Quantization dimension of probability measures supported on Cantor-like sets

J. Math. Anal. Appl. 338 (2008) 742 750 Note www.elsevier.com/locate/jmaa Quantization dimension of probability measures supported on Cantor-like sets Sanguo Zhu Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China Received 13 October 2006 Available online 22 May 2007 Submitted by M. Laczkovich Abstract Let μ be an arbitrary probability measure supported on a Cantor-like set E with bounded distortion. We establish a relationship between the quantization dimension of μ and its mass distribution on cylinder sets under a hereditary condition. As an application, we determine the quantization dimensions of probability measures supported on E which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied. 2007 Elsevier Inc. All rights reserved. Keywords: Quantization dimension; Cantor-like sets; Bounded distortion; Hereditary condition 1. Introduction The quantization problem consists in studying the L r -error induced by the approximation of a given probability measure with discrete probability measures of finite supports. This problem originated in information theory and some engineering technology. Its history goes back to the 1940s (cf. [1,6,12,13]). Graf and Luschgy studied this problem systematically and gave a general mathematical treatment of it (cf. [3]). Two important objects in the quantization theory are the quantization coefficient and the quantization dimension. Let μ be a Borel probability measure on R d and let 0 <r<.thenth quantization error of μ of order r is defined by { } V n,r (μ) = inf min x a α a r dμ(x): α R d, card(α) n. (1) If the infimum in (1) is attained at some α R d with card(α) n, we call α an n-optimal set of μ of order r. The collection of all the n-optimal sets of order r is denoted by C n,r (μ). The upper and lower quantization dimension of μ of order r are defined by D r (μ) := lim sup r log n log V n,r (μ) ; D r(μ) := lim inf r log n log V n,r (μ). E-mail address: sgzhu2006@163.com. 0022-247X/$ see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2007.05.004

S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 743 If D r (μ), D r (μ) coincide, we call the common value the quantization dimension of μ of order r and denote it by D r (μ). In recent years, the quantization problem for self-similar distributions have been extensively studied. Let {f 1,...,f N } be an iterated function system of contractive similitudes on R d with contraction ratios c 1,...,c N. The corresponding self-similar set refers to the unique non-empty compact set E satisfying E = N i=1 f i (E). The self-similar measure associated with {f 1,...,f N } and a given probability vector (p 1,...,p N ) is the unique Borel probability measure satisfying μ = N i=1 p i μ fi 1. We say that {f 1,...,f N } satisfies the strong separation condition if f i (E), 1 i N, are pairwise disjoint. We say that {f 1,...,f N } satisfies the open set condition if there exists a non-empty open set U such that f i (U) U for all i = 1,...,N and f i (U) f j (U) = for any pair i, j with 1 i j N. Under the open set condition, Graf and Luschgy (cf. [2,4]) proved that the quantization dimension of μ exists and equals s r which is the solution of the following equation: N ( pi ci r ) sr sr +r = 1. i=1 The above result was extended by Lindsay and Mauldin to the F -conformal measures associated with finitely many conformal maps (cf. [10]). Pötzelberger showed that, if the strong separation condition is satisfied and the corresponding vector (log(p 1 c1 r),...,log(p N cn r )) is non-arithmetic, the quantization coefficient of a self-similar measure exists (cf. [11]). By using different methods, Graf and Luschgy extended this result to the cases where only the open set condition is satisfied (cf. [5, Theorem 4.1]). In this paper, we study the quantization dimensions of arbitrary probability measures μ supported on certain Cantorlike sets under a hereditary condition. We will establish a relationship between the quantization dimension of μ and its distribution on cylinders. This generalizes Graf and Luschgy s result on the quantization dimension of self-similar distributions. As an application, we determine the quantization dimension of the product measures supported on the Cantor-like sets provided that a hereditary condition is satisfied. The paper is organized as follows. In the next section, we give some definitions and notations. In Section 3, we define some separation conditions and a hereditary condition and then state our main theorem. In Section 4, we first establish some lemmas, on the basis of these lemmas, we give the proof of our main result. The last section is devoted to the application to product measures. 