multivalued fractals and hyperfractals

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MULTIVALUED FRACTALS AND HYPERFRACTALS JAN ANDRES Dept. of Math. Anal. and Appl. of Math., Faculty of Science, Palack´ y University, 17. listopadu 12, 77146 Olomouc, Czech Republic [email protected] MIPOSLAV RYPKA Dept. of Math. Anal. and Appl. of Math., Faculty of Science, Palack´ y University, 17. listopadu 12, 77146 Olomouc, Czech Republic [email protected] Received (to be inserted by publisher)

A new class of fractals called hyperfractals is systematically investigated. Their underlying fractals in supporting spaces are multivalued fractals developed by ourselves in our earlier papers [Andres & Fiˇser, 2004], [Andres et al., 2005]. In the case of uniqueness, they also coincide with the supports of invariant measures of the associated Markov–Feller hyperoperators. We prove their existence by means of fixed point theorems in hyperspaces and calculate, in particular cases of generating hypermaps, their Hausdorff dimension. Two illustrative examples are supplied. Keywords: Hyperfractal, hyperIFS, hyperattractor, multivalued fractal, IMS, invariant measure, Hausdorff dimension, correlation dimension.

1. Introduction Stimulated both theoretically (see Sections 4 and 5 below) and practically (see [Andres& Rypka, 2011], where a particular application to quantitative linguistics was already done), we will study the new class of geometric objects called hyperfractals. In order to justify this study and to understand in a better way their essence, we will firstly explain the relationship to superfractals, investigated systematically in [Barnsley, 2006], [Barnsley et al., 2005]– [Barnsley et al., 2003], [Singh et al., 2009], and to multivalued fractals, treated by ourselves in a series of papers [Andres, 2001], [Andres, 2004], [Andres & Fiˇser, 2004], [Andres et al., 2005], [Andres& Górniewicz, 2001], [Andres& Górniewicz, 2003], [Andres & V¨ ath, 2007] (cf. also [Boriceanu et al., 2010], [Chifu & Petru¸sel, 2008], [Colapinto & La Torre, 2008], [Fiˇser, 2004], [Glavan & Gut¸u, 2004], [Kunze et al., 2007], [Kunze et al., 2008], [Le´sniak, 2002]–[Le´sniak, 2005], [Lasota & Myjak, 2000], [Lasota & Myjak, 2002], [Llorens-Fuster et al., 2009], [La Torre & Mendivil, 2008], [Mihail, 2008], [Mihail & Miculescu, 2010], [Ok, 2004], [Petru¸sel, 2001], [Petru¸sel, 2002], [Petru¸sel & Rus, 2001], [Petru¸sel & Rus, 2004], [Petru¸sel et al., 2007], [Singh et al., 2009]). The notion of a superfractal is strictly related to V -variable iterated function systems (shortly, V variable IFSs) and V -variable fractals. Hence, the V -variable IFS is a quadruple (K V (X), Fi , Pi , i ∈ I), where K V (X) is the set of all V -tuples of compact subsets of a complete metric space (X, d), Fi : K V (X) → 0

Supported by the Council of the Czech Government (MSM 6198959214) 1

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Jan Andres, Miroslav Rypka

K V (X) is a one-parameter set of contractions such that the maps Fi can be considered as the induced Hutchinson–Barnsley operators, i.e. 1 2 V fij , fij , . . . , fij : K V (X) → K V (X), (1) Fi := j∈Ji1

