have µ K gk < c unless gk = K. Proof. Suppose that g n I K that s, g n s an sometry of the affne span of K such that g n K K n N and µ K g n K 1. We can clearly assume that K g n K for each n and so the whole sequence g n s n a compact subset of I K. Thus, after choosng a subsequence f necessary, we can also assume that g n converges to an h I K. By Lemma 2.24 we must have K hk. It s well known and not hard to prove that no compact set n R d can have an sometrc proper subset, so K hk mples that hk = K. Applyng Theorem 3.2 we get a c < 1 and an open neghborhood U A K of the dentty such that g U \ {dentty} = µ K gk < c. Snce g n h we get g n h 1 dentty. Let n be large enough to have g n h 1 U and µ K g n K > c. Snce g n K K but hk = K we cannot have g n = h and so g n h 1 U \ {dentty}. Then, by the prevous paragraph, we get µ K g n K < c, contradctng µ K g n K > c. 4. Self-smlar sets wth the strong separaton property Our frst goal n ths secton s to prove the followng theorem.

18 Self-smlar and self-affne sets 17 Theorem 4.1. Let K = ϕ 1 K... ϕ r K be a self-smlar set satsfyng the strong separaton condton and µ be a self-smlar measure on t. There exsts c < 1 such that for every smltude g ether µ gk K < c or K gk. Now, for the sake of transparency we outlne the proof. At frst we need a new notaton. From S K we excluded those smlarty maps whch map everythng to a sngle pont. So let SK be the metrc space of all degenerate and all non-degenerate smlarty maps n the affne span A K of K; that s, S K = S K {f f : A K {y}, y A K }. 10 Frst we show that there exsts a compact set G SK of smlarty maps such that for every g SK there exsts g G for whch g K K = gk K. Then clearly t suffces to prove the theorem for g G. It s easy to see that no such compact set G n S K exsts. Let µ H be a constant multple of Hausdorff measure of approprate dmenson so that µ H K = 1. The restrcton of ths measure to K s a self-smlar measure. Let us consder those h G for whch K hk holds. Usng Hausdorff measures and Theorem 3.2 we prove that there are only fntely many such h, and also that the theorem holds n small neghbourhoods of each such h for the measure µ H. The maxmum of the correspondng fntely many values c s stll strctly smaller than 1. Let us now cut these small neghbourhoods out of G. Usng upper semcontnuty of our measure Lemma 2.24 we produce a c < 1 such that for the remanng smlarty maps g we have µ H gk K < c. Then clearly the same holds for all elements of G, possbly wth a larger c < 1, fnshng the proof for the measure µ H. Applyng the theorem for µ H, and also n a small open neghbourhood U of the dentty for every self-smlar measure µ, we show that f h G, K hk, and g s n a small neghbourhood of h then µ gk K < c. Then the same argument as above usng upper semcontnuty yelds the theorem, possbly wth a larger constant agan. Proposton 4.2. Let K = ϕ 1 K... ϕ r K be a self-smlar set satsfyng the strong separaton condton. Then there exsts a compact set G SK such that for every smlarty map g SK there s a g G for whch g K K = gk K holds. Proof. Let D denote the dameter of K, let δ = mn 1 <j r dstϕ K, ϕ j K and let G = {g S K : gk K, the smlarty rato of g s at most D/δ} {g 0 }, where g 0 SK s an arbtrary fxed smlarty map such that gk K =. It s easy to check that G SK s compact. Let g SK. If g G or g K K = then we can choose g = g or g = g 0, respectvely. So we can suppose that g K K and the smlarty rato of g s greater than D/δ. Then the mnmal dstance between the frst generaton