2. Definitions and notations Let (n k ) be a sequence of integers with n k 2 for all k 1. Let Ξ 0 denote the set containing only the empty word. We define n Ω k := {1, 2,...,n k }, Ξ n := Ω k, Ξ := Ω k, Ξ := Ξ k. k=1 For σ = (σ (1),...,σ(n)) Ξ n, we call the number n the length of σ and denote it by σ. For any σ Ξ Ξ with σ n, we write σ n := ( σ(1),...,σ(n) ). If σ,τ Ξ and σ τ, σ = τ σ, we call σ a predecessor of τ and denote this by σ τ. The empty word is a predecessor of any finite or infinite word. We say σ,τ are incomparable if we have neither σ τ nor τ σ. A finite set Γ Ξ is called a finite anti-chain if any two words σ, τ in Γ are incomparable. A finite anti-chain Γ is called maximal if any word σ Ξ has a predecessor in Γ.Forn 2, σ = (σ (1),..., σ (n)) Ξ n and i Ω n+1, we define σ := σ n 1, σ i = ( σ(1),...,σ(n),i ). Let f kj, 1 j n k,k 1, be contractive similitudes on R d of contraction ratios 0 <c kj < 1. We assume that {f kj } satisfies the open set condition: there exists a bounded non-empty open set V such that for k 1 and 1 j n k we have f kj (V ) V and f ki (V ) f kj (V ) = for 1 i j n k. Set k=1 f σ := f 1σ(1) f nσ (n), c σ := c 1σ(1) c nσ (n), σ Ξ n. k=1

744 S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 Let V denote the closure of V in R d. We call the non-empty compact set E = f σ (V), k 1 σ Ξ k the Cantor-like set determined by {f kj }.Forσ Ξ k, we call the set E σ := f σ (V) a cylinder set of order k; forthe empty word φ, wetakee φ = V. The Hausdorff and packing dimension of this type of sets have been discussed in [7,8]. 3. Statement of the main theorem We will need the following three conditions, the first two of which is on the separation property of the Cantor-like set and the third is on the mass distribution of a given probability measure. (a) Bounded distortion (BD): we say that {f kj } satisfies the bounded distortion property if c kj c>0 for all 1 j n k and all k 1. (b) Extra Strong Separation Condition (ESSC): for k 0 and σ Ξ k, we define Λ(σ ) := {τ Ξ k+1 : σ τ}; we say that {f kj } satisfies the extra strong separation condition if there exists a constant β>0 such that for any σ Ξ Ξ 0,wehave min { dist(e τ,e ρ ): τ,ρ Λ(σ ) } β max { E τ : τ Λ(σ ) }, (2) where A denotes the diameter of a set A and E σ = f σ (V). (c) Hereditary Condition (HC): Let μ be a probability measure supported on the Cantor-like set E; we say that μ satisfies the hereditary condition if (I) for any σ Ξ Ξ 0 with μ(e σ )>0, there exist at least two distinct words τ,ρ Λ(σ ) with μ(e τ ), μ(e ρ )>0; (II) there exists a constant 0 <p<1 which is independent of σ such that for all τ Λ(σ ) with μ(e τ )>0, we have μ(e τ ) pμ(e σ ). Before we state our main theorem, we need to give some more definitions and notations. For σ Ξ, we define := μ(e σ )c r σ. Set l := min { μ(e i )c r 1i : μ(e i)>0 }, where E i = f 1i (V),1 i n 1. For each n 1, we define Γ n := {σ Ξ : h ( σ ) ln } >h(σ). (3) The set Γ n is crucial in the calculation of the quantization dimension. We remark that the definition of Γ n is motivated by Graf and Luschgy s work on the quantization for self-similar distributions (cf. [4]). For each n N, according to the definition of l, thesetγ n is non-empty; and for any σ Γ n,bythe(hc),wehave l/n h ( σ ) = μ(e σ )cσ r μ(e σ ) (1 p) σ 2. This means σ 2 + log(l/n)/ log(1 p) =: K(n) and Γ n is a finite set. Moreover, for each n, Γ n is a finite antichain, but Γ n may fail to be a finite maximal anti-chain since, in general, μ(e σ ) = 0 is possible. Let d n,r, n 1 and s, s be defined by ( ) dn,r dn,r +r = 1; σ Γ n s := lim sup d n,r, s := lim inf d n,r. (4) By considering the continuous function g(t) := σ Γ n () t, one easily see that the numbers d n,r, n 1areall well defined. We are now in the position to state our main result. Theorem 1. Let E be the Cantor-like set determined by {f kj } which satisfies the conditions (BD) and (ESSC). Let μ be a probability measure supported on E satisfying the condition (HC). Then we have D r (μ) = s, D r (μ) = s, where s and s are as defined in (4).