j∈Ji2

j∈JiV

associated, for each i ∈ I, with the individual IFSs {X, fijkV , j ∈ JikV }, Pi are their respective probabilities being appropriately chosen and I is the index set. Observe that, unlike at the standard IFS operating on points in X, the V -variable IFS operates on points in K V (X), i.e. on V −tuples of compact subsets of X. It follows from the theory in [Barnsley, 2003], [Barnsley et al., 2005], [Barnsley et al., 2008], [Barnsley et al., 2003] that there is again a unique superfractal set, together with an associated superfractal invariant probability measure, which corresponds to an attractor of the V -variable IFS, i.e. a compact positively invariant subset of K V (X) and the stationary measure in the space P (K V (X)) of probability measures on K V (X). In terms of the fixed point theory, there are unique fixed points in K(K V (X)) and in P (K V (X)). They can be approximated via the random iteration algorithm. The correspondence with V -variable fractals consists in the fact that V -variable fractals are simply components of points in a superfractal. In [Kunze et al., 2008], a special case was considered, where V = 1 and JikV = J, for all i ∈ I, kV ∈ {1, . . . , V }. We will consider the case, where V = 1 and, moreover, JikV = J = {1}, for all i ∈ I = {1, 2, . . . , n}, and kV ∈ {1, . . . , V }. On the other hand, in our system {K(X), Fi , Pi , i = 1, 2, . . . , n}, Fi : K(X) → K(X) will be induced by continuous multivalued maps with compact values Fi : X → K(X), i = 1, 2, . . . , n, which are more general than the Hutchinson–Barnsley maps n

1 fi1 : X → K(X),

i=1

as in (1). Because of these differences, we decided to call the fixed points of the Hutchinson–Barnsley operators1 n

Fi : K(K(X)) → K(K(X))

i=1

as hyperfractals. By multivalued fractals, we understand the fixed points of the Hutchinson–Barnsley operators n

φi : K(X) → K(X),

i=1

where φi are induced by the continuous multivalued maps with compact values φi : X → K(X), i = 1, 2, . . . , n. Thus, for the same system of continuous multivalued maps with compact values, hyperfractals (considered as fixed points in K(K(X))) are related to multivalued fractals (fixed points in K(X)) as follows. The union of the point restrictions of these hyperfractals into X, i.e. the hyperfractal “shadows” (called the underlying fractals in [Barnsley, 2006]) on X, coincides with the union of multivalued fractals in X. 1

Formally, it will be convenient to define these operators here as n

Fi (A) :=

i=1

n

Fi (x), for each A ∈ K(K(X)),

i=1 x∈A

rather than in the usual way (for the standard IFSs) by means of the Hutchinson–Barnsley maps F (x) :=

n i=1

Fi (x), i.e.

x∈A

F (x) =

n x∈A i=1

Fi (x), for each A ∈ K(K(X)).

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Multivalued fractals and hyperfractals

3

Summing up, hyperfractals can be regarded as a generalization of trivial (V = 1) superfractals whose “shadow” on the supporting (underlying) space coincides with multivalued fractals. The paper is organized as follows. The preliminaries are divided into two sections. In Section 2, we recall the basic properties of auxiliary hyperspaces and hypermaps. We will also use the prefix “hyper” when speaking about the related iterated function systems in hyperspaces, etc. In Section 3, we present a panorama of the adequate fixed point theorems in hyperspaces which we state in the form of propositions. The main results are contained in Section 4 and Section 5. Section 4 deals with multivalued fractals, hyperfractals and invariant measures in terms of fixed points from the foregoing section. The terminological Remark 22 reinterpretes and clarifies the relationship between the obtained results in terms of fractals. In Section 5, the Hausdorff dimension of self-similar hyperfractals is calculated, by means of the Moran– Hutchinson formula, for the generating similitudes induced by a special class of multivalued maps. Two illustrative examples are supplied. The second one, in the concluding Section 6, refers to the case when (unlike in the first example) the Hausdorff dimension of the related hyperfractal cannot be calculated by means of our theorems developed in Section 5, but the correlation dimension can be applied there.

