On the geometric structure of the limit set of conformal iterated function systems

Publ. Mat. 47 (2003), 133–141 ON THE GEOMETRIC STRUCTURE OF THE LIMIT SET OF CONFORMAL ITERATED FUNCTION SYSTEMS Antti K¨ aenm¨ aki Abstract We consider infinite conformal function systems on Rd.Westudy the geometric structure of the limit set of such systems. Suppose this limit set intersects some l-dimensional C1-submanifold with positive Hausdorff t-dimensional measure, where 0 <l<dand t is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some l-dimensional affine subspace or geometric sphere whenever dexceeds 2 and analytic curve if dequals 2. 1. Introduction We work on the setting introduced by Mauldin and Urba´nski in [6]. There they consider uniformly contractive countable collections of conformal injections defined on some open, bounded and connected set Ω ⊂ Rd.Itisneeded that there exists some compact set X⊂Ω with nonempty interior such that each contraction maps this set to some subset of itself. For this kind of setting, even without the conformality assumption, there is always so called limit set associated. We denote it with Eand we are particularly interested in the properties of this set. The conformality assumption is basically needed for good behavior of the derivatives. As usual, open set condition (OSC), introduced by Moran in [8], is used for getting decent separation for those previously mentioned subsets of X. We also need bounded distortion property (BDP), which says that the value of the norm of derivatives cannot vary too much. This is actually 2000 Mathematics Subject Classification. Primary: 28A80; Secondary: 37C45. Key words. Iterated function system, self-conformal set, local geometric structure. Author is supported by the Academy of Finland, project 23795, and during his stay in Universitat Aut`onoma of Barcelona he was supported by the Marie Curie Fellowship, grant HPMT-CT-2000-00053. He also wishes to thank the UAB, where this paper was completed, for warm hospitality. 134 A. K¨ aenm¨ aki just a consequence of our previous assumptions. And finally the boundary of Xis assumed to be “smooth” enough. For example all convex sets are like that. From this property one may verify that the limit set is a Borel set. In the finite case it is always compact. Together with OSC it also guarantees some properties for the natural Borel regular probability measure, so called conformal measure, associated to this set. This is one way to generalize similar kind of situation for finite collection of similitudes; the setting introduced by Hutchinson in [2]. Mattila proved in[3] for the limit set Eof this kind of setting the following result: Either Elies on an l-dimensional affine subspace or Ht(E∩M)=0 for every l-dimensional C1-submanifold M⊂Rd. Here 0 <l<dand Htdenotes the t-dimensional Hausdorff measure, where t= dimH(E), the Hausdorff dimension of the set E. The main result in this note is a generalization for this. Our approach is based on an extensive use of tangents. Before going into more detailed preliminaries we should mention that Springer has proved in [10] similar result in the plane and Mauldin, Mayer and Urba´nski have studied similar behavior for connected limit sets in [5] and [7]. As usual, let Ibe a countable set with at least two elements. Put I∗=∞ n=1 Inand I∞=IN={(i1,i 2,...):ij∈Ifor j∈N}.Thus, if i∈I∗there is k∈Nsuch that i=(i1,...,i k), where ij∈Ifor all j=1,...,k.Wecall this kas the length of iand we denote |i|=k.If i∈I∞we denote |i|=∞.Fori∈I∗∪I∞we put i|k=(i1,...,i k) whenever 1 ≤k<|i|. Choose Ω to be open, bounded and connected set on Rd.Nowfor each i∈Iwe define an injective mapping