uniform distribution theory

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1 Uniform Distribution Theory 4 (2009), no.2, uniform distribution theory UIFORM DISTRIBUTIO O FRACTALS Maria Infusino Aljoša Volčič ABSTRACT. In this paper we introduce a general algorithm to produce u.d. sequences of partitions and of points on fractals generated by an IFS consisting of similarities which have the same ratio and which satisfy the open set condition (OSC). Moreover we provide an estimate for the elementary discrepancy of van der Corput type sequences constructed on this class of fractals. Communicated by Robert F. Tichy Introduction In this paper we will extend to certain fractals the concept of uniformly distributed (u.d.) sequences of partitions introduced by Kakutani in 976 for the interval [0, ] (cfr. [9]). Moreover, we will study in this setting the relation which has been recently established in [5] for [0, ] between u.d. sequences of partitions and the classical theory of u.d. sequences of points, a theory which goes back to Weyl, [6]. The classical concept of u.d. sequences of points is more natural when we deal with the interval [0,] and with manifolds. On the other hand, when we work on fractals, in particular with fractals generated by Iterated Function Systems (IFS), partitions become a convenient tool for introducing a uniform distribution theory. The advantage of considering partitions was implicitely used by Grabner and Tichy in [6] and by Cristea and Tichy in [3] even if they treated u.d. sequences of points. In these papers various concepts of discrepancies were introduced on the planar Sierpiński gasket and on the multidimensional Sierpiński carpet respectively, by using different kinds of partitions on these two fractals. In [6] an analogon of the classical van der Corput sequence has been constructed on the 2000 Mathematics Subject Classification: K06, K38, 28A080. Keywords: Keywords: Discrepancy, fractals, uniform distribution. 47

2 MARIA IFUSIO ALJOŠA VOLČIČ planar Sierpiński gasket. Similarly, in a succesive paper of Cristea, Pillichshammer, Pirsic and Scheicher [2] a sequence of van der Corput type has been defined on the s-dimensional Sierpiński carpet by exploiting the IFS-addresses of the carpet points. In all these papers the authors gave estimates for the elementary discrepancy of van der Corput type sequences, finding that is of the order O ( ). The idea to study this special kind of sequences in relation to uniform distribution on IFS fractals is the starting point for this paper, too. In fact, we will introduce a procedure to define u.d. sequences of partitions and of points on the class of fractals generated by a system of similarities on R d having the same ratio c and verifying the open set condition. Moreover, our estimate for the elementary discrepancy in this wider class of fractals is of the order O ( ), too. Our results include those due to Cristea, Pillichshammer, Pirsic and Scheicher in [2], giving a more trasparent proof and taking in consideration the whole class of fractals described above. The choice of the elementary discrepancy is convenient because the family of elementary sets is obtained in the most natural way by the construction of the fractal and because the elementary sets can be constructed for every IFS fractal regardless of the complexity of its geometric structure. Let us give a brief outline of the paper. In Section we introduce some basic definitions and some preliminaries on uniform distribution in compact Hausdorff spaces. The books [0] and [4] are excellent general references. In Section 2 we recall some notions about IFS fractals and we present the main results of this paper. In fact, we prove in Theorem 2.5 and in Theorem 2.6, that it is possible to construct explicitly u.d. sequences of partitions and of points on selfsimilar fractals generated by similarities with the same ratio and which satisfy the open set condition. This approach also allows to produce a different construction of the known van der Corput sequences on [0,], using an observation which goes back to Mandelbrot. Finally, in Section 3 we provide an estimate for the elementary discrepancy of the sequences of van der Corput type generated by our explicit algorithm and this result is presented in Theorem 3... Preliminaries Let X be a compact Hausdorff space and let us denote by B the σ-algebra of Borel subsets of X. Suppose µ is a regular probability on B. By C(X) we will denote the continuous real valued functions defined on X. 48

