pacific journal of mathematics Vol. 183,No. 1,1998 SELF-AFFINE MULTIFRACTAL SIERPINSKI SPONGES IN Rd L. Olsen Westudyself-a(cid:14)nemultifractalsinRd usingtheformalism introduced in [Olsen, A multifractal formalism, Advances in Math- ematics, 116 (1996), 82-196]. We prove that new multifractal phenomena, not exhibited by self-similar multifractals in Rd, appear in the self-a(cid:14)ne case. 1. Introduction. We analyze the multifractal structure of self-a(cid:14)ne invariant measures in Rd supported by a particular type of self-a(cid:14)ne sets usually called Sierpinski Sponges. Our analysis is based on the multifractal formalism introduced by Olsen in [Ol1]. For a metric space X we denote the family of Borel probability measures on X by P(X). For (cid:22) 2 P(X) and x 2 X we de(cid:12)ne the upper and lower local dimension of (cid:22) at x by log(cid:22)B(x;r) (1.1) (cid:11) (x) = limsup (cid:22) logr r&0 resp. log(cid:22)B(x;r) (1.2) (cid:11) (x) = liminf (cid:22) r&0 logr where B(x;r) denotes the closed ball with centre x and radius r. If (cid:11) (x) (cid:22) and (cid:11) (x) agree we refer to the common value as the local dimension of(cid:22) (cid:22) at x and denote it by (cid:11) (x). For each (cid:11) (cid:21) 0 de(cid:12)ne (cid:1) ((cid:11)) by (cid:22) (cid:22) (1:3) (cid:1) ((cid:11)) = fx 2 supp(cid:22) j (cid:11)(cid:22)(x) = (cid:11)g (cid:22) where supp(cid:22) denotes the topological support of (cid:22). The main problem in multifractal analysis is to estimate the size of the sets (cid:1) ((cid:11)); this is done (cid:22) by introducing the functions f ;F : R+! R de(cid:12)ned by (cid:22) (cid:22) + (1.4) f ((cid:11)) = dim(cid:1) ((cid:11)) (cid:22) (cid:22) (1.5) F ((cid:11)) = Dim(cid:1) ((cid:11)) (cid:22) (cid:22) 143 144 L.OLSEN where dim and Dim denote Hausdor(cid:11) dimension and packing dimension respectively. These and similar functions are generically known as \the multifractal spectrum of (cid:22)", \the singularity spectrum of (cid:22)", \the spectrum of scaling indices" or simply \the f((cid:11))-spectrum". The function f((cid:11)) = f ((cid:11)) (cid:22) was (cid:12)rst explicitly de(cid:12)ned by the physicists Halsey et al. in 1986 in their seminalpaper[HJKPS]. ThereaderisreferredtoCawley&Mauldin[CM] or Olsen [Ol1] for a more detailed historical account of multifractality, and an extensive list of references. Many recent papers have studied the multifractal structure of self-similar measures. Cawley & Mauldin [CM] analyzed the multifractal structure of (non-random) self-similar measures, and Edgar & Mauldin [EM] and Olsen [Ol1]investigatedthemultifractalstructureof(non-random)graphdirected self-similar measures. Riedi [Re2] has studied self-similar multifractals generated by a countable in(cid:12)nite number of similarities. A recent research monograph by Olsen [Ol2] presents a detailed multifractal analysis of random graph directed self-similar measures based on the formalism introduced in[Ol1]. IndependentlyFalconer[Fa4]andlaterArbeiter&Patzschke[PA] have studied random self-similar multifractals. Figure 