ϕi:Ω→Ω such that it is contractive, that is, there exists 0 <s i<1 such that |ϕi(x)−ϕi(y)|≤si|x−y|(1.1) whenever x, y ∈Ω. A mapping with equality in (1.1) is called similitude.Weassume also that ϕiis conformal, that is, |ϕ i|d=|Jϕi|, where Jstands for normal Jacobian and the norm in the left side is just the normal “sup-norm” for linear mappings. Here the derivative exists Hd-almost all points using Rademacher’s theorem. This definition for conformality is usually better known as 1-quasiconformality; see [12]. Note that a conformal mapping is always C∞by [9, Theorem 4.1] of Reshetnyak. We assume also that there exists a compact set Xwith int(X)=∅such that ϕi(X)⊂Xfor every i∈I. The use of the bounded set Ω here is essential, since conformal mappings On the Structure of the Limit Set of CIFS 135 contractive in whole Rdare similitudes, as dexceeds 2. We call a collection {ϕi:i∈I}as conformal iterated function system (CIFS) if the following conditions (1)–(4) are satisfied: (1) Mappings ϕiare uniformly contractive, that is, s:= supi∈Isi<1. Denoting ϕi=ϕi1◦···◦ϕi|i|for each i∈I∗,weget from this property that for every n∈N dϕi|n(X)≤snd(X)(1.2) whenever i∈I∞. Here dmeans the diameter of a given set. Now we may define a mapping π:I∞→Xsuch that π(i)= ∞  n=1 ϕi|n(X).(1.3) We set E=π(I∞)=  i∈I∞ ∞  n=1 ϕi|n(X)(1.4) and we call this set as the limit set of the corresponding CIFS. Our aim is to study this set. Observe that Esatisfies the natural invariance equality, E=i∈Iϕi(E). (2) Bounded distortion property (BDP), that is, there exists K≥1 such that |ϕ i(x)|≤K|ϕ i(y)|for every i∈I∗and x, y ∈Ω. Forafinite collection of mappings, this is just a consequence of smoothness (see [6, Lemma 2.2]) and in the infinite case it follows from Koebe’s distortion theorem as dequals 2 and [11, Theorem 1.1] whenever dexceeds 2. Using these assumptions we may prove that each mapping ϕi is a diffeomorphism and that there exists D≥1 such that D−1≤dϕi(E) ||ϕ i|| ≤D(1.5) for every i∈I∗. Here ||ϕ i|| = supx∈Ω|ϕ i(x)|. (3) Open set condition (OSC) holds for int(X), that is, ϕiint(X)∩ ϕjint(X)=∅for every i=j. (4) There exists r0>0 such that inf x∈∂X inf 0<r<r0 HdB(x, r)∩int(X) HdB(x, r)>0,(1.6) where ∂X is the boundary of the set X. 136 A. K¨ aenm¨ aki We should mention that in [6], instead of assumption (4), it was used so called cone condition, which says that for each boundary point x there exists some “cone” in the interior of Xwith vertex x.However, assumption (4) is sufficient for our setting, as was remarked also in [6]. Using these assumptions it follows that Eis a Borel set. Suppose there exists a Borel regular probability measure mon Esuch that mϕi(A)=A|ϕ i(x)|tdm(x),(1.7) where t= dimH(E), A⊂Xis a Borel set and i∈I∗. Then OSC and assumption (4) are crucial to derive that mϕi(X)∩ϕj(X)=0for every i=j(see [6, Section 3] for details). If this measure exists, we call it a t-conformal measure and the corresponding CIFS regular. Observe that finite CIFS’s are always regular (see [6, Section 3] for details). If we consider measure theoretical Jacobian Jmfor function ϕidefined as Jmϕi(x)=lim r0 mϕiB(x, r) mB(x, r) (1.8) for each point x∈E,wenotice using conformality of ϕiand (1.7) that Jmϕ−1 iϕi(x)=Jmϕi(x)−1=|ϕ i(x)|−t=ϕ−1 iϕi(x) tfor m-almost all x∈Eand for all i∈I∗(recall also [4, Theorem 2.12]). Thus for example, using BDP mϕ−1 i(A)=A|(ϕ−1 i)(x)|tdm(x) ≤||(ϕ−1 i)||tm(A)≤Kt||ϕ i||−tm(A) (1.9) whenever A⊂ϕi(X)isBorel. Furthermore by setting Φ|ϕi(X)(x)= ϕ−1 i(x) for all i∈Iwe get at m-almost every point defined mapping Φ: i∈Iϕi(X)→Xfor which mΦ(A)=A|Φ(x)|tdm(x).