3 UIFORM DISTRIBUTIO O FRACTALS Ò Ø ÓÒ.º A sequence {x i } of elements in X is said to be uniformly distributed (u.d.) with respect to µ, if lim f(x i ) = f(t) dµ(t) i= X for all f C(X). Ò Ø ÓÒ.2º A Borel set M X is called a µ-continuity set if µ( M) = 0, where M denotes the boundary of M with respect to the topology on X. Ò Ø ÓÒ {.3º } Let {π n } be a sequence of partitions of X, where π n = A n,an 2,...,An k(n) and the A n i s are µ-continuity sets. The sequence {π n} is said to be µ-uniformly distributed if for any f C(X), and any choice t n i An i we have k(n) lim f(t n i ) = f(t) dµ(t). n k(n) i= The existence of u.d. sequences of partitions in separable metric spaces has been addressed, but not completely solved, in []. However, this will not be an issue in this paper, since known results on fractals will assure us the existence of all the partitions we need. Given a class of real valued integrable functions F, we say that it is determining for the uniform distribution of sequences of points if for any sequence {x i } in X the validity of the relation lim f(x i ) = i= X X f(t) dµ(t), () for all f F already implies that {x i } is u.d. Similarly, we say that F is determining for the uniform{ distribution of sequences of partitions if for any sequence {π n }, where π n = A n,an 2,...,An k(n) }, the validity of the relation k(n) lim f(t n i n k(n) ) = f(t) dt i= for all f F and for any choice t n i An i already implies that {π n } is u.d. Observe the determining classes for the sequences of points play the same role for the sequences of partitions and viceversa. For a family of real valued functions F, we will denote by span(f) the linear space generated by F and by span(f) its closure. The construction of many important determining classes is based on the following theorem. X 49

4 MARIA IFUSIO ALJOŠA VOLČIČ Ì ÓÖ Ñ.4º If F is a class of real valued functions defined on X such that () holds for all f F and span(f) C(X), then F is a determining class. It is also convenient to define determining classes of sets: if G is a class of sets, we say that it is determining if the corresponding class of characteristic functions {χ G : G G} is determining. One more important definition is needed. Ò Ø ÓÒ.5º Let G be a determining class of µ-continuity sets and ω = (x,...,x ) a finite set of points in X. Then the discrepancy with respect to G (or G discrepancy) is defined by D G (ω) = sup A G χ A (x i ) µ(a). i= In particular situations traditional terms are used instead of G-discrepancy. For instance, if X = [0,] and G is the class of all subintervals of [0,], it is simply called discrepancy, while the term star-discrepancy is used if G is reduced to the class of all intervals of the type [0,a[. If {y n } is a sequence of points, we associate to it the sequence of positive real numbers D G (Y ), where Y = {y,y 2,...y }. It follows from the definition that {y n } is u.d. if and only if D G (Y ) tends to zero when tends to infinity. In the next section we will introduce the van der Corput sequences on [0,]. They play an important role because they are the best distributed sequences in the sense that their discrepancy is the smallest possible. In fact, Schmidt proved that in general the order O ( log ) cannot be improved, [3]. 2. Fractals and Van der Corput sequences Fromnow on wewill be concernedwith uniform distribution on a special class of fractals, namely those which are generated by an Iterated Function System (IFS) of similarities having the same ratio c ]0,[ and satisfying the Open Set Condition (OSC). Let us denote by K(R d ) the space of all the non-empty compact subsets of R d endowed with the Hausdorff distance, which makes it complete. Let ψ,...,ψ m be similarities on R d with ψ i (x) ψ i (y) = c x y, for all x,y R d, with 0 < c <. 50