1.1. The (cid:12)rst two stages in the construction of a self-a(cid:14)ne measure. In this example d = 2; n = 2; n = 5; B consists of 3 boxes and 1 2 p = (p ;p ;p ): 1 2 3 SELF-AFFINEMULTIFRACTALSIERPINSKISPONGESINRd 145 We note that all the papers [AP, CM, EM, Fa4, Ol1, Ol2, Re2] analyze the multifractal structure of self-similar (or graph directed self-similar) measures. In this paper we focus on the multifractal structure of self-a(cid:14)ne measures (cid:22) in Rd supported by a particular type of self-a(cid:14)ne sets usually called Sierpinski Sponges { thus we will call the measures (cid:22) that we study for self-a(cid:14)ne multifractal Sierpinski Sponges. Let 1 < n (cid:20) n (cid:20) (cid:1)(cid:1)(cid:1) (cid:20) n 1 2 d beintegers. Byconsideringn −1(d−1)-dimensionalhyperplanesparallelto (cid:0)l the hyperplane spanned by ((cid:14) ) ;:::;((cid:14) ) ;((cid:14) ) ;:::; (cid:1) i1 i=1;:::;d i;l−1 i=1;:::;d i;l+1 i=1;:::;d ((cid:14) ) (here (cid:14) denotes the Kronecker delta) for each l = 1;:::;d, we id i=1;:::;d ij partition the unit cube [0;1]d into n (cid:1)(cid:1)(cid:1)n congruent boxes with sidelengths 1 d 1 ;:::; 1 . LetBbeasubcollectionoftheseboxesandletp = (p ) bea pnr1obabilnidtyvectorindexedbyB. ErasealltheboxesnotcontainedBinB2BB, and divide a unit mass between the remaining boxes in the ratio determined by the probability vector p. Next partition each of the remaining boxes B into n (cid:1)(cid:1)(cid:1)n congruent subboxes of B with sidelengths 1 ;:::; 1 , again keeping 1 d n2 n2 only those which corresponds to B, and divide the 1mass ofdB between the remaining subboxes of B in the ratio determined by the probability vector p. Continuing this process in(cid:12)nitely, a compact set K and a probability measure (cid:22) supported on K are obtained, see Figure 1.1. The set K is called aself-a(cid:14)neSierpinskiSponge, andthemeasure(cid:22)iscalledaself-a(cid:14)nemulti- fractal Sierpinski Sponge. In this paper we study the multifractal structure of the measure (cid:22). Our analysis will be based on the multifractal formalism introduced by Olsen [Ol1]. In particular we (cid:12)nd, assuming separation condition (II) (introduced in Section 4), 1) the Hausdor(cid:11) spectrum f of (cid:22); (cid:22) 2) the multifractal box dimensions of (cid:22); 3) the generalized Renyi dimensions of (cid:22); 4) themultifractaldimensionfunctionsb ,B and(cid:3) introducedin[Ol1]; (cid:22) (cid:22) (cid:22) 5) a su(cid:14)cient condition guaranteeing that the multifractal Hausdor(cid:11) and multifractal packing measures, Hq;b(cid:22)(q)(supp(cid:22)) and Pq;B(cid:22)(q)(supp(cid:22)) (cid:22) (cid:22) introduced in [Ol1], are positive and (cid:12)nite. In the 2-dimensional case, separation condition (II) states that if a column of rectangles contains a box from B, then the two immediately adjacent columns of rectangles do not contain any boxes from B, c.f. Figure 1.1. This separation condition is unfortunately very strong and it would be very desirable if it could be weakened or omitted. Weprovethatself-a(cid:14)nemultifractalSierpinskiSpongespossesssomewell- known