On the Dimension of Self-Affine Fractals

Transcription

1 On the Dimension of Self-Affine Fractals Ibrahim Kirat and Ilker Kocyigit Abstract An n n matrix M is called expanding if all its eigenvalues have moduli >1. Let A be a nonempty finite set of vectors in the n-dimensional Euclidean space. Then there exists a unique nonempty compact set F satisfying MF D F C A. F is called a self-affine set or a self-affine fractal. F can also be considered as the attractor of an affine iterated function system. Although such sets are basic structures in the theory of fractals, there are still many problems on them to be studied. Among those problems, the calculation or the estimation of fractal dimensions of F is of considerable interest. In this work, we discuss some problems about the singular value dimension of self-affine sets. We then generalize the singular value dimension to certain graph directed sets and give a result on the computation of it. On the other hand, for a very few classes of self-affine fractals, the Hausdorff dimension and the singular value dimension are known to be different. Such fractals are called exceptional self-affine fractals. Finally, we present a new class of exceptional self-affine fractals and show that the generalized singular value dimension of F in that class is the same as the box (counting) dimension. 1 Introduction Let S 1 ;:::;S q, q>1, be contractions on R n, i.e., js j.x/ S j.y/j c j jx yj for all x;y 2 R n with 0<c j <1. It is well known [3] that there exists a unique nonempty compact set F R n such that I. Kirat ( ) Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey ibkst@yahoo.com I. Kocyigit Department of Mathematics, University of Washington, Seattle, WA , USA ilkerk@gmail.com S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI / , Springer-Verlag Berlin Heidelberg

2 152 I. Kirat and I. Kocyigit F D q[ S j.f /: Let M n.r/ denote the set of nn matrices with real entries. We will assume that S j.x/ D T j.x C a j /; x 2 R n ; where a j 2 R n, called digits, andt j 2 M n.r/ are contractions. In that case, F is called a self-affine set or a self-affine attractor, and can be viewed as the invariant set or the attractor of the (affine) iterated function system (IFS) fs j.x/g (in the terminology of dynamical systems). Further, if A WD fa 1 ;:::;a q gz n and T 1 D ::: D T q D T; where T 1 2 M n.z/, it is called an integral self-affine set.here M n.z/ is the set of n n integer matrices. In addition to this, if j det.t 1 /jdq, and F WD F.T;A/ has positive Lebesgue measure, then F is called an integral self-affine tile [11]. For an invertible matrix T 2 M n.r/, the singular values i.1 i n/ of T are the positive square roots of the eigenvalues of T T,whereT is the adjoint of T. Assume that 0< n 2 1 <1.Thesingular value function s.t / is defined for 0<s n as s.t / D 1 2 ::: m 1 s mc1 m ; (1) where m is the integer such that m 1 < s m. Fors > n,itisdefinedas j det.t /j n s. Let J k Df.j 1 ;j 2 ;:::;j k / W 1 j i qg and let j D.j 1 ; ;j k / denote a multi-index in J k so that jjjdk is the length of j. ByT j we mean the matrix product T j1 T jk. Then there exists a unique positive number denoted by d or d.t 1 ;:::;T q / (if we wish to emphasize the dependence on T 1 ;:::;T q ) and defined by lim ŒX d.t j / k 1 D 1: k!1 j2j k d is sometimes called the singular value dimension of F. The main result of [2] is dim H F dim B F dim B F d (2) where dim H, dim B F, dim B F denote the Hausdorff, lower and upper box dimensions respectively. When dim B F D dim B F, this common value is called the box dimension of F and denoted by dim B F [3]. Solomyak [16] showed that equality holds throughout (2) for almost all.a 1 ;a 2 ;:::;a q / 2 R nq provided that max j kt j k < 1 2 ; but this almost sure result doesn t extend to the case max j kt j k < 1 C for any >0. The difficulty with this approach is that it is not 2 easy to determine if a given family of affine transformations fs j.x/g is exceptional (i.e., dim H F<d) as far as this estimate is concerned. When dim H F<d,wesimply say F is exceptional.

3 On the Dimension of Self-Affine Fractals 153 For an exceptional case, besides the upper bound d, it is desirable to give a lower bound too. We say that the IFS fs j g q satisfies the open set condition (OSC) if there exists a bounded non-empty open set U such that [ q S j.u / U with the union disjoint. Concerning the lower bound, in [4], Falconer showed that if OSC is satisfied and F is totally disconnected, then d dim H F d; (3) where d WD d.t 1 ;:::;T q / is the unique positive number such that lim ŒX. d.tj 1 // 1 k 1 D 1: k!1 j2j k There are some limitations to the use of the bounds in (3): 1. Falconer [4] mentioned that it is not possible to replace the hypothesis F is totally disconnected for the lower bound by OSC is satisfied. 2. It is stated in [2] that the singular value dimension d is not easy to calculate. Although, there has been some progress in certain cases [5], the general case remains as a difficult problem and there is still need for further study. 3. Even if we can compute d or d, wemayhaved n, which doesn t give any information on dim H F. In this work, we want to remedy the listed deficiencies for integral self-affine sets. Other related work can be found in [1, 4, 6 8, 14 16]. Very few classes of self-affine sets are known to be exceptional. For example, McMullen-type fractals give such a class [13]. It is the motivation for our study to give new examples of exceptional cases. The purpose of this note is to report some of our recent results in that direction. The details and other examples are in [10]. 2 The Graph Directed Representation The dimension of graph-directed constructions was first studied by Mauldin and Williams [12] foraspecialclass offractals. Togivea lowerboundforthedimension of an arbitrary integral self-affine set, we need to consider graph-directed sets. We present the necessary material now. Let V be a set of vertices which we label f1;2;:::;mg, and E be a set of directed edges with each edge starting and ending at a vertex so that.v; E/ is a directed graph. We write E i;j for the set of edges from vertex i to vertex j,ande r i;j for the set of sequences of r edges.e 1;e 2 ;:::;e r / which form a directed path from vertex i to vertex j. We assume that S e.x/ D T.xC e/ for e 2 E i;j Z n,wheret 1 2 M n.z/ is expanding or expansive (i.e. all eigenvalues have moduli >1). It is well known that there exists a unique family of nonempty compact sets F 1 ;:::;F m R n such that

