Typical Rényi dimensions of measures. The cases: q = 1andq =

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1 J. Math. Anal. Appl Typcal Rény dmensons of measures. The cases: q = 1andq = L. Olsen Department of Mathematcs Unversty of St. Andrews St. Andrews Ffe KY16 9SS Scotland UK Receved 5 May 006 Avalable onlne 7 October 006 Submtted by B.S. Thomson Abstract We study the typcal behavour n the sense of Bare s category of the q-rény dmensons D μ q and D μ q of a probablty measure μ on R d for q [ ]. Prevously we found the q-rény dmensons D μ q and D μ q of a typcal measure for q 0. In ths paper we determne the q-rény dmensons D μ q and D μ q of a typcal measure for q = 1andforq =. In partcular we prove that a typcal measure μ s as rregular as possble: for q = the lower Rény dmenson D μ q attans the smallest possble value and for q = 1andq = the upper Rény dmenson D μ q attans the largest possble value. 006 Elsever Inc. All rghts reserved. Keywords: Multfractals; Rény dmensons; Bare category; Resdual set 1. Statement of results For a Borel probablty measure μ on R d and q [ ] we defne the lower and upper q-rény dmensons of μ by D μ q = lm nf D μ q = lm sup 1 log supp μ μbx rq 1 dμx q 1 1 log supp μ μbx rq 1 dμx q 1 for q R \{1} for q R \{1} E-mal address: lo@st-and.ac.uk X/$ see front matter 006 Elsever Inc. All rghts reserved. do: /j.jmaa

2 146 L. Olsen / J. Math. Anal. Appl and D μ 1 = lm nf D μ 1 = lm sup D μ = lm nf D μ = lm sup supp μ supp μ log μbx r dμx log μbx r dμx log nf x supp μ μbx r log nf x supp μ μbx r log sup x supp μ μbx r D μ = lm nf log sup x supp μ μbx r D μ = lm sup. The Rény dmensons were essentally ntroduced by Rény [910] n 1960 as a tool for analyzng varous problems n nformaton theory. Indeed for a probablty vector p = p 1...p n and q R Rény defned the q-entropy H p q of p by H p q = 1 q 1 log pq for q 1 and H p 1 = p log p. Observe that l Hosptal s rule shows that H p q H p 1 as q 1 and the q-entropes H p q can therefore be regarded as natural generalzatons of the usual entropy H p 1 = p log p of p. The entropes H p q are dscussed n detal by Rény n [11 Chapter 9]. The man sgnfcance of the Rény dmensons s ther relatonshp wth the multfractal spectrum of μ. For a probablty measure μ on R d the local dmenson μ at the pont x s defned by log μbx r dm loc x; μ = lm. 1.1 We defne the Hausdorff multfractal spectrum functon f μ ofμ as the Hausdorff dmenson of the level sets of the local dmenson of μ.e. we put { } f μ α = dm x R d log μbx r lm = α α 0 1. where dm denotes the Hausdorff dmenson. Next recall that the Legendre transform ϕ of a functon ϕ : R R s defned by ϕ x = nf y xy + ϕy. In the 1980s t was conjectured n the physcs lterature [45] that for good measures the followng result relatng the multfractal spectrum functon f μ to the Legendre transform of the Rény dmensons holds: namely 1 that the upper and lower Rény dmensons concde.e. D μ q = D μ q for all q R and that the multfractal spectrum functon f μ concdes wth the Legendre transform of the functon τ μ : R R defne by τ μ q = 1 qd μ q = 1 qd μ q.e. { } dm x R d log μbx r lm = α = τμ α