multifractal characteristica; in particular we prove the following: 1) Thereexisttwonumbers0 (cid:20) a (cid:20) asuchthat(cid:1) ((cid:11)) = ?for(cid:11) 2= [a;a], (cid:22) and f ((cid:11)) > 0 for (cid:11) 2]a;a[. (cid:22) 146 L.OLSEN 2) the multifractal spectrum function f equals the Legendre transform (cid:22) (cid:12)(cid:3) of a certain auxiliary function (cid:12) de(cid:12)ned explicitely in terms of the numbers n ;:::;n and the probability vector p. In particular, f is 1 d (cid:22) concave on its support. However, we also prove that new multifractal phenomena, not exhibited by self-similar multifractals in Rd, appear in the self-a(cid:14)ne case; in particular we prove the following: 1) TheHausdor(cid:11)multifractaldimensionfunctionb andthepackingmul- (cid:22) tifractal dimension function B do not necessarily coincide; in fact, for (cid:22) a (cid:12)xed q 2 R, b (q) and B (q) coincide if and only if condition (I ) (cid:22) (cid:22) q introduced in Section 4 is satis(cid:12)ed. This phenomenon is in sharp contrast to the self-similar case in which b = B by [Ol1, Theorem 5.1]. (cid:22) (cid:22) 2) WeconjecturethatthemultifractalHausdor(cid:11)measureHq;b(cid:22)(q)(supp(cid:22)) (cid:22) and the multifractal packing measure Pq;B(cid:22)(q)(supp(cid:22)) are not neces- (cid:22) sarily positive and (cid:12)nite. For a (cid:12)xed q 2 R, condition (I ) implies q that 0 < Hq;b(cid:22)(q)(supp(cid:22)) < 1 and 0 < Pq;B(cid:22)(q)(supp(cid:22)) < 1, and (cid:22) (cid:22) we conjecture that if (Iq) is not satis(cid:12)ed, then H(cid:22)q;b(cid:22)(q)(supp(cid:22)) = Pq;B(cid:22)(q)(supp(cid:22)) = 1; cf. Conjecture 4.1.10 and Conjecture 4.1.11. (cid:22) This phenomenon is in sharp contrast to the self-similar case in which Hq;b(cid:22)(q)(supp(cid:22)) and Pq;B(cid:22)(q)(supp(cid:22)) are positive and (cid:12)nite for all q 2 (cid:22) (cid:22) R by [Ol1, Theorem 5.1]. 3) The Legendre transform B(cid:3) of B does not necessarily attain the con- (cid:22) (cid:22) stant value −1 outside the set [a;a]. There exist self-a(cid:14)ne measures (cid:22) such that 0 < B(cid:3)((cid:11)) for all (cid:11) 2 [A;a] [ [a;A] where at least one (cid:22) of the intervals [A;a] or [a;A] is non-degenerate. This phenomenon is in sharp contrast to the self-similar case in which B(cid:3)((cid:11)) = −1 for (cid:22) all (cid:11) 2= [a;a] by [Ol1, Theorem 5.1]. (We remark that a very similar situation arises in multifractal analysis of random (graph directed) self-similar measures: generically there exist (cf. [Ol2]) numbers 0 (cid:20) a (cid:20) a (cid:20) a (cid:20) a such that for each (cid:12)xed (cid:11) 2]a;a [[]a ;a[, min max min max almost all self-similar measures (cid:23) satisfy −1 < b(cid:3)((cid:11)) = B(cid:3)((cid:11)) < 0 = (cid:23) (cid:23) f ((cid:11)) = F ((cid:11)).) (cid:23) (cid:23) Wenotethatourresults, duetotheuseofthegeneralizedmultifractalHaus- dor(cid:11)andpackingmeasuresintroducedin[Ol1], appearasnaturalmultifrac- tal generalizations of some of the main results on self-a(cid:14)ne sets by Bedford [Be],McMullen[McM],Kenyon&Peres[KP]andPeres[Pe1,Pe2],inpar- ticular Kenyon & Peres [KP, Theorem 1.2 and Proposition 1.3] and Peres [Pe2, Theorem 1.1.