5 UIFORM DISTRIBUTIO O FRACTALS The unique fixed point F of the contraction ψ(e) m i= ψ i(e) is called the attractor of the IFS, [8]. The set F is called a self-similar set and we have m F = ψ i (F). i= Moreover, if F 0 K(R d ) is such that ψ i (F 0 ) F 0 for i m, then the sequence of iterates {ψ n (F 0 )} is decreasing and convergentto F in the Hausdorff metric as n, with F = ψ n (F 0 ) n=0 (where ψ 0 (F 0 ) = F 0 and ψ n+ (F 0 ) = ψ(ψ n (F 0 )) for n 0). The set F 0 = ψ 0 (F 0 ) is called initial set and the iterates ψ n (F 0 ) are called pre-fractals for F. Ò Ø ÓÒ 2. (OSC)º A class of similarities ψ,...,ψ m : R d R d satisfies the open set condition if there exists a non-empty bounded open set V such that m V ψ i (V), where ψ i (V) are pairwise disjoint. i= If OSC holds the Hausdorff dimension of the attractor F is s = logm logc and its s-dimensional Hausdorff measure H s is positive and finite (cfr. [2], [5]). The OSC ensures that the components ψ i (F) of the invariant set F cannot overlap too much. In fact, the following result holds (cfr. [8]). Ì ÓÖ Ñ 2.2º Let ψ,...,ψ m be similarities on R d with ratio 0 < c < and let F be the attractor. If the OSC holds, then H s (ψ i (F) ψ j (F)) = 0 for i j. Our class of fractals includes the most popular kind of fractals as for instance the Cantor set, the Sierpiński triangle, the Sierpiński carpet, the von Koch curve andsoon.butalso[0,]canbeseenastheattractorofanifs,infactofinfinitely many IFS s. Fix a positive integer m and consider the mappings ϕ,...,ϕ m, from R to R, where ϕ k (x) = k m + x for k m. (2) m Then [0,] is the attractor of this IFS. This observation goes back to Mandelbrot (cfr. []). ow, we will see how this observation can be used to define on the kind of fractals we are considering (and also on [0,]) the van der Corput sequences of base m >. 5

6 MARIA IFUSIO ALJOŠA VOLČIČ Let ψ = {ψ,...,ψ m } be our IFS and F its attractor. Assume that F 0 is the initial set such that ψ i (F 0 ) F 0 for i =,...,m. Fix a point x 0 F and apply ψ,...,ψ m in that order to x 0 getting so the points x,...,x m. At the second step, we apply the m mappings first to x, then to x 2 and so on, getting finally m 2 points ordered in a precise manner. ow we keep going, applying the functions of the IFS first to x and continue so until we reach the point x m 2, getting so m 3 points in the order determined by the construction. Iterating this procedure we get a sequence of points {x n } on F which will be called the van der Corput sequence. Observe that if [0,] is seen as the attractor of the IFS described in (2), and if x 0 = 0, this is the classical van der Corput sequence (cfr. [4], [7], [0]). Let us come back to the general situation, showing now how a similar construction produces sequences of uniformly distributed partitions. Observe that if we apply the ψ i s to F, in the same order of before, we construct a sequence {π k } of partitions of F π k = { ψ jk ψ jk ψ j (F) : j,...,j k {,...,m} }. Each of the m k sets E k j of the partition π k contains exactly one point of the van der Corput sequence {x n } constructed above for n = m k. We order the sets E k j accordingly. Let us denote by E k the collection of the m k sets in π k and by E the union of the families E k, for k. The sets of the class E are called elementary sets. Consider on F the normalized s-dimensional Hausdorff measure P, i.e. P(A) = Hs (A) H s (F) for any Borel set A F which is a regular probability. Ä ÑÑ 2.3º The elementary sets are P-continuity sets. Proof. Consider an elementary set E i = ψ i (F) E. Let x E i. By definition, every neighbourhood U of x in the relative topology is such that U E j for some j {,2,...,m} and j i. But each E j is closed, therefore x E j. 52