S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 745 4. Main results Throughout this section, f kj,1 j n k, k 1 are contractive similitudes with contraction ratios c kj satisfying the conditions (BD) and (ESSC); E denotes the Cantor-like set determined by {f kj } and μ is an arbitrary probability measure supported on E satisfying the condition (HC). Let [x] denote the largest integer less than or equal to x. We begin with the following simple lemma which is an immediate consequence of the definitions. Lemma 2. (See [14, Lemma 6].) Let l,ζ,ξ > 0.Forφ(n):= [ζ(n/l) ξ ], we have D r (μ) = lim sup log V φ(n),r (μ), Proof. By the definition, it is easy to see that D r (μ) = lim inf r log([n/2]) = lim inf log V [n/2],r (μ) D r(μ) = lim inf Now let (n i ) be an arbitrary subsequence of N. Weset [ 1/ξ ] [ 1/ξ ] ln i ln i W i := ζ 1/ξ, Z i := ζ 1/ξ + 1. Then for large i, wehave n i 2 ξ+1 φ(w i) n i, log V φ(n),r (μ). r log n log V [n/2],r (μ). (5) n i 2 φ(z i) 2 ξ+1 n i. (6) By (5), (6) and the definition of the quantization dimension, one easily gets D r (μ) lim sup log V φ(n),r (μ), The reverse inequalities are clear. D r(μ) lim inf log V φ(n),r (μ). Let Γ n be as defined in (3). For σ Γ n, it could happen that h(σ ) l/n but = 0. This will bring us much inconvenience when considering the quantization dimension. For this reason, we need to pick out those words σ Γ n with positive μ-measure, i.e., for each n 1, we set Γ n := { σ Γ n : μ(e σ )>0 }. Let (A) ɛ denote the ɛ-neighborhood of a set A. Forα C m,r (μ) and, we define α σ := α (E σ ) β Eσ /8. The following lemma will be crucial in the proof of the main theorem. It is a generalization of [14, Lemma 9]. Lemma 3. There exists a constant L 1 such that for any m card( Γ n ), α C m,r (μ) and all we have card(α σ ) L. Proof. Let c>0 be as in the condition (BD). We set C = pc r and let M be a constant with M r > 2/C. Then for, by (HC) and (BD), we have = μ(e σ )cσ r pcr μ(e σ )cσ r = Ch ( σ ). (7) By the condition (ESSC), for any distinct words σ,τ Ξ,wehave (E σ ) β Eσ /4 (E τ ) β Eτ /4 =. By estimating the volumes, we know that there exist two constants L 1,L 2 1 which are independent of σ, τ such that (E σ ) β Eσ /4 can be covered by L 1 closed balls with radii β E σ /(8M) and E τ can be covered by L 2 closed balls with radii β E τ /(8M). Indeed, we may take

746 S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 L 1 := [( 16Mβ 1 + 8M + 1 ) d], L2 := [( 16Mβ 1 + 1 ) d]. Set L := L 1 + L 2. Suppose card(α σ )>Lfor some. Then there exists some τ Γ n such that card(α τ ) = 0 since card(α) card( Γ n ). We assume that b 1,...,b L α σ.letq 1,...,q L1 be the centers of the L 1 closed balls with radii β E σ /(8M) which cover (E σ ) β Eσ /4. Lete 1,...,e L2 be the centers of the L 2 closed balls with radii β E τ /(8M) which cover E τ. Set γ := ( α \{b 1,...,b L } ) {q 