19 18 M. Elekes, T. Kelet, A. Máthé elementary peces g ϕ j K of g K s larger than D. So there exsts ϕ such that g K K = g ϕ K K. Therefore g can be replaced by g ϕ, whch has smlarty rato α tmes smaller than the smlarty rato of g, where α denotes the smlarty rato of ϕ. Snce maxα 1,..., α r < 1, ths way n fntely many steps we get a g wth smlarty rato at most D/δ such that gk K = g K K, whch completes the proof. Proposton 4.3. Let K = ϕ 1 K... ϕ r K be a self-smlar set satsfyng the strong separaton condton. Then {g S K : gk K} s dscrete n S K, hence countable, and also closed n S K. Let µ H be a constant multple of Hausdorff measure of approprate dmenson so that µ H K = 1. There exsts c < 1 such that for every smltude g ether µ H gk K < c or K gk. Proof. By Lemma 2.24 {g S K : gk K} s closed. Snce every dscrete subset of a subspace of R d2 +d s countable, n order to prove t s enough to prove that {g S K : gk K} s dscrete. Let ε be a postve number to be chosen later, and h be a smltude for whch K hk. Denote by K δ the δ-neghbourhood of K. As µ H hk s fnte, there s a small δ > 0 such that µ H Kδ hk \ K < ε. Applyng Theorem 3.2 to K and µ H we obtan an open neghbourhood U A K and a constant c H. There exsts an open neghbourhood W ε S K of the dentty such that a W ε = W 1 ε U, b dstgx, x < δ for every x K, c µ H gb 1 + εµh B for every g W ε and Borel set B, where for c we use that a smltude of rato α multples the s-dmensonal Hausdorff measure by α s. Let g W ε h and g h. Clearly W ε h s an open neghbourhood of h and g h 1, h g 1 W ε \ {dentty}, and h g 1 K K δ. Hence µ H K gk 1 + εµh h g 1 K gk = = 1 + εµ H h g 1 K hk = = 1 + εµ H h g 1 K K εµ H h g 1 K hk \ K 1 + εc H εµ H Kδ hk \ K 1 + εc H εε. 11 The last expresson s clearly smaller than 1 f ε s small enough, so let us fx such an ε. Therefore f g W ε h and g h then gk K, whch shows that {g S K : gk K} s dscrete fnshng the proof of. In order to prove suppose towards a contradcton that sup {µ H gk K : g SK, gk K} = 1. Then we also have sup {µ HgK K : g

20 Self-smlar and self-affne sets 19 G, gk K} = 1. Let g n be a convergent sequence n G so that g n K K, µ H gn K K 1, g n h. Lemma 2.24 yelds hk K, hence g n h. If n s large enough then g n W ε h and, by 11, µ H K gn K 1 + εc H εε, contradctng µ H gn K K 1. Proof of Theorem 4.1. By Proposton 4.2 we can assume g G. Let c H be the constant yelded by Proposton 4.3. Fx h G wth hk K. There are only fntely many such h by Proposton 4.3 and the compactness of G. Let us now apply Lemma 2.12 to the self-smlar set hk, µ H, 0 < ε 1 c H and B = K hk. We obtan ϕ I such that µ H K hϕi K 1 εµ H hϕi K. Hence Proposton 4.3 appled to the self-smlar set hϕ I K and the smltude h ϕ I 1 gves K hϕ I K. Snce hϕ I K s open n hk, t s also open n K and so t can be wrtten as a unon of elementary peces of K. Snce hϕ I K s compact ths mples that hϕ I K s a fnte unon of elementary peces of K. Let ϕ J K be one of these elementary peces. So ϕ J K hϕ I K K hk. As ϕ J K s open n K, t s also open n hϕ I K, hence also n hk. Therefore dstϕ J K, hk \ ϕ J K > 0, and so for every g that s close enough to h we have g h 1 hk \ ϕ J K ϕ J K =. Thus, as ϕ J K hk, for every such g we have gk ϕ J K = g h 1 hk ϕ J K = g h 1 ϕ J K ϕ J K. On the other hand, Theorem 3.2 yelds that there exsts a c < 1 such that f g s close enough to h and g h then µ g h 1 ϕ J K ϕ J K < c µ ϕ J K = c p J. Therefore µ gk ϕ J K = µ g h 1 ϕ J K ϕ J K < c p J and µ gk K = µ gk ϕ J K + µ gk K \ ϕ J K < c p J + 1 p J = 1 1 cp J. 12 As we only consdered fntely many h s, there exsts c < 1 such that f g s close to one of these h s, but dstnct from t, then µ gk K < c. Ths, together wth Lemma 2.24 provdes a c < 1 such that for every g G ether µ gk K < c or gk K. Just lke at the end of the proof of Proposton 4.3. Fnally, by Proposton 4.2 ths also holds outsde G. We wll apply ths theorem to elementary peces of K nstead of K tself. It s easy to see that the same c works for every elementary pece; that s, we have the followng corollary of Theorem 4.1.