(1.10) In fact, mis ergodic and equivalent to some invariant (with respect to the function Φ) measure (see [6, Section 3] for details). In the finite case, the t-conformal measure is Ahlfors regular, that is, there exists C≥1 such that C−1≤mB(x, r) rt≤C(1.11) for all x∈Eand r>0 small enough (see [6, Lemma 3.14]). Let 0 <l<dbe an integer and G(d, l)bethe collection of all l-dimensional subspaces of Rd.Wedenote by PVthe orthogonal projection On the Structure of the Limit Set of CIFS 137 onto V∈G(d, l) and we put QV=PV⊥, where V⊥is the orthogonal complement of V.Forx∈Rdwe denote V+x={v+x:v∈V}.If a∈Rd,V∈G(d, l) and 0 <δ<1weset X(a, V, δ)={x∈Rd:|QV(x−a)|<√δ|x−a|} ={x∈Rd:d(x, V +a)<√δ|x−a|}. (1.12) Note that the closure of X(a, V, δ) equals to the complement of X(a, V ⊥,1−δ). We say that Visa(t, l)-tangent plane for Eat a if lim r0 mB(a, r)\X(a, V, δ) rt=0(1.13) for all 0 <δ<1. Furthermore we say that Vis a strong l-tangent plane for given set A⊂Rdat aif for every 0 <δ<1 there exists r>0 such that A∩B(a, r)⊂X(a, V, δ).(1.14) For example, an l-dimensional C1-submanifold has a strong l-tangent plane at all of it’s points. Observe that these two tangents are exactly the same for Ein the case of Ahlfors regular m.However, we shall not need this fact here. Recall also that t-dimensional upper density of a measure µat ais defined as Θ∗t(µ, a)=lim sup r0 µB(a, r) rt.(1.15) 2. Main result The main result of this note is the following theorem. Theorem 2.1. Suppose CIFS is given, t= dimH(E)and 0<l<d. Then either Ht(E∩M)=0for every l-dimensional C1-submanifold M⊂Rdor the closure of Eis contained in some l-dimensional affine subspace or l-dimensional geometric sphere whenever dexceeds 2and analytic curve if dequals 2. Using this theorem we are able to find minimal amount of essential directions for where the set Eis spread out. It also follows that if t is an integer, then the limit set is always either t-rectifiable or purely t-unrectifiable. See [4] for definitions and properties for these concepts. Provided that dexceeds 2, it is also easy to see that if at least one of our conformal mappings is similitude, the latter case of the theorem concerns only affine subspaces. 138 A. K¨ aenm¨ aki The proof is divided into three parts. We start with an easy lemma which provides a useful dichotomy for our purposes. Lemma 2.2. Suppose CIFS is given, t= dimH(E)and 0<l<d. Then either Ht(E∩M)=0for every l-dimensional C1-submanifold M⊂Rd or the system is regular and at least one point of Ehas a (t, l)-tangent plane. Proof: Assume Ht(E∩M)>0 for some M. Since Ht(E)<∞, the regularity of the system is guaranteed by [6, Theorem 4.17]. From [1, 2.10.19(4)] of Federer we get that Θ∗t(m|E\M,x)=0for Ht-almost every x∈E∩M. Let a∈E∩Mbe such point and V∈G(d, l)beastrong l-tangent plane for Mat a. This means that for any given 0 <δ<1 there exists rδ>0 such that M∩B(a, r)⊂X(a, V, δ)(2.1) whenever r<r δ.Thusfor every 0 <δ<1 lim sup r0 mB(a, r)\X(a, V, δ) rt≤lim sup r0 mB(a, r)\M∩B(a, r) rt =Θ∗t(m|E\M,a)=0 (2.2) and Vis a (t, l)-tangent plane for Eat a. With this dichotomy in mind it is enough to study what happens if one point of Ehas a (t, l)-tangent plane. Theorem 2.3. Suppose CIFS is regular, t= dimH(E)and 0<l<d. If one point of Ehas a (t, l)-tangent plane then m-almost all of Eis contained in the set f(V), where V∈G(d, l)and fis some conformal mapping. Proof: Suppose a∈Ehasa(t, l)-tangent plane V∈G(d, l) and let ε>0. Now for each j∈Nthere exists rj,0>0 such that mB(a, r)\X(a, V, 1 j)≤εrt (2.3) whenever r<r j,0. Let i∈I∞be such that π(i)=a.Foreach j∈N choose some fixed radius rj<r j,0and nj∈Nsuch that ϕi|nj(E)⊂B(a, rj) and ϕi|nj(E)\B(a, rj 2)=∅.(2.4) Since trivially a∈ϕi|n(E) for all n,wehave rj 2≤dϕi|nj(E)≤2rj.