3 L. Olsen / J. Math. Anal. Appl for all α 0. Ths result s known as the Multfractal Formalsm. Durng the 1990 s there has been an enormous nterest n verfyng the Multfractal Formalsm and computng the multfractal spectra of measures n the mathematcal lterature and wthn the last 8 or 9 years the multfractal spectra of varous classes of measures n Eucldean space R d exhbtng some degree of selfsmlarty have been computed rgorously cf. [1] and the references theren. In partcular t has been proved that many nce measures ncludng determnstc and stochastc self-smlar measures and certan classes of nvarant measures of dynamcal systems satsfy the Multfractal Formalsm cf. [18]. In ths paper we study the Rény dmensons of a typcal measure n the sense of Bare. One of the consequences of our man results s that a typcal measure fals to satsfy part 1.3 of the Multfractal Formalsm n a very spectacular way. For a compact subset K of R d we denote the famly of Borel probablty measures on K by PK and we equp PK wth the weak topology. We wll say that a typcal probablty measure on K has property P f the set of probablty measures that do not have property P.e.ftheset { μ PK μ does not have property P } s of the frst category wth respect to the weak topology on PK. In [6] we found the q-rény dmensons of a typcal measure for q 0 and the purpose of ths paper s to complement ths result by determnng the q-rény dmensons of a typcal measure for q = 1 and q =. However before we state ths result t s nstructve to recall the result from [6] gvng the q-rény dmensons of a typcal measure for q 0. To state ths result we begn wth a few defntons. For a subset E of R d we denote the lower box dmenson of E and the upper box dmenson of E by dm B E and dm B E respectvely; the reader s referred to [1] for the defntons of the box dmensons. Also for a subset K of R d and x K we defne the lower local box dmenson of K at x and the upper local box dmenson of K at x by dm Bloc x K = lm dm B K Bxr and dm Bloc x K = lm dm B K Bxr respectvely. We can now state the result from [6] gvng the q-rény dmensons of a typcal measure for q 0. Theorem A. [6] Let K be a compact subset of R d. Wrte s = nf dm Blocx K x K s = nf dm Blocx K x K s = dm B K. Observe that s s s. Assume that s = s = s ths condton s clearly satsfed f for example K s the closure of an open and bounded set or f K s a self-smlar set satsfyng the open set condton. 1 For all measures μ PK we have 0 D μ q D μ q s for all q 1.

4 148 L. Olsen / J. Math. Anal. Appl A typcal measure μ PK satsfes the followng D μ q = 0 D μ q = s for all q 1. The purpose of the paper s to show that ths result extends to the followng two lmtng cases namely for q = 1 and for q =. However we frst gve a general result provdng the relatonshp between the q-rény dmensons for dfferent values of q [0 ]. Ths result wll be useful later. Proposton 1. Let K be a compact subset of R d and wrte s = dm B K. For all measures μ PK and all q 1 we have and 0 D μ D μ q D μ 1 s 0 D μ D μ q D μ 1 s. Proof. It s clear that 0 D μ D μ q and 0 D μ D μ q and t follows easly from Jensens s nequalty that D μ q D μ 1 and D μ q D μ 1. Hence t suffces to show that D μ 1 s. For a postve real number r>0 let N r K denote the smallest number of balls of radus equal to r that s needed to cover the set K. Then s = dm B K = lm sup log N r K cf. [1]. Fx r>0. For brevty wrte N = N r E. We can thus choose balls Bx 1 r...bx N r such that K Bx r. Put K 1 = Bx 1 r and K = Bx r \ 1 j=1 Bx j r for =...N. Next observe that f x K then K Bxr. We conclude from ths and Jensen s nequalty appled to the functon Φ : 0 R defned by Φt = t log t that μ Bxr dμx = K K K K log μ Bxr dμx log μk Kdμx = μk Klog μk K = N 1 N Φ μk K 1 NΦ N μk K 1 = NΦ N = log N. 1.5 The desred concluson now follows from 1.5 by dvdng by.