1,...,q L1,e 1,...,e L2 }. Then using the condition (HC) we have min x a α a r dμ(x) βr E τ r 8 r μ(e τ ) Cβr V r 8 r h ( τ ) Cβr V r l 8 r. n E τ On the other hand, by the definition of γ,wehave min x c γ c r dμ(x) βr E σ r 8 r M r μ(e σ ) + βr E τ r 8 r M r μ(e τ ) = βr V r ( ) Cβ r V r l + h(τ) < 8 r M r 8 r. n E σ E τ Combining the above inequalities, we have min x a α a r dμ(x) > min x c γ c r dμ(x). (8) E σ E τ E σ E τ For any x E \ (E σ E τ ) and for any a α σ, we denote by B(a,β E σ /8) the closed ball centered at a and of radius β E σ /8. Let y be the intersection of the surface of B(a,β E σ /8) and the line determined by x and a. Then we have y (E σ ) β Eσ /4. Recall that (E σ ) β Eσ /4 is covered by L 1 balls of radii β E σ /(8M) centered at q j,1 j L 1. Hence there exists some point q i such that y B(q i,β E σ /(8M)). By the triangular inequality, we have x q i x y +β E σ /(8M) < x y +β E σ /8 = x a. From the above inequalities, we deduce min x a min x q i. a α σ 1 i L 1 Observing the difference between α and γ, this implies min x a min a α c γ for all x E \ (E σ E τ ). (9) Hence (8) and (9) yields V m,r (μ) = min a α a r dμ(x) > min c γ c r dμ(x) = min c γ c r dμ(x). E σ E σ This contradicts the optimality of α. The lemma follows. Lemma 4. Let L 1 be an integer and α an arbitrary subset of R d with cardinality L. Then there exists a constant D>0 such that for any σ Ξ with μ(e σ )>0, we have min x a α a r dμ(x) D. (10) E σ Proof. We assume that σ Ξ k. Choose j 1 such that 2 j > L. Set Λ j (σ ) := {τ Ξ k+j : σ τ}. By the condition (HC), there exist τ (i) Λ j (σ ), 1 i L + 1 such that

S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 747 μ(e τ (i))>0, 1 i L + 1. Suppose that for some a α there exist 1 i 1 i 2 L + 1 such that dist(a, E τ (i h ))< β 2 min{ E τ,τ Λ j (σ ) }, h= 1, 2. (11) Then by the triangular inequality we have dist(e τ (i 1 ),E τ (i 2 ))<βmin { E τ,τ Λ j (σ ) }. This contradicts the condition (ESSC). Therefore, for each a α there is at most one cylinder set E τ, τ Λ j (σ ) such that (11) holds. On the other hand, we have card(α) < L + 1; thus there exists some E τ (i), 1 i L + 1 such that μ(e τ (i))>0, min dist(a, E a α τ (i)) β 2 min{ E τ,τ Λ j (σ ) }. (12) Using (12) and the conditions (ESSC) and (HC), we deduce E σ min x a α a r dμ(x) E τ (i) min x a α a r dμ(x) μ(e τ (i))2 r β r( min { E τ,τ Λ j (σ ) }) r p j μ(e σ )2 r β r c jr E σ r =: D, where D = 2 r p j β r c jr V r. This completes the proof of the lemma. Proof of Theorem 1. Let S>d r. Then d n,r <Sfor large n. Letc>0 be as in the condition (BD) and let C = pc r. By (3) and (4), we have 1 = ( ) dn,r r+dn,r C dn,r r+dn,r ( ( )) dn,r h σ r+dn,r C dn,r r+dn,r ( ( h σ )) ( ) S r+s S l r+s C card( Γ n ), n Hence we have card( Γ n ) C 1 (n/l) r+s S.Letφ(n):= [C 1 (n/l) r+s S ]. For each, we choose an arbitrary point of E σ and denote