2. Spaces and hyperspaces, maps and hypermaps In the entire text, all topological spaces will be at least metric. Hence, let (X, d) be a metric space. By the hyperspace (H(X), dH ), we will understand as usually a certain class H(X) of nonempty subsets of X endowed with the induced Hausdorff metric dH , i.e.2 dH (A, B) := inf{r > 0|A ⊂ Or (B) and B ⊂ Or (A)} = max{sup d(a, B), sup d(b, A)} a∈A

b∈B

= max{sup( inf d(a, b)), sup( inf d(a, b))}, a∈A b∈B

b∈B a∈A

where Or (A) := {x ∈ X|∃a ∈ A : d(x, a) < r)} and A, B ∈ H(X). The following typical classes C(X) := {A ⊂ X|A is nonempty and closed}, B(X) := {A ⊂ X|A is nonempty, closed and bounded}, K(X) := {A ⊂ X|A is nonempty and compact} satisfy the obvious inclusions X ⊂ K(X) ⊂ B(X) ⊂ C(X). If (X, ||.||) is a Banach space, then we can also consider their subclasses, namely CCo (X) := C(X) ∩ Co(X), BCo (X) := B(X) ∩ Co(X), KCo (X) := K(X) ∩ Co(X), where Co(X) := {A ⊂ X|A is nonempty and convex}, and obviously X ⊂ KCo (X) ⊂ BCo (X) ⊂ CCo (X) ⊂ Co(X). The hyperspace (B(X), dH ) is a closed subset of (C(X), dH ) (cf. e.g. [Hu & Papageorgiu, 1997, Proposition 1.7]) and if (X, d) is complete, then also (K(X), dH ) is a closed subset of (C(X), dH ) (cf. e.g. [Klein 2

For the hyperspace (C(X), dH ), we in fact often employ the metric min{1, dH } (cf. e.g. [Banakh & Voytsitskyy, 2007]) but, for the sake of simplicity, we will use the same notation here.

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& Thompson, 1984, Theorem 4.3.9], [Hu & Papageorgiu, 1997, Proposition 1.6], [Beer, 1993, Exercise 3.2.4 (b)]). Thus, for a complete (X, d), (C(X), dH ) is complete (cf. e.g. [Beer, 1993, Theorem 3.2.4]), and (K(X), dH ) ⊂ (B(X), dH ) are both complete subspaces of (C(X), dH ) (cf. e.g. [Hu & Papageorgiu, 1997, Propositions 1.6 and 1.7]). If (X, ||.||) is a normed space, then (KCo (X), dH ) ⊂ (BCo (X), dH ) ⊂ (CCo (X), dH ) are all closed subsets of (C(X), dH ) (cf. e.g. [Hu & Papageorgiu, 1997, Corollary 1.9]), and subsequently, for a Banach space (X, ||.||), all the above subsets are complete subspaces of (C(X), dH ) (cf. e.g. [Hu & Papageorgiu, 1997, Remark 1.10]). · · × E, where (E, ||.||) is a Banach If X ⊂ E n is nonempty, compact convex subset of E n = E × · n-times

space, then (KCo (X), dH ) is, according to [Hu & Huang, 1997] and [Liu et al., 2007], a compact and convex subspace of (K(X), dH ). Besides the mentioned completeness, the list of some further induced properties of (K(X), dH ) by those of (X, d) can be found in Table 1. Except last two properties in Table 1, the implications hold in both directions, i.e. these properties are in fact equivalent on the lines. Since the Vietoris topology, called a finite topology in [Michael, 1951], coincides in K(X) with the Hausdorff metric topology (cf. e.g. [Hu & Papageorgiu, 1997, Theorem 1.30 on p. 14], [Beer, 1993, Exercise 3.2.9]), we could also employ for Table 1 the equivalences proved in [Michael, 1951].

Remark 1.

(X, d) compact

(K(X), dH ) compact

complete

complete

separable

separable

Polish locally compact connected locally connected locally continuum-connected locally continuum-connected and connected

Polish locally compact connected locally connected ANR

references [Beer, 1993], [Klein & Thompson, 1984], [Michael, 1951] [Beer, 1993], [Hu & Papageorgiu, 1997], [Klein & Thompson, 1984] [Hu & Papageorgiu, 1997], [Klein & Thompson, 1984] [Hu & Papageorgiu, 1997] [Michael, 1951] [