(2.5) On the Structure of the Limit Set of CIFS 139 For each jdefine mapping ψj:Rd→Rdsuch that ψj(x)=||ϕ i|nj||−1(x− a)+a. Then ψjX(a, V, 1 j)=X(a, V, 1 j)(2.6) and |ψj(x)−ψj(y)|=||ϕ i|nj||−1|x−y|(2.7) for every x, y ∈Rd.NowFj:= ψj◦ϕi|njis clearly a conformal mapping from Ω to Rd. Since for arbitrary x, y ∈Ω K−1|x−y|≤||ϕ i|nj||−1||(ϕ−1 i|nj )||−1|x−y| ≤||ϕ i|nj||−1|ϕi|nj(x)−ϕi|nj(y)| =|Fj(x)−Fj(y)|≤|x−y| (2.8) using BDP and mean value theorem, we notice that Fjis bi-Lipschitz with constants K−1and 1 for every j∈N. Using now Ascoli-Arzela’s theorem we shall find an uniformly converging subsequence, say, Fjk→F, as k→∞. According to [12, Corollaries 37.3 and 13.3] of V¨ais¨al¨awe notice that F−1is conformal. Since m E\F−1 jX(a, V, 1 j)=mϕ−1 i|njϕi|nj(E)\X(a, V, 1 j) ≤Kt||ϕ i|nj||−t mϕi|nj(E)\X(a, V, 1 j) ≤(DK)tdϕi|nj(E)−tmB(a, rj)\X(a, V, 1 j) ≤(2DK)tr−t jεrt j (2.9) for every j∈Nusing (1.9), (1.5), (2.4), (2.3) and (2.5), we conclude mE\F−1(V+a)≤(2DK)tε.(2.10) We finish the proof by letting ε0. Notice that the inclusion in the previous theorem holds for the closure of E, since any set of full m-measure is dense in Eand any l-dimensional C1-submanifold is closed in Rd.Now the main theorem follows as a corollary just by recalling that conformal mappings are complex analytic in the plane and by Liouville’s theorem M¨obius transformations elsewhere (see [9, Theorem 4.1] of Reshetnyak). 140 A. K¨ aenm¨ aki Remark. The proof of the main theorem was found in January ’01 and it was supposed to be part of the author’s thesis. Since recently there has been some interest for similar kind of questions (particularly [7]), it was decided to be published independently. Acknowledgement. Author likes to thank professor Pertti Mattila for many useful discussions during the preparation of this note and the referee for pointing out couple of excellent remarks. References [1] H. Federer,“Geometric measure theory”, Die Grundlehren der mathematischen Wissenschaften 153, Springer-Verlag New York Inc., New York, 1969. [2] J. E. Hutchinson,Fractals and self-similarity, Indiana Univ. Math. J. 30(5) (1981), 713–747. [3] P. Mattila,Onthe structure of self-similar fractals, Ann. Acad. Sci. Fenn. Ser. A I Math. 7(2) (1982), 189–195. [4] P. Mattila,“Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability”, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995. [5] R. D. Mauldin, V. Mayer and M. Urba´ nski, Rigidity of connected limit sets of conformal IFSs, Michigan Math. J. 49(3) (2001), 451–458. [6] R. D. Mauldin and M. Urba´ nski, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73(1) (1996), 105–154. [7] V. Mayer and M. Urba´ nski, Finer geometric rigidity of connected limit sets of conformal IFS, Preprint (2001), Proc. Amer. Math. Soc. (to appear). [8] P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc. 42 (1946), 15–23. [9] Yu. G. Reshetnyak,“Stability theorems in geometry and analysis”, Mathematics and its Applications 304, Kluwer Academic Publishers Group, Dordrecht, 1994. [10] O. B. Springer, Order two density and self-conformal sets, Ph. D. thesis, University of Bristol (1993). [11] M. Urba´ nski, Rigidity of multi-dimensional conformal iterated function systems, Nonlinearity 14(6) (2001), 1593–1610. [12] J. V¨ ais¨ al¨ a,“Lectures on n-dimensional quasiconformal mappings”, Lecture Notes in Mathematics 229, Springer-Verlag, BerlinNew York, 1971. On the Structure of the Limit Set of CIFS 141 Department of Mathematics and Statistics P. O. Box 35 (MaD) FIN-40014 University of Jyv¨askyl¨a Finland E-mail address:[email protected] Primera versi´o rebuda el 12 de febrer de 2002, darrera versi´o rebuda el 19 de setembre de 2002.