5 L. Olsen / J. Math. Anal. Appl We wll now state the man results n the paper extendng the results n Theorem A to the cases q = 1 and q =. Theorem The case: q =. Let K be a compact subset of R d. Let s s and s be defned as n Theorem A. 1 For all measures μ PK we have 0 D μ D μ s. A typcal measure μ PK satsfes the followng D μ = 0 s D μ s. In partcular f s = s = s ths condton s clearly satsfed f for example K s the closure of an open and bounded set or f K s a self-smlar set satsfyng the open set condton then a typcal measure μ PK satsfes the followng D μ = 0 D μ = s. Theorem 3 The case: q = 1. Let K be a compact subset of R d. Let s s and s be defned as n Theorem A. 1 For all measures μ PK we have 0 D μ 1 D μ 1 s. A typcal measure μ PK satsfes the followng s D μ 1 s. In partcular f s = s = s ths condton s clearly satsfed f for example K s the closure of an open and bounded set or f K s a self-smlar set satsfyng the open set condton then a typcal measure μ PK satsfes the followng D μ 1 = s. Observe that part 1 of Theorem follows mmedately from Proposton 1 and that Theorem 3 follows mmedately from Proposton 1 and Theorem. Part of Theorem s proved n Secton 3. Note that the second half of part of Theorem A follows mmedately from Proposton 1 and Theorem namely snce D μ D μ q s for all μ PK and all q 1 we conclude from Theorem that f s = s = s then a typcal measure μ PK satsfes the followng D μ q = s for all q 1. Ths provdes an alternatve proof of the second half of part of Theorem A. Comparng the statements n part 1 and part of Theorem A Theorem and Theorem 3 we see that a typcal measure μ s as rregular as possble: for all q 1 ] thelowerq-rény

6 1430 L. Olsen / J. Math. Anal. Appl dmenson D μ q attans the smallest possble value and for all q [1 ] the upper q-rény dmenson D μ q attans the largest possble value. In partcular Theorems A and 3 show that a typcal measure fals to satsfy part 1.3 of the Multfractal Formalsm n a very spectacular way. The typcal behavour of varous other quanttes related to multfractal analyss has also been studed. In partcular the local dmenson dm loc x; μ of a typcal measure has been studed by Haase [3] and nvestgated further by Genyuk [].. Proof of part of Theorem Wrte Γ = { μ PK D μ = 0 } Δ u = { μ PK s D μ } Δ l = { μ PK Dμ s }..1 We must prove that the three sets Γ Δ u and Δ l are resdual. In Secton.1 we prove that the set Γ s resdual n Secton. we prove that the set Δ u s resdual and fnally n Secton.3 we prove that the set Δ l s resdual. It s well known cf. for example [7 p. 51 Theorem 6.8] that the weak topology on PK s nduced by the metrc L on PK defned as follows. Let LpK denote the famly of Lpschtz functons f : K R wth f 1 and Lpf 1 where Lpf denotes the Lpschtz constant of f. The metrc L s now defned by Lμ ν = sup fdμ fdν f LpK for μ ν PK. We wll always equp PK wth the metrc L and all balls n PK wll be wth respect to the metrc L.e.fμ PK and r>0 we wll wrte Bμr ={ν PK Lμ ν < r} for the ball wth centre at μ and radus equal to r. Forx K and r>0 defne f xr : K R by f xr t = { r f x t r t x +r f r< x t < r 0 f r x t. Observe that f r 1 then f xr s Lpschtz wth f xr 1 and Lpf xr = 1. In partcular ths mples that f r 1 then f xr dμ f xr dν Lμ ν.3 for all μ ν PK. Ths nequalty wll be used frequently n Sectons.1.3. Fnally for a probablty measure μ and r>0 wrte I μ ; r = sup x supp μ. μ Bxr..4