7 Hence E i is contained in m UIFORM DISTRIBUTIO O FRACTALS j= j i (E i E j ). By Theorem 2.2, we have 0 H s ( E i ) H s m ( ) Ei E j = H s m ( ψi (F) ψ j (F) ) j= j i m H s( ψ i (F) ψ j (F) ) = 0. j= j i ow, a generic elementary set A E k with k 2 is a homothetic image of an elementary set in E and therefore H s ( A) = 0, too. j= j i Ä ÑÑ 2.4º The class consisting of all elementary sets is determining. Proof. Let M be the class consisting of all characteristic functions of sets E E and f C(F). By uniform continuity, for every ε > 0 there exists δ > 0 such that f(x ) f(x ) < ε whenever x x < δ. Chose n such that every Ek n E n has diameter smaller than δ. Take for any Ek n E n a point t k and consider the function m n g(y) = f(t k )χ E n k (y), y F. k= If y E n k, then g(y) f(y) = f(t k) f(y) < ε. Hence, span(m) is uniformly dense in C(F) and the conclusion follows by Theorem.4. Ì ÓÖ Ñ 2.5º The sequence {π n } of partitions of F generated by the algorithm is u.d. with respect to the probability P. Proof. Let us fix E k h π k. By the previous lemma, we have to show that lim n m n m n χ E k h (t n j ) = j= F χ E k h (t) dp(t) for every Ej n π n and for every choice of t n j En j. This is equivalent to lim n m n χ E k h (t n j ) = m k m n j= 53

8 because F MARIA IFUSIO ALJOŠA VOLČIČ χ E k h (t) dp(t) = P(E k h ) = csk P(F) = c sk = m k. ow, observe that for n > k, among the m k sets generated by the algorithm, exactly one set of π n is contained in the fixed set E k h. Since there are mn k sets of π n which are contained in E k h, then lim n m n m n j= χ E k h (t n j) = mn k m n = m k. Ì ÓÖ Ñ 2.6º The sequence {x i } of points of F generated by the algorithm is u.d. with respect to P. Proof. ByLemma2.4,theclass E isdetermining. Hence,forafixedsetE E k, we have to prove that lim Let m t < m t+, then χ E (x i ) = i= = χ E (x i ) = i= E m+m m t ( m ) t m m + ( m i= t m m ( m t m m ) χ E dp = mk. (3) χ E (x i )+ ) i= mt m m m+m m t i= ( m t m m ) χ E (x i ) i= mt m m χ E (x i ) χ E (x i ) (4) since +m+m m t = mt m. Observe that for i > mt m, because of the order of the points x i, among the first m k points exactly one point of the sequence {x i } is contained in the fixed set E. Hence, for t we have 54 ( mt m m ) m+m m t i= χ E (x i ) mk. (5)

9 UIFORM DISTRIBUTIO O FRACTALS Writing as = ( ) m t m m +Mm k +r with 0 r < m k, we have ( ) m t m m i= mt m m χ E (x i ) = By the previous remarks we obtain that Mm k + r i= mt m m Mm k ( m t m m r ( m t m m ) χ E (x i ) = m k, ) r while for and hence for t we have r ( ) χ m t m E (x i ) 0 m i= r+ Mm k r i= mt m m i= r+ χ E (x i ) χ E (x i ). (6) because 0 r < mk m. t Using the last two relations in (6) and taking the limit for (and hence for t ) we have ( m t m m ) i= χ E (x i ) mk. (7) Finally, (4) is a convex combination of two terms which both tend to m k for by (5) and (7) and therefore the conclusion (3) holds. 3. Order of convergence of the elementary discrepancy In this section, F and P are asdefined previously. The next theorem evaluates the elementary discrepancy. Ì ÓÖ Ñ 3.º Let {x i } be the sequence of points generated on F by the algorithm described in the previous section and let. Then the elementary discrepancy of the sequence ω = (x,...,x ) is D E (ω) =. 55