(ii)]. King[Ki]havedeterminedtheHausdor(cid:11)spectrumf forself-a(cid:14)nemulti- (cid:22) fractalSierpinskiSpongesinR2. InthispaperweextendKing’sresultstoRd SELF-AFFINEMULTIFRACTALSIERPINSKISPONGESINRd 147 (itshouldbenotedthattheextensionfromR2 toRd isnotmerelyatechnical extension, c.f. also Kenyon & Peres [KP, remark just above Theorem 1.2]) and, in addition, investigate the multifractal box dimensions of (cid:22), the generalized multifractal Hausdor(cid:11) and packing measures, Hq;t and Pq;t, and the (cid:22) (cid:22) generalized multifractal dimension functions b and B introduced in [Ol1]. (cid:22) (cid:22) Schmeling & Siegmund-Schultze [SS] have studied certain self-a(cid:14)ne multifractals. However, Schmeling & Siegmund-Schultze’s approach is di(cid:11)erent from our approach. Schmeling & Siegmund-Schultze consider a probability vector (p ) and a family of a(cid:14)ne maps (x ! A x+a ) where i i=1;:::;N i i i=1;:::;N A are linear contractions of Rd with kA k < 1 and a 2 Rd. They study, i i 3 i for Lebesgue almost all translation vectors (a ) 2 (Rd)N, a part of i i=1;:::;N the Hausdor(cid:11) spectrum function f of the self-a(cid:14)ne measure (cid:22) generated (cid:22) by the maps (x ! A x + a ) and the probabilities (p ) (i.e. i i i=1;:::;N i i=1;:::;N (cid:22) is the unique probability measure on Rd satisfying the self-a(cid:14)ne equa- P tion (cid:22) = p (cid:22)(cid:14)S−1 where S (x) = A (x)+a ). Schmeling & Siegmund- i i i i i i Schultze’s approach can be viewed as an attempt to generalize some of Fal- coner’s[Fa1,Fa3]resultsonself-a(cid:14)nesetstothemultifractalcase. Falconer also considers a family of a(cid:14)ne maps (x ! A x+a ) where A are i i i=1;:::;N i linear contractions of Rd with kA k < 1 and a 2 Rd. Falconer then shows i 3 i that, for Lebesgue almost all translation vectors (a ) 2 (Rd)N, the i i=1;:::;N self-a(cid:14)ne set K generated by the maps (x ! A x+a ) (i.e. K is the i i i=1;:::;N unique non-empty compact subset of Rd satisfying the self-a(cid:14)ne equation K = [ S (K) where S (x) = A (x)+a ) has equal Hausdor(cid:11) dimension and i i i i i box dimension and presents an asymptotic formula for this dimension. Fi- nallywenotethatRiedi[Ri1]hascomputedthemultifractalboxdimensions of a class of self-a(cid:14)ne multifractal. We will now give a brief description of the organization of the paper. In Section 2 we recall the multifractal formalism introduced in [Ol1], and de(cid:12)ne the notion of a self-a(cid:14)ne set and a self-a(cid:14)ne measure. In Section 3 we introduce two auxiliary functions, (cid:12) and γ, and study their properties. Section 4 contains the statements of our main results formulated in terms of the auxiliary functions (cid:12) and γ. Section 5 contains an example. In Section 6 we present the proofs of our main results. 2. The Setting. 