7 L. Olsen / J. Math. Anal. Appl The set Γ s resdual In ths secton we prove that the set Γ s resdual. It clearly suffces to construct a set M PK satsfyng the followng three condtons: 1 M Γ ; M s dense n PK; 3 M s G δ. For a postve nteger wrte { Λ n = λ PK λ {x 0 } 1 } n for some x 0 K. Next put G n = 1 B λ 9n n+1 λ Λ n and defne the set M PK by M = G n. m n m Below we show that the set M has the followng three propertes: 1 M Γ M s dense n PK and 3 M s G δ.thesetm s clearly G δ and t thus suffces to show that M Γ and that M s dense n PK. Ths s done n Proposton.1.1 and Lemma.1.. Proposton.1.1. We have M Γ. Proof. Let μ M and fx a postve nteger m. Snce μ M there exsts n m and a measure λ Λ n such that Lμ λ 1. Also snce λ Λ 9n n+1 n we can fnd a pont x 0 K wth λ{x 0 } n 1. For brevty wrte r n = n 1 n. Now observe that for all x Bx 0 r n 3 we have usng.3 μ Bxr n = 1 Bxrn dμ and usng.3 once more fx0 rn 3 r n3 dμ 3 Lλ μ + r n f x0 rn 3 dλ 3 1 r n 9n n+1 + f x 0 rn x 0λ {x 3 0 } 3 1 r n 9n n+1 + r n 3 1 = n 3n.5

8 143 L. Olsen / J. Math. Anal. Appl μ B x 0 r n = 1 3 Bx0 rn 3 dμ fx0 rn 6 r n6 dμ 6 Lλ μ + f r x0 rn dλ n r n 9n n+1 + f x 0 rn x 0λ {x 6 0 } 6 1 r n 9n n+1 + r n 6 1 n = 6 7n. In partcular ths mples that μbx 0 r n 3 > 0 and we therefore conclude that there exsts y n Bx 0 r n 3 supp μ. Snce y n Bx 0 r n 3 t follows from.5 that μby n r n 3n. Hence I μ ; r n = Ths mples that sup x supp μ D μ = lm nf lm nf n lm nf n μ Bxr n μ By n r n 3n. log I μ ; r log I μ ; r n n log 3n n log n = 0. Ths completes the proof of Proposton.1.1. Lemma.1.. The set M s dense n PK. Proof. Snce PK s a complete metrc space because K s compact and each set n m G n s open t suffces by Bare s Theorem to show that n m G n s dense for all m. In order to show that n m G n s dense t suffces to show that the subset n m Λ n s dense for all m. Therefore fx a postve nteger m. Letμ PK and 0 <ε 1. Pck any x 0 K. Next choose a postve nteger n 0 m wth n 1 0 ε and put λ = ε δ x ε μ. Then λ{x 0} ε n 1 0 whence λ Λ n0 n m Λ n.alsolμ λ = sup f LpK fdμ ε fdλ =sup f LpK fdμ fx 0 sup f LpK ε = ε. Ths shows that n m Λ n s dense for all m... The set Δ u s resdual In ths secton we prove that the set Δ u s resdual. For a real number t wrte Δ u t = { μ PK t Dμ }. Snce Δ u = Δ u t t Q t<s

