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DISERTAˇCN´I PR´ACE Multivalued Fractals and Hyperfractals

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DISERTAˇCN´I PR´ACE Multivalued Fractals and Hyperfractals UNIVERZITA PALACKEHO´ V OLOMOUCI PRˇ´IRODOVEDECKˇ A´ FAKULTA KATEDRA MATEMATICKE´ ANALYZY´ A APLIKAC´I MATEMATIKY DISERTACNˇ ´I PRACE´ Multivalued fractals and hyperfractals Vedouc´ıdiplomov´epr´ace: Vypracoval: prof. RNDr. dr hab. Jan Andres, DSc. Mgr. Miroslav Rypka Rok odevzd´an´ı: 2012 Matematick´aanal´yza, IV. roˇcn´ık Prohl´aˇsen´ı Prohlaˇsuji, ˇze jsem disertaˇcn´ı pr´aci zpracoval samostatnˇepod veden´ım prof. RNDr. dr hab. Jana Andrese, DSc. s pouˇzit´ım uveden´eliteratury. V Olomouci dne 20. srpna 2012 Podˇekov´an´ı M˚uj d´ık patˇr´ıpˇredevˇs´ım profesoru Andresovi za rady a trpˇelivost. Chci podˇekovat sv´ym rodiˇc˚um a pˇr´atel˚um za podporu nejen pˇri psan´ıdisertaˇcn´ıpr´ace a profesoru Bandtovi za vstˇr´ıcnost a laskavost bˇehem moj´ıst´aˇze v Greifswaldu. Contents 1 Introduction 5 1.1 Currentstateoftheart........................ 5 1.2 Aimsofthethesis........................... 7 1.3 Theoreticalframework . 9 1.4 Appliedmethods ........................... 9 1.5 Mainresults.............................. 9 2 Spacesandhyperspaces,mapsandhypermaps 11 3 Iteratedfunctionsystemsandinvariantmeasures 20 3.1 Iteratedfunctionsystems. 20 3.2 Measure ................................ 25 3.3 Dimensionandself-similarity. 31 3.4 LiftedIFSandsuperfractals . 34 4 Fixedpointtheoryinhyperspaces 38 5 Multivaluedfractalsandhyperfractals 42 5.1 Existenceresults ........................... 42 5.2 Address structure of multivalued fractals . .. 51 5.3 Ergodicapproach ........................... 57 5.4 Chaosgame .............................. 62 6 Visualizationanddimensionofhyperfractals 67 6.1 Convexsetsandsupportfunctions . 67 6.2 Visualizationofhyperfractals . 71 6.3 Dimension and self-similarity of hyperfractals . ..... 83 7 Measureonmultivaluedfractals 93 8 Fuzzy approach 103 8.1 Fuzzysets ............................... 103 8.2 Fuzzyfractals ............................. 111 8.3 Visualization of fuzzy fractals and measures . 119 9 Summary 125 List of figures 127 References 128 Abstract Fractals generated by iterated function systems are systematically studied from 1981 (see [Hu]). The theory of multivalued fractals is developed from 2001 (see [AG1]) almost separately. In the thesis, we treat multivalued fractals and structures supported by them by means of hyperfractals. We deal with structure, self-similarity of multivalued fractals and their visualization. We discuss relation- ship of multivalued fractals to fractals generated by iterated function systems. We also extend the theory of hyperfractals. We proceed in the following way. First, we review hyperspaces, maps and hypermaps, since fractals are fixed points of hypermaps in hyperspaces. We also remind iterated function systems and related notions of code space, self-similarity and invariant measure. Although we usually construct fractals by means of the Banach theorem, we discuss other fixed point theorems, which can be applied in hyperspaces. We describe generalizations of the Banach (metric) and Schauder (topological) fixed point theorems. Existence results for metric and topological multivalued fractals and hyperfractals are supplied. We study multivalued fractals and associated hyperfractals generated by the same iterated multifunction system. We prove that they possess the same address structure. Since we can also regard fractals generated by iterated function systems as attractors of chaotic dynamical systems, we can draw them by means of the chaos game. We remind the theory related to the chaos game, particularly the ergodic theory and chaos. Next, we investigate visualization and dimension of hyperfractals. Hyperfrac- tals lie in hyperspaces, which are nonlinear and infinite-dimensional spaces. For a particular class of hyperfractals, we construct projections of their structure by means of support functions. We apply the Moran formula to calculate dimension of self-similar hyperfractals. We also show that self-similar fractals form a subset of shadows of self-similar hyperfractals. Since hyperfractals are attractors of iterated function systems, we visualize also an invariant measure. Moreover, we construct a shadow of the invariant measure supported by the underlying multivalued fractal by means of ergodic theorem. Finally, our results are generalized to spaces of fuzzy sets. We remind the theory of fuzzy sets. Then, we define fuzzy fractals and fuzzy hyperfractals, which are related in the same way as multivalued fractals and hyperfractals. We find their address structure, which helps us to visualize these fractals and calculate their Hausdorff dimension. 