4 154 I. Kirat and I. Kocyigit F i D m[ [ e2e i;j S e.f j / D m[ T.F j C E i;j / Note that each S e is a contraction in some norm [11]. The set fs e W e 2 Eg is called a graph directed iterated function system and the sets F 1 ;:::;F m form a family of graph directed sets. Surely,m here represents something different form the one in (1). Hence there shouldn t be any confusion about that. For any integral self-affine set F D F.T;A/ with at least two digits, there exist graph directed sets F 0 ;F 1 ;:::;F m such that F 0 D F. This graph directed system is obtained from an auxiliary self-affine tile WD F.T;C/ D T. C C/[7]. For our purposes,we need a modification of the graph in [7] with the following properties: (I) F, C A T 1,andC D C ˇD,where ; ˇ 2 Z and D is a complete residue system for T 1 2 M n.z/, (II) F 1 ;:::;F m and F is a finite union of the affine copies of some of F 1 ;:::;F m. We now give the detail of the new graph directed system. Select a digit set C D fc 1 ;c 2 ; c N g Z n and form an auxiliary self-affine tile D T. C C/ with property (I). Let j.x/ D T.x C c j /, j D 1;:::;N, with N Djdet.T 1 /j. We also define the index sets k Dfi D.i 1 ;i 2 ;:::;i k / W 1 i j N g. Forj 2 J k and i 2 k, weseta j D kid1 T.k i/ a ji and c i D k T.k j/ c ij. For each i 2 k, let.i/ WD fa j c i W j 2 J k ; i. /\ ı S j. / ;g ı and S k Dfi2 k W.i/ ;g. Finally, we define V F D[ 1 kd1 S k.sincev F is a finite set, we write V F Dfv 1 D.i 1 /; v 2 D.i 2 /;:::;v m D.i m /g: We will also assume that first l. m/ vertices in V F form S 1,i.e.,S 1 D fv 1 ; v 2 ;:::;v l g.for1 i;j m, edges are defined by E i;j Dfc s W.i i s/ D.i j /; 1 s N g: If S m E i;j D; for some i, we discard v i and all paths going to v i. Let E F WD S m i;jd1 E i;j. Then, similar to Proposition 3.3 in [7], one can prove that F D[ l i j.f j / so that (II) holds. 3 Statement of the Results For our results, we give the necessary definitions. Let B D Œb ij WD Œ#E i;j, 1<i;j <m,betheadjacency matrix of the directed graph.v F ;E F / in the previous section. By permuting the indices, we may assume that

5 On the Dimension of Self-Affine Fractals B D 4 B 1 0 B 2 : : : :: : :: 0 0 Bs where each B i is irreducible with dimension n i n i (1 i s). Let i denote the spectral radius of B i.thenmaxf i W 1 i sg D B is the spectral radius of B. Let D minf i W 1 i sg. Before stating our results, we define two sequences fu r g and fv r g by the following equations: 3 5; Œ X m X i;jd1 e2ei;j r ur.t r / 1 r D 1; Œminf X m X e2e r i;j. vr.t r // 1 j 1 i mg 1 r D 1: Also, we define a unique number u WD u. / by lim r!1 Œ P m P i;jd1 e2ei;j r u.t r / 1 r D1; and another number v WD v. / by lim ŒminfX m X. v.t r // 1 j 1 i mg 1 r!1 e2ei;j r r D 1; where min stands for minimum. To distinguish the m in (1) foru, v, we write m u, m v, respectively. Our first result below is about the estimation of fractal dimensions. The second result gives a new class of exceptional fractals. Another exceptional class, the proof techniques and more detail are given in [10]. Proposition 1. Let F D F.T;A/be an integral self-affine set. ı (i) We always have v dim H F u n. Further, F D;if and only if u <n: In particular, integral self-affine sets with empty interior are exceptional when #A jdet.t 1 /j. (ii) For each r,u r is an upper bound for u, and v r is a lower bound for v. (iii) Suppose that T is a triangular matrix or a normal matrix with singular values satisfying 0< n n 1 1 <1. Then we have m v 1 log. n n 1 n.mv 2// log. n.mv 1// for some integer m n. dim H F m 1 log. 1 2 m 1 B / log. m/ (4) Working with diagonal matrices is relatively easy [7]. Therefore, we study the non-diagonal triangular case as in [9] and, for the integers t p 2, consider the following. T 1 D p0 ; A D 1t p ; 0 0 i ; W 0 i p 1; 0 j t 1 n t j p 1 : t 1 (5) Proposition 2. Let F D F.T;A/be an integral self-affine set with T 1, A as in (5). Then (4) holds. Further, there exists a graph.v F ;E F / such that

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