9 L. Olsen / J. Math. Anal. Appl t clearly suffces to prove that the set Δ u t s resdual for each ratonal number t wth t<s. Therefore fx a ratonal number t wth t<s. To prove that the set Δ u t s resdual t clearly suffces to construct a set M u PK satsfyng the followng three condtons: 1 M u Δ u t ; M u s dense n PK; 3 M u s G δ. Lemma..1. Assume that x 0 K r 0 > 0 and t 0 satsfy t<dm B K Bx0 r 0. Then there exsts c>0 such that for each r>0 there exsts a measure μ PK wth 1 supp μ K Bx 0 r 0 ; for all x K we have μbx r cr t. Proof. For r>0 let M r K Bx 0 r 0 denote the largest number of parwse dsjont balls of log M radus r wth centres n K Bx 0 r 0. Then dm B K Bx 0 r 0 = lm nf r K Bx 0 r 0 cf. [1]. We can thus fnd 0 <δ 1 such that log M r K Bx 0 r 0 >t for all 0 <r δ whence M r K Bx0 r 0 >r t.6 for all 0 <r δ. Now put c = 1 δ t 1. We must prove that for each r>0there exsts a measure μ PK satsfyng condtons 1 and. Therefore fx r>0. We dvde the proof nto two cases. Case 1: δ<r.pckanyμ PK wth supp μ K Bx 0 r 0. For example we may put μ = δ x0. For all x K we clearly have μbx r 1 = cδ t <cr t. Case : 0 <r δ. For brevty wrte M = M r K Bx 0 r 0. By defnton of M there exst M parwse dsjont balls Bx 1 r...bx M rwth centres x 1...x M n K Bx 0 r 0.Now put μ = M 1 M=1 δ x. Then clearly supp μ K Bx 0 r 0. Next let x K and observe that the ball Bxr can at most contan one of the x s. Indeed otherwse there exst two dstnct ndces and j such that x x j Bxr whence x Bx r Bx j r contradctng the fact that the balls Bx 1 r...bx M r are parwse dsjont. Snce r δ and the ball Bxr contans at most one of the x s we conclude from.6 that μ Bxr 1 M = 1 M r K Bx 0 r 0 <rt cr t. Ths completes the proof of Lemma..1. Let x n n be a dense sequence n K. Fxn and = 1...n. Snce t<s= nf dm Blocx K dm B K B x 1 x K n t follows from Lemma..1 that there exsts a constant c n such that for all r>0 there exsts a measure μ PK wth

10 1434 L. Olsen / J. Math. Anal. Appl supp μ K Bx 1 n ; for all x K we have μbx r c n r t. Now put c n = max t c n1... t c nn n and r n = 1 e cn. We can thus choose a measure μ n PK wth 1 supp μ n K Bx 1 n ; for all x K we have μ n Bx r n c n r n t. For a postve nteger n wrte Λ u n = { n =1 p μ n p 0 Next put G u n = λ Λ u n B λrn t+1 } n p = 1. =1 and defne the set M u PK by M u = G u n. m n m Below we show that the set M u has the followng three propertes: 1 M u Δ u t Mu s dense n PK and 3 M u s G δ.thesetm u s clearly G δ and t thus suffces to show that M u Δ u t and that M u s dense n PK. Ths s done n Proposton.. and Lemma..4. Proposton... We have M u Δ u t. Proof. Let μ M u and fx a postve nteger m. Snce μ M u there exsts n m and a measure λ Λ u n such that Lμ λ rt+1 n. Also snce λ Λ u n we can fnd p 1...p n wth p 0 and λ = p μ n. Now observe that for all x K we have usng.3 μ Bxr n = 1 Bxrn dμ fxrn dμ r n 1 Lμ λ + r n f xrn dλ 1 r n Lμ λ + rn λ Bxr n 1 r n 1 r n 1 r n rn t+1 rn t+1 rn t+1 + r n p μ n Bxrn + r n p c n r n t + r n p c n rn t = 1 + c n r t n.