2.1. The multifractal measures Hq;t and Pq;t. (cid:22) (cid:22) This section gives a brief summary of the main results in [Ol1]. We (cid:12)rst recall the de(cid:12)nition of the Hausdor(cid:11) measure, the centered Hausdor(cid:11) measure and the packing measure. Let X be a metric space, E (cid:18) X and (cid:14) > 0. A countable family B = (B(x ;r )) of closed balls in X is called a i i i 148 L.OLSEN centered (cid:14)-covering of E if E (cid:18) [ B(x ;r ), x 2 E and 0 < r < (cid:14) for all i. i i i i i The family B is called a centered (cid:14)-packing of E if x 2 E, 0 < r < (cid:14) and i i B(x ;r )\B(x ;r ) = ? for all i 6= j. Let E (cid:18) X, t (cid:21) 0 and (cid:14) > 0. Now i i j j put ( ) X [1 Ht(E) = inf diam(E )t j E (cid:18) E ; diamE < (cid:14) : (cid:14) i i i i i=1 The t-dimensional Hausdor(cid:11) measure Ht(E) of E is de(cid:12)ned by Ht(E) = supHt(E): (cid:14) (cid:14)>0 The reader is referred to [Fa2] for more information on Ht. We will now de(cid:12)ne the packing measure. Write ( ) X1 Pt(E) = sup (2r )t j (B(x ;r )) is a centered (cid:14)-packing of E : (cid:14) i i i i i=1 t The t-dimensional prepacking measure P (E) of E is de(cid:12)ned by t t P (E) = inf P (E): (cid:14) (cid:14)>0 t The set function P is not necessarily countable subadditive, and hence not t necessarily an outer measure, c.f. [TT] or [Fa2]. But P give rise to a Borel measure, namely the t-dimensional packing measure Pt(E) of E, as follows X1 Pt(E) = inf Pt(E ): i E(cid:18)[1i=1Ei i=1 The packing measure was introduced by Taylor and Tricot in [TT] using centered(cid:14)-packingsofopenballs, andbyRaymondandTricotin[RT]using centered (cid:14)-packings of closed balls. Also recall that the Hausdor(cid:11) dimension dim(E), the packing dimension Dim(E) and the logarithmic index (cid:1)(E) of E is de(cid:12)ned by dim(E) = supft (cid:21) 0 j Ht(E) = 1g Dim(E) = supft (cid:21) 0 j Pt(E) = 1g t (cid:1)(E) = supft (cid:21) 0 j P (E) = 1g: We refer the reader to [Tr] and [RT] for more information on the centered Hausdor(cid:11) measure, the packing measure and the packing dimension. SELF-AFFINEMULTIFRACTALSIERPINSKISPONGESINRd 149 Olsen [Ol1] suggested that some multifractal generealizations of the (centered) Hausdor(cid:11) measure and the packing measure might be useful in multifractal analysis. For q 2 R de(cid:12)ne ’ : [0;1[ ! R = [0;1] by q + ( 1 for x = 0 ’ (x) = for q < 0 q xq for 0 < x ’ (x) = 1 for q = 0 q ( 0 for x = 0 ’ (x) = for 0 < q: q xq for 0 < x For (cid:22) 2 P(X), E (cid:18) X, q;t 2 R and (cid:14) > 0 write ( X Hq;t(E) = inf ’ ((cid:22)(B(x ;r )))(2r )t j (B(x ;r )) (cid:22);(cid:14) q i i i i i i i ) is a centered (cid:14)-covering of E ; E 6= ? q;t H (?) = 0 (cid:22);(cid:14) q;t q;t H (E) = supH (E) (cid:22) (cid:22);(cid:14) (cid:14)>0 Hq;t(E) = supHq;t(F): (cid:22) (cid:22) F(cid:18)E We also make the dual de(cid:12)nitions ( X Pq;t(E) = sup ’ ((cid:22)(B(x ;r )))(2r )t j (B(x ;r )) (cid:22);(cid:14) q i i i i i i i ) is a centered (cid:14)-packing of E ; E 6= ? q;t P (?) = 0 (cid:22);(cid:14) q;t q;t P (E) = inf P (E) (cid:22) (cid:22);(cid:14) (cid:14)>0 X Pq;t(E) = inf Pq;t(E ): (cid:22) (cid:22) i E(cid:18)[iEi i It is proven