by α the set of these points. Note that φ(n) card( Γ n ). We deduce V φ(n),r (μ) E σ r min x a α a r dμ(x) μ(e σ ) E σ r C 1 (n/l) σ Γ n where C r = C 1 V r l r+s. Thus by Lemma 2, we have D r (μ) = lim sup S. log V φ(n),r (μ) S r+s l/n V r =: C r n S+r r, By the arbitrariness of S, wehaved r (μ) d r. One can show D r (μ) d r in a similar manner by considering subsequences. Next we show the reverse inequalities. Let s<d r. Then s<d n,r for large n. By (3) and (4) we have 1 = ( ) dn,r r+dn,r ( ) s ( ) s r+s l r+s card( Γ n ). n Hence card( Γ n ) (n/l) s r+s.letα C [(n/l) s r+s ],r (μ). For each,letw 1,...,w L2 be the centers of the L 2 closed balls with radii β E σ /(8M) which cover E σ and define α σ := α σ {w 1,...,w L2 }. Thus for and all x E σ,wehave min x a min x a. a α a α σ By Lemma 3, card( α σ ) L + L 2 =: L. By Lemma 4 and (7), we deduce

748 S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 V [(n/l) s r+s ],r (μ) = E σ min x a α a r dμ(x) E σ DC(n/l) r+s s l/n =: C1 r n r+s r, where C1 r = DClr/(s+r). Thus by Lemma 2, for φ(n):= [(n/l) r+s s ], wehave D r (μ) = lim inf s. log V φ(n),r (μ) min x a r dμ(x) D a α σ By the arbitrariness of s, we finally get D r (μ) d r. The inequality D r (μ) d r follows similarly by considering subsequences. Remark 5. As we mentioned above, our definition of the crucial set Γ n is motivated by Graf and Luschgy s work for self-similar measures. We note that Graf and Luschgy s method to get the lower bound of the quantization dimension is based on Hölder s inequality with exponent less than one and is dependent on the existence of the quantization dimension. As a result, their method is not applicable here. 5. Application to product measures In this section, we use Theorem 1 to determine the quantization dimension of the product measures supported on a Cantor-like set E which is determined by contractive similitudes f kj s of contraction ratios c kj,1 j n k, k 1. Let p kj,1 j n k, k 1 be positive real numbers satisfying inf min p kj > 0, k 1 1 j n k n k p kj = 1, k 1. (13) j=1 By Kolmogorov consistency theorem, there exists a unique probability measure μ supported on E such that μ(e σ ) = p 1σ(1) p nσ (n), σ = ( σ(1),...,σ(n) ) Ξ n. We call this measure the product measure on E associated with {p kj }. A special product measure is the uniform probability measure μ on E which assigns a weight (n 1 n k ) 1 to each of the cylinder sets of order k, i.e., μ(e σ ) = (n 1 n k ) 1, σ Ξ k. (14) For σ Ξ n,wesetp σ = p 1σ(1) p nσ (n).letd n,r, d r,d r be defined by ( pσ cσ r ) dn,r dn,r +r = 1, σ Γ n d r := lim inf d n,r, where Γ n is as defined in (3). Then we have d r := lim sup d n,r, Theorem 6. Let E be the Cantor-like set determined by {f kj } which satisfies the (BD) and (ESSC). Let μ be the product measure associated with {p kj } satisfying (13). Then D r (μ) = d r, D r (μ) = d r. In particular, if c kj c k for all k 1 and μ is the uniform probability measure on E, then D r (μ) = lim sup k log(n 1 n k ) log(c 1 c k ), D r(μ) = lim inf k log(n 1 n k ) log(c 1 c k ). Proof. Since (13) implies the (HC), the first part of the theorem follows immediately from Theorem 1. In the following we show the remaining part. Let μ be the uniform probability measure on E. We write s k,r := log(n 1 n k ) log(c 1 c k ), ξ := lim sup s k,r, k ξ := lim inf k s k,r. Let l = n 1 1 cr 1. For each n 1, there exists some k 1 such that (n 1 n k+1 ) 1 (c 1 c k+1 ) r <l/n (n 1 n k ) 1 (c 1 c k ) r. (15)

S. Zhu / J. Math. Anal. Appl. 338 (2008) 742 750 749 By (14), for σ Ξ k, = (n 1 n k ) 1 (c 1 c k ) r. Thus by (15), d n,r = s k+1,r for the above k. By Theorem 1, this implies D r (μ) ξ, D r (μ) ξ. On the other hand, for any k 2, there exists some n 1, such that l/(n + 1) (n 1 n k ) 1 (c 1 c k ) r <l/n. For this n,wehave (n 1 n k 1 ) 1 (c 1 c k 1 ) r n k ck r l/(n + 1) l/n. This implies Ξ k = Γ n and s k,r = d n,r for the above n. It follows that D r (μ) ξ,d r (μ) ξ. This completes the proof of the theorem which generalizes [9, Theorem 1.6(2)]. In the following, we give an example to illustrate Theorem 6. Example 7. Let f kj, p kj, j = 1, 2, k 1, be defined as follows: { x6 { if k = 1, x3 + 2 3 if k = 1, f k1 (x) = x f if k>1, k2 (x) = x + 1 1 if k>1, 6 1+2 k 3 1+2 k 3 { { 1+2 k 1 (1 + 2 r ) 1 if k = 1, (1 + 2 r ) 1 if k = 1, p k1 = p k2 = 1 (1 + 2 (1+2 k)r ) 1 if k>1, (1 + 2 (1+2 k)r ) 1 if k>1. Let E be the Cantor-like set on R determined by {f kj } and let μ be the product measure supported on E determined by {p kj }.Wehave D r (μ) = r log 2 r log 3 + log(1 + 2 r =: s. ) log 2 In fact, by our construction, we have p 11 c r 11 = p 12c r 12, p k1c r k1 = p k2c r k2 = ( p 11 c r 11) Ak, k 1, where A k = (1 + 2 r )(1 + 2 (1+2 k )r ) 1 3 2 kr. Thus for each n 1, we have Γ n = Ξ k for some k 1. For any σ Ξ k, we have = μ(e σ )c r σ = p 1σ(1) p kσ(k) c r 1σ(1) cr kσ(k) = ( p 11 c r 11 Let d n,r be the solution of the following equation: ( (1 2 k + 2 r ) ) k 1 dn,r dn,r +r k 3 kr A i = 1. i=1 k 1 ) k Using the fact that A k 1 (k ), we deduce lim n r d n,r = s. By Theorem 6, D r (μ) exists and equals s. Acknowledgment I thank the referee for some helpful comments and nice suggestions. References [1] J.A. Bucklew, G.L. Wise, Multidimensional asymptotic quantization with rth power distortion measures, IEEE Trans. Inform. Theory 28 (1982) 239 247. [2] S. Graf, H. Luschgy, The quantization of the Cantor distribution, Math. Nachr. 183 (1997) 113 133. [3] S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math., vol. 1730, Springer-Verlag, 2000. [4] S. Graf, H. Luschgy, The quantization dimension of self-similar probabilities, Math. Nachr. 241 (2002) 103 109. i=1 A i.

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