11 L. Olsen / J. Math. Anal. Appl Ths mples that Hence I μ ; r n = sup x supp μ D μ = lm sup lm sup n lm sup n = t. μ Bxr n 1 + c n r t n. log I μ ; r log I μ ; r n n log1 + c n + t n n Ths completes the proof of Proposton... Lemma..3. Let F R d be a bounded Borel set and r>0. Then there exsts fntely many parwse dsjont Borel sets F 1...F N wth dam F j r such that F j F j and such that for each j there exsts an x j F satsfyng B x j r 4 F j. Proof. Frst construct a sequence of balls Bx 1 r Bx r... such that x F and x x j > r for all j. Because F s totally bounded ths process must termnate at some fnte stage gvng balls Bx 1 r Bx r...bx N r such that any x F must satsfy mn j x x j r and consequently F N j=1 Bx j r. Note that the smaller balls Bx 1 4 r Bx 4 r...bx N 4 r are parwse dsjont. Now set F 1 = B x 1 r N B x r 4 = F j = B x j r j 1 N F =1 F N = B x N r N 1 F. =1 =j+1 B x 4 r for j =...N 1 It s clear that the sets F 1 F...F N are parwse dsjont and snce Bx 1 4 r Bx 4 r... Bx N 4 r are parwse dsjont we conclude that Bx j 4 r F j and F j F j. Lemma..4. The set M u s dense n PK. Proof. Snce PK s a complete metrc space because K s compact and each set k m Gu k s open t suffces by Bare s Theorem to show that k m Gu k s dense for all m. In order to show that k m Gu k s dense t suffces to show that the subset k m Λu k s dense for all m. Therefore fx a postve nteger m. Letμ PK and 0 <ε 1. Accordng to Lemma..3 we

12 1436 L. Olsen / J. Math. Anal. Appl may choose fntely many parwse dsjont Borel sets K 1...K N wth dam K j ε such that K j K j and such that for each j there exsts an y j K satsfyng B y j ε K j. 4 Snce the sequence x k k s dense n K we can also choose a postve nteger n m such that 1 n 8 ε and {x 1...x n } By j 8 ε for all j. Hence for each j = 1...N we can pck a not necessarly unque j wth x j B y j ε. 8 Now put { μk Kj f = j for some j = 1...N; p = 0 f j for all j = 1...N. Fnally wrte λ = p μ n. We wll now show that λ k m Λu k and that Lμ λ ε. Indeed we clearly have that λ Λ u n k m Λu k. Next we prove that Lμ λ ε. Wehave Lμ λ = sup fdμ fdλ f LpK sup fdμ fdλ..7 f LpK j K K j K K j Frst observe that f f : K R s a real valued functon wth Lpf 1 and f 1 then μk K j nf fx fdμ μk K j sup fx..8 x K K j x K K j K K j Next observe that snce supp μ nj K Bx j n 1 K Bx j 8 ε K By j 4 ε K K j and the sets K 1...K N are parwse dsjont we have fdλ= p j fdμ nj. K K j K K j It follows from ths that fdλ p j μ nj K K j sup fx= μk K j sup fx.9 x K K j x K K j K K j and that fdλ p j μ nj K K j nf fx= μk K j nf fx..10 x K K j x K K j K K j

13 L. Olsen / J. Math. Anal. Appl Fnally combnng.8.10 show that fdμ fdλ μk K j sup fx nf fx x K K j x K K j K K j K K j It now follows from.7 and.11 that Lμ λ sup μk K j damk K j f LpK ε j j μk K j μk K j damk K j..11 = εμ K j K j = ε. Ths completes the proof..3. The set Λ l s resdual In ths secton we prove that the set Δ l s resdual. For a real number t wrte Δ l t = { μ PK D μ s }. Snce Δ l = Δ l t t Q s<t t clearly suffces to prove that the set Δ l t s resdual for each ratonal number t wth s<t. Therefore fx a ratonal number t wth s<t. To prove that the set Δ l t s resdual t clearly suffces to construct a set M l PK satsfyng the followng three condtons: 1 M l Δ l t ; M l s dense n PK; 3 M l s G δ. Put { Λ l = λ PK there exsts x 0 K and r 0 > 0 such that dm B K Bx0 r 0 <t and λ B x 0 r } 0 > 0. Hence for λ Λ l there exst x 0 K and r 0 > 0 such that dm B K Bx 0 r 0 < t and λbx 0 r 0 > 0; we now wrte r λ = r 0 4 λbx 0 r 0. Put M l = λ Λ l Bλr λ.