in [Ol1] that Hq;t and Pq;t are measures on the family of Borel (cid:22) (cid:22) subsets of X. The measure Hq;t is of course a multifractal generalisation of (cid:22) the (centered) Hausdor(cid:11) measure, whereas Pq;t is a multifractal generalisa- (cid:22) tion of the packing measure. In fact, it is easily seen that the follwing holds for t (cid:21) 0, (2:1) 2−tH0;t (cid:20) Ht (cid:20) H0;t; Pt = P0;t; Pt = P0;t: (cid:22) (cid:22) (cid:22) (cid:22) 150 L.OLSEN The next result shows that the measures Hq;t, Pq;t and the pre-measure (cid:22) (cid:22) q;t P in the usual way assign a dimension to each subset E of X. (cid:22) Proposition 2.1.1. There exist unique extended real valued numbers (cid:1)q(E) 2 [−1;1], Dimq(E) 2 [−1;1] and dimq(E) 2 [−1;1] such (cid:22) (cid:22) (cid:22) that ( 1 for t < (cid:1)q(E) Pq;t(E) = (cid:22) (cid:22) 0 for (cid:1)q(E) < t (cid:22) ( 1 for t < Dimq(E) Pq;t(E) = (cid:22) (cid:22) 0 for Dimq(E) < t (cid:22) ( 1 for t < dimq(E) Hq;t(E) = (cid:22) (cid:22) 0 for dimq(E) < t: (cid:22) Proof. See [Ol1, Proposition 1.1]. The number dimq(E) is an obvious multifractal analogue of the Hausdor(cid:11) (cid:22) dimension dim(E) of E whereas Dimq(E) and (cid:1)q(E) are obvious multifrac- (cid:22) (cid:22) tal analogues of the packing dimension Dim(E) and the logarithmic index (cid:1)(E) of E respectively. In fact, it follows immediately from the de(cid:12)nitions that (2:2) dim(E) = dim0(E); Dim(E) = Dim0(E); (cid:1)(E) = (cid:1)0(E): (cid:22) (cid:22) (cid:22) Nextwede(cid:12)nemultifractaldimensionfunctionsb ;B ;(cid:3) : R ! [−1;1] (cid:22) (cid:22) (cid:22) by b (q) = dimq(supp(cid:22)); B (q) = Dimq(supp(cid:22)); (cid:3) (q) = (cid:1)q(supp(cid:22)): (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) We will now give a brief list of some of the most important properties of themeasuresHq;t andPq;t, andthecorrespondingdimensionfunctions. The (cid:22) (cid:22) reader is referred to Olsen [Ol1] for a detailed study of the measure Hq;t and (cid:22) Pq;t, and the dimension functions b , B and (cid:3) . For (cid:22) 2 P(X) and a > 1 (cid:22) (cid:16) (cid:22) (cid:22) (cid:17) (cid:22) write T ((cid:22)) = limsup sup (cid:22)B(x;ar) and de(cid:12)ne the family P (X) a r&0 x2supp(cid:22) (cid:22)B(x;r) F of Federer probability measures on X by P (X) = f(cid:22) 2 P(X) j T ((cid:22)) < F a 1 for some a > 1g. It follows from [Ol1] that the de(cid:12)nition of P (X) is F independent of the number a > 1, i.e. T ((cid:22)) < 1 for all a > 1 if and only if a T ((cid:22)) < 1 for some a > 1. a Proposition 2.1.2. Let (cid:22) 2 P(Rd) and q;t 2 R. Then: q;t i) Hq;t (cid:20) Pq;t for (cid:22) 2 P (Rd), and Pq;t (cid:20) P for (cid:22) 2 P(Rd). (cid:22) (cid:22) F (cid:22) (cid:22) SELF-AFFINEMULTIFRACTALSIERPINSKISPONGESINRd 151 ii) dimq (cid:20) Dimq (cid:20) (cid:1)q, in particular b (cid:20) B (cid:20) (cid:3) . (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) iii) b is decreasing, and B and (cid:3) are convex and decreasing. (cid:22) (cid:22) (cid:22) Proof. See [Ol1]. Write b (q) b (q) (2:3) a = sup− (cid:22) ; a = inf − (cid:22) : (cid:22) 0 0 and q;t 2 R with 0 < (cid:11)q+t. Then i) Hq;t((cid:1) ((cid:11))) (cid:20) 2tH(cid:11)q+t−(cid:14)((cid:1) ((cid:11))). (cid:22) (cid:22) (cid:22) ii) Pq;t((cid:1) ((cid:11))) (cid:20) 2−(cid:11)q+(cid:14)P(cid:11)q+t−(cid:14)((cid:1) ((cid:11))). (cid:22) (cid:22) (cid:22) Proof. See [Ol1]. 