21 20 M. Elekes, T. Kelet, A. Máthé Corollary 4.4. Let K = ϕ 1 K... ϕ r K be a self-smlar set satsfyng the strong separaton condton and µ be a self-smlar measure on t. There exsts c < 1 such that for every smltude g and every elementary pece ak of K ether µ gk ak < c µ ak or ak gk. Now we are ready to prove the second man result of ths secton. Theorem 4.5. Let K = ϕ 1 K... ϕ r K be a self-smlar set satsfyng the strong separaton condton, µ be a self-smlar measure on t, and g be a smltude. Then µ gk K > 0 f and only f the nteror n K of gk K s nonempty. Moreover, µ nt K gk K = µ gk K. Proof. If the nteror n K of gk K s nonempty then clearly t s of postve measure, snce the measure of every elementary pece s postve. Let c be the constant gven by Corollary 4.4, and let g be a smltude such that µ gk K > 0. Applyng Lemma 2.13 for B = gk K and ε = 1 c we obtan countably many dsjont elementary peces a K of K such that µ gk a K = µ gk K a K > c µ a K 13 and gk K \ a K s of µ-measure zero. By Corollary 4.4, 13 mples that a K gk. Snce a K s open n K, t s open n gk K, so a K nt K gk K. Hence µ gk K = µ gk K a K + µ gk K \ a K provng the theorem. As an mmedate consequence we get the followng. = µ a K µ nt K gk K, Corollary 4.6. Let K R d be a self-smlar set satsfyng the strong separaton condton, and let µ 1 and µ 2 be self-smlar measure on K. Then for any smltude g of R d, µ 1 gk K > 0 µ2 gk K > 0. We also get the followng farly easly. Corollary 4.7. Let K R d be a self-smlar set satsfyng the strong separaton condton, let A K be the affne span of K and let µ be a self-smlar measure on K. Then the set of those smltudes g : A K R d for whch µ gk K > 0 s countably nfnte. Proof. It s clear that there exst nfntely many smltudes g such that µ gk K > 0 snce the elementary peces of K are smlar to K and have postve µ measure. By Lemma 2.6, µ gk K > 0 mples that g S K and, by Theorem 4.5, that gk contans an elementary pece of K. Therefore t s enough to show that

22 Self-smlar and self-affne sets 21 for each fxed elementary pece ak of K there are only countably many g S K such that gk ak, whch s the same as a 1 gk K. By the frst part of Proposton 4.3 there are only countably many such a 1 g S K, so there are only countably many such g S K. From the frst part of Proposton 4.3 we get more results about those smlarty maps that map a self-smlar set nto tself. These results wll be used n the next secton and they are also related to a theorem and a queston of Feng and Wang [8] as t wll be explaned before Corollary Lemma 4.8. Let K = ϕ 1 K... ϕ r K be a self-smlar set wth strong separaton condton. There exsts only fntely many smltudes g for whch gk K holds and gk ntersects at least two frst generaton elementary peces of K. Proof. The smlarty ratos of these smltudes g are strctly separated from zero. Thus the smlarty rato of ther nverses have some fnte upper bound, and also K g 1 K holds. The set of smltudes wth the latter property form a dscrete and closed set accordng to the frst part of Proposton 4.3. Those h SK smlarty maps cf. 10 whose smlarty rato s under some fxed bound and for whch hk K holds form a compact set n SK see proof of Proposton 4.2. Snce a dscrete and closed subspace of a compact set s fnte, the proof s fnshed. Theorem 4.9. Let K = ϕ 1 K... ϕ r K be a self-smlar set wth strong separaton condton and let λ be a smltude for whch λk K. There exst an nteger k 1 and mult-ndces I, J such that λ k ϕ I = ϕ J. Proof. For every nteger k 1 there exsts a smallest elementary pece ϕ I K whch contans λ k K. For ths mult-ndex I, ϕ 1 I λ k K s a subset of K and ntersects at least two frst generaton elementary peces of K. There are only fntely many smltudes wth ths property accordng to Lemma 4.8, hence there exst k < k, I, I such that ϕ 1 I λ k = ϕ 1 I. By rearrangement we obtan ϕ λk I ϕ 1 I = λ k k and λ k k ϕ I = ϕ I. Feng and Wang [8, Theorem 1.1 The Logarthmc Commensurablty Theorem] proved that f K = ϕ 1 K... ϕ r K s a self-smlar set n R satsfyng the open set condton wth Hausdorff dmenson less than 1 and such that each smlarty map ϕ s of the form ϕ x = bx + c wth a fxed b and ak + t K for some a, t R then log a / log b Q. They also posed the problem Open Queston 2 of generalzng ths result to hgher dmensons. If we assume the strong open set condton nstead of the open set condton then the above Theorem 4.9 tells much more about the maps ϕ 1,...,ϕ r and ax + t and mmedately gves the followng hgher dmensonal generalzaton of the Logarthmc Commensurablty Theorem of Feng and Wang, n whch we can also allow non-homogeneous self-smlar sets.