14 1438 L. Olsen / J. Math. Anal. Appl Below we show that the set M l has the followng three propertes: 1 M l Δ l t Ml s dense n PK and 3 M l s G δ.thesetm l s clearly G δ and t thus suffces to show that M l Δ l t and that M l s dense n PK. Ths s done n Propostons.3. and.3.3. Lemma.3.1. Let μ PK and E K wth μe > 0. Then D μ dm B E. Proof. For a postve real number r>0 let N r E denote the smallest number of balls of radus log N equal to r that s needed to cover the set E. Then dm B E = lm sup r E cf. [1]. We wll now show that 1 I μ ; r μe.1 N r E for all r>0. Therefore fx r>0. For brevty wrte N = N r E. We can thus choose balls Bx 1 r...bx N r such that E Bx r. Put E 1 = Bx 1 r and E = Bx r \ 1 j=1 Bx j r for =...N. Next observe that f x E then E Bxr. We conclude from ths that I μ ; r = sup μ Bxr x supp μ = max sup μ Bxr x supp μ E E = max sup μe E x supp μ E E = max μe E 1 μe E N = 1 N μ E E = 1 N μe. Ths completes the proof of.1. Snce μe > 0 the desred concluson now follows from.1 by takng logarthms and dvdng by. Proposton.3.. We have M l Δ l t. Proof. Let μ M l. We can thus choose λ Λ l such that Lμ λ r λ where r λ = r 0 4 λbx 0 r 0 for some x 0 K and r 0 > 0 wth dm B K Bx 0 r 0 < t and λbx 0 r 0 > 0. It now follows that usng.3 μ K Bx 0 r 0 = 1 Bx0 r 0 dμ fx0 r 0 r 0 dμ

15 L. Olsen / J. Math. Anal. Appl Lλ μ + r 0 r λ + r 0 r 0 r λ + r 0 = 1 λ B Bx 0 r 0 x 0 r 0 λ B. f x0 r 0 dλ f x0 r 0 dλ x 0 r 0 Ths shows that μk Bx 0 r 0 > 0 and we therefore nfer from Lemma.3.1 that D μ dm B K Bx 0 r 0 < t. Proposton.3.3. The set M l s dense n PK. Proof. Let μ PK and 0 <ε<1. Snce s<t there exst x 0 K and r 0 > 0 such that dm B K Bx 0 r 0 < t. Now put λ = ε δ x ε μ. Snce λbx 0 r 0 ε > 0 we conclude that λ Λ l M l.also Lμ λ = sup fdμ fdλ = sup ε fdμ f0 sup ε = ε. f LpK f LpK f LpK Ths shows that M l s dense n PK. References [1] K.J. Falconer Technques n Fractal Geometry John Wley & Sons Chchester [] J. Genyuk A typcal measure typcally has no local dmenson Real Anal. Exchange / [3] H. Haase A survey on the dmenson of measures n: Topology Measures and Fractals Warnemünde 1991 n: Math. Res. vol. 66 Akademe-Verlag Berln 199 pp [4] T.C. Halsey M.H. Jensen L.P. Kadanoff I. Procacca B.J. Shraman Fractal measures and ther sngulartes: The characterzaton of strange sets Phys. Rev. A [5] H. Hentschel I. Procacca The nfnte number of generalzed dmensons of fractals and strange attractors Phys. D [6] L. Olsen Typcal L q -dmensons of measures Monatsh. Math [7] K.R. Parthasarathy Probablty Measures on Metrc Spaces Academc Press New York [8] Y. Pesn Dmenson Theory n Dynamcal Systems. Contemporary Vews and Applcatons The Unversty of Chcago Press [9] A. Rény Some fundamental questons of nformaton theory Magyar Tud. Akad. Mat. Fz. Oszt. Közl [10] A. Rény On measures of entropy and nformaton n: Proceedngs 4th Berkeley Symposum on Mathematcal Statstcs and Probablty 1960 Unv. of Calforna Press Berkeley 1961 pp [11] A. Rény Probablty Theory North-Holland Amsterdam 1970.