2.2. Multifractal box dimensions. Webeginbyrecallingthede(cid:12)nitionoftheupperandlowerbox-dimension. Let E (cid:18) Rd be a bounded set and N (E) denote the largest number of (cid:14) disjoint balls of radius (cid:14) with centres in E. Then the lower and upper box- dimension of E are de(cid:12)ned as logN (E) logN (E) C(E) = liminf (cid:14) ; C(E) = limsup (cid:14) : (cid:14)&0 −log(cid:14) (cid:14)&0 −log(cid:14) If C(E) = C(E) we refer to the common value as the box-dimension and denote it by C(E). The reader is referred to [Fa2] for more information about box-dimensions. We will now de(cid:12)ne multifractal box-dimensions. Multifrac- tal box dimensions were introduced for example by Falconer [Fa2, p. 225], 152 L.OLSEN Olsen [Ol1, Ol2], Riedi [Ri1] and Strichartz [St]. Here we follow the approach in Olsen [Ol1]. Let (cid:22) 2 P(Rd) and q 2 R. For E (cid:18) Rd and (cid:14) > 0 write ( ) X Sq (E) = sup (cid:22)(B(x ;(cid:14)))q j (B(x ;(cid:14))) is a centered packing of E : (cid:22);(cid:14) i i i2N i Theupperrespectivelylowermultifractalq-boxdimensionCq(E)andCq(E) (cid:22) (cid:22) of E (with respect to the measure (cid:22)) is de(cid:12)ned by logSq (E) logSq (E) Cq(E) = limsup (cid:22);(cid:14) ; Cq(E) = liminf (cid:22);(cid:14) : (cid:22) (cid:14)&0 −log(cid:14) (cid:22) (cid:14)&0 −log(cid:14) If Cq(E) = Cq(E) we refer to the common value as the q-box dimension of (cid:22) (cid:22) E (with respect to the measure (cid:22)) and denote it by Cq(E). Also observe (cid:22) that C0(E) = C(E) and C0(E) = C(E). Now write (cid:22) (cid:22) C (q) = Cq(supp(cid:22)); C (q) = Cq(supp(cid:22)); C (q) = Cq(supp(cid:22)): (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) (cid:22) The next theorem is proven in Olsen [Ol1], Proposition 2.19-Proposition 2.23. Theorem 2.2.1. b (cid:20) C (cid:20) C = (cid:3) for (cid:22) 2 P (X). (cid:22) (cid:22) (cid:22) (cid:22) F 2.3. Generalized R(cid:19)enyi dimensions. Generalised R(cid:19)enyi dimensions were introduced for example by the physicists Hentschel & Procaccia [HP] and Grassberger & Procaccia [GP] in 1983, and later by mathematicians, e.g. Cutler [Cu1], Olsen [Ol1, Ol2], Pesin [Pes1, Pes2] and Strichartz [St]. For (cid:22) 2 P(X) and q 2 R we de(cid:12)ne the upper and lower generalized R(cid:19)enyi q-dimensions of (cid:22) by (cid:16) (cid:17) R log (cid:22)(B(x;r))qd(cid:22)(x) q supp(cid:22) D = limsup for q 6= 0 (cid:22) qlogr r&0 R log(cid:22)(B(x;r))d(cid:22)(x) D0 = limsup supp(cid:22) for q = 0 (cid:22) logr r&0 (cid:16) (cid:17) R log (cid:22)(B(x;r))qd(cid:22)(x) Dq = liminf supp(cid:22) for q 6= 0 (cid:22) r&0 qlogr R log(cid:22)(B(x;r))d(cid:22)(x) D0 = liminf supp(cid:22) for q = 0: (cid:22) r&0 logr

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