10 MARIA IFUSIO ALJOŠA VOLČIČ Proof. The lower bound is trivial. In fact, if we fix k, then D E k (ω) m k. In order to find an upper bound for D E (ω) let us consider DE k (ω) for any k. Fix k and let E E k. D E k (ω) = sup χ E (x i ). E E k m k Among the first m k points of the sequence {x i } exactly one point is contained in the fixed set E because of the special order induced by the algoritm. Let us distinguish two different cases:. For m k, the set E contains at most one point of ω. Hence χ E (x i ) m k = max{ m k, 0 } m k. i= i= 2. If > m k, we can write as follows = Q m k +r with 0 r < m k and Q. Therefore, every E E k contains either Q points or (r of them) Q + points and hence χ E (x i ) Q m k max{ m k, Q+ } m k. i= ote that Q m k = Qm k m k = r m k < mk m k =, while Q+ m k = Qm k +m k m k = m k r m k = r m k <. So we have that χ E (x i ) m k <. i= Hence, it follows that for any k we have D E k (ω) <. This implies that D E (ω), as we wanted to prove. 56

11 UIFORM DISTRIBUTIO O FRACTALS 4. Conclusions This paper leaves open several questions. The most relevant concerns a unified approach to the discrepancy on a wider class of fractals. This appears to be difficult due to different geometrical features of the various fractals. Another problem concerns the extension of our results to IFS with OSC consisting of similarities which do not have the same ratio, and of course to more general fractals. ÒÓÛÐ Ñ Òغ The authors wish to thank Kenneth Falconer for helpful discussions. REFERECES [] CHERSI, F. VOLČIČ, A.: λ-equidistributed sequences of partitions and a theorem of the de Bruijn-Post type, Ann. Mat. Pura Appl. (IV) 62 (992), [2] CRISTEA, L.L. PILLICHSHAMMER, F. PIRSIC, G. SCHEICHER, K.: Discrepancy estimates for point sets on the s-dimensional Sierpiński carpet, Quaest. Math. 27 (2004), no. 4, [3] CRISTEA, L.L. TICHY, R.F.: Discrepancies of point sequences on the Sierpiński carpet, Math. Slovaca 53 (2003), [4] DRMOTA. M. TICHY, R.F.: Sequences, Discrepancies and Applications, in: Lecture otes in Math. 65, Springer, Berlin, 997. [5] FALCOER, K.J.: The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 985. [6] GRABER, P. TICHY, R.: Equidistribution and Brownian motion on the Sierpiński gasket, Monatsh. Math. 25 (998), [7] HALTO, J.H.: On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals, umer. Math. 2 (960), [8] HUTCHISO, J.E.: Fractals and self-similarity, Indiana Univ. Math. J. 30 (98), [9] S. KAKUTAI: A problem of equidistribution on the unit interval [0,], in:proceedings of Oberwolfach Conference on Measure Theory (975), Lecture otes in Math. 54, Springer, Berlin, 976. pp [0] KUIPERS, L. IEDERSEITER, H.: Uniform Distribution of Sequences, John Wiley & and Sons, ew York, 974. [] MADELBROT, B.B.: The Fractal Geometry of ature, W. H. Freeman and Co., ew York, 982. [2] MORA, P.A.P.: Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc. 42 (946),

12 MARIA IFUSIO ALJOŠA VOLČIČ [3] SCHMIDT, W.M.: Irregularities of distribution.vii, Acta Arithm. 2 (972), [4] VA DER CORPUT, J.G.: Verteilungsfunktionen, Proc. Akad. Wet. Amsterdam 38 (935), [5] VOLČIČ, A.: A generalization of Kakutani s splitting procedure, (submitted). [6] WEYL, H.: Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (96), Received September 7, 2009 Accepted October 22, 2009 Maria Infusino Department of Mathematics University of Calabria IT Arcavacata di Rende (CS) ITALY infusino@mat.unical.it Aljoša Volčič Department of Mathematics University of Calabria IT Arcavacata di Rende (CS) ITALY volcic@mat.unical.it 58