23 22 M. Elekes, T. Kelet, A. Máthé Corollary Let K = ϕ 1 K... ϕ r K be a self-smlar set wth strong separaton condton and suppose that λ s a smltude for whch λk K. If a 1,...,a r and b denote the smlarty ratos of ϕ 1,..., ϕ r and λ, respectvely, then log b must be a lnear combnaton of log a 1,..., loga r wth ratonal coeffcents. 5. Isometry nvarant measures In ths secton all self-smlar sets we consder wll satsfy the strong separaton condton. Before we start to study and characterze the sometry nvarant measures on a self-smlar set of strong separaton condton, we have to pay some attenton to the connecton of a self-smlar set and the self-smlar measures lvng on t. We have called a compact set K self-smlar wth the SSC f K = ϕ 1 K... ϕ r K holds for some smltudes ϕ 1,..., ϕ r. A presentaton of K s a fnte collecton of smltudes {ψ 1,...,ψ s }, such that K = ψ 1 K... ψ s K and s 2. Clearly, a self-smlar set wth SSC has many dfferent presentatons. For example, f {ϕ 1, ϕ 2,...,ϕ r } s a presentaton of K, then {ϕ ϕ j : 1, j r} s also a presentaton. As we wll see n the next secton, t s possble that a self-smlar set has no smallest presentaton. We say that a presentaton F 1 = {ψ 1, ψ 2,..., ψ s } s smaller than the presentaton F = {ϕ 1, ϕ 2,..., ϕ r } f for every 1 r there exsts a mult-ndex I such that ϕ = ψ I. Ths defnes a partal order on the presentatons; let us wrte F 1 F when F 1 s smaller than F. We call a presentaton mnmal, f there s no smaller presentaton excludng tself. We call a presentaton smallest, f t s smaller than any other presentaton. There exsts a self-smlar set wth the SSC whch has more than one mnmal presentatons; that s, t has no smallest presentaton see Secton 6. The noton of a self-smlar measure on a self-smlar set depends on the presentaton. Thus, when we say that µ s a self-smlar measure on K, we always mean that µ s self-smlar wth respect to the gven presentaton of K. Clearly f F 1 F, then there are less self-smlar measures wth respect to F 1 than to F. It wll turn out that the sometry nvarant self-smlar measures are the same ndependently of the presentatons. Notaton 5.1. For the sake of smplcty, for a smltude λ wth λk K let µλ denote µ λk. In the composton of smltudes we mght omt the mark, so g 1 g 2 stands for g 1 g 2, and by g k we wll mean the composton of k many g s. Clearly, gven any self-smlar measure µ, µ ϕ I = µϕ I µ holds for the smltudes ϕ I arsng from the presentaton of K. Accordng to the next proposton, f for a gven self-smlar measure µ the congruent elementary peces are of equal measure, then the same holds for any smltude λ satsfyng λk K; that s, we have µ λ = µλ µ as well. Proposton 5.2. Let K = ϕ 1 K... ϕ r K be a self-smlar set wth strong separaton condton, and µ be a self-smlar measure on K for whch the congruent