4 1. Introduction 1.1. Current state of the art Fractals are extensively studied objects without an exact definition (see [Ma1], [Ma2], [Ma3]). However, fractals have usually some of the following features [Fa1] Fractals have a “natural” appearance. • Fractals have a fine structure, that is irregular detail at arbitrarily small • scales. Fractals are too irregular to be described by calculus or traditional geomet- • rical language, either locally or globally. Fractals have often some sort of self-similarity or self-affinity, perhaps in a • statistical or approximate sense. The “fractal dimension” (defined in some way) is strictly greater than the • topological dimension of a fractal. In many cases of interest, fractals have a very simple, perhaps recursive, • definition. We will meet all the features of fractals but the last three are crucial for us. We will consider self-similar fractals with a noninteger Hausdorff (=fractal) dimension generated by iterated function systems (cf. [Hu], [Ba1]). An iterated function system (IFS) is usually a system of a finite number of contractions on a complete metric space X; fi,i = 1, 2,...,n . Then the transformation F : K(X) K(X) defined by{ } → n F (B) := fi(B), i=1 [ for all B K(X), is a contraction mapping on the complete metric space ∈ (K(X), dH ), i.e. the space of compact subsets of X. Its unique fixed point A∗ K(X) obeys ∈ n A∗ = fi(A∗). i=1 [ The fixed point A∗ is called an attractor of the IFS and it is a union of its contracted copies. If the copies are separated and contractions are in addition similitudes, we talk about self-similarity of an attractor. For self-similar attrac- tors, the Hausdorff dimension can be calculated by means of the Moran formula. These results were stated by Hutchinson [Hu] (see also [B], [BG], [BK], [E], [Fa1], [Fa2], [PJS], [Sc], [Wc] and [Wi]). 5 Barnsley explores more ways of drawing fractals and invariant measures (see [Ba1], [BD]). He also came out with the idea of fractal image compression. It- erated function systems enable us to store image-like data in a few parameters of functions comprising the IFS whose attractor is close to the image. Barns- ley developed techniques for image encoding and decoding by means of IFS (cf. [BEHL], [BH]). His method provides a great compression ratio but it is very time consuming. Jacquin developed the approach based on the idea of IFS which use domain and range blocks of a picture (cf. [Ja1]–[Ja4]). See also [F] or [WJ] for further references. Iterated function systems were extended in many ways, for instance to R∞ in [CR]. Infinite IFS were described in [GJ] and [L3]. One of the most signifi- cant generalizations of fractals generated by IFSs are multivalued fractals. They are generated by iterated multifunction systems (shortly IMSs). By multivalued fractals, we understand the fixed points of operators F : K(X) K(X), such that → n F (A)= Fi(A), i [ where F are induced by continuous multivalued maps : X K(X) from an i Fi → IMS (X, d), 1, 2, ..., n . Multivalued{ F fractalsF wereF } extensively investigated in the last ten years. They were developed independently by three groups. Andres and co-workers repre- sent one group which also introduced the terminology [A1], [A2], [AF], [AFGL], [AG1], [AG2], [AV], [Fi]. Their work was inspired by relationships between maps, multivalued maps and hypermaps. Problems in differential inclusions motivated Petru¸sel and co-workers ([LPY], [P1], [P2], [PR1], [PR2]). For the similar ap- proach see also [BBP], [CP], [CL], [GG], [KLV1], [KLV2], [LtM], [Mh], [MM], [Ok], [SPK]. The articles are devoted mainly to existence results by means of the fixed point theorems and structure of attractors. Drawing and approximation of attractors is also investigated. Lasota and Myjak studied the related notion of semifractals in [L1]-[L4], [LM1]–[LM4]. Our work was inspired by superfractals [Ba2] (see also [BHS1]–[BHS4]). Barns- ley found out that it is more effective to treat some sets like sets of fractal sets. Thus, he investigated iterated function systems of iterated function systems. In the easiest case, he considered IFSs F1, F2,...,Fm, F = (X, d), f i, f i,...,f i , i { 1 2 in } where f i : X X are contractions. The operators F : K(X) K(X), j → i → i Fi(A)= fj (A), j [ 6 comprise the IFS (K(X), dH ), F1, F2,...,Fm and generate the contracting operator φ : K(K(X{ )) K(K(X)), } → φ(α)= F (A) . { j } j A α [ [∈ The fixed point of φ in a hyperhyperspace is called a superfractal. At the end of [BHS1], it is suggested to investigate the previous case with i multivalued mappings fj . Attractors of such systems will be called hyperfractals (see [AR1] and [AR2]). Hyperfractals are fixed points of operators φ in K(K(X)). Given a system (X, d), 1, 2, ..., n , of multivalued maps i : X K(X) we induce the maps{ to FF: KF(X) KF(X} ), F → i → F (A)= (x). i Fi x A [∈ The mappings F : K(X) K(X) comprise an IFS φ = (K(X), d ),F , F , ..., i → { H 1 2 Fn , like in the case of superfractals. Then we obtain the contracting operator φ :}K(K(X)) K(K(X)), → n n φ(α)= φ (α)= F (A) . i { i } i=1 i=1 A α [ [ [∈ If i are contractions in a complete metric space, φ is also a contraction. The fixedF point of the contraction is called a hyperfractal. Hyperfractals are attractors of IFS in hyperspaces. Generalizing IFS to fuzzy sets, fuzzy fractals can be obtained (see e.g. [CFMV], [FLV], [FMV] and [DK]). This research is also motivated by image compression. 1.2. Aims of the thesis The aim of the thesis is to understand multivalued fractals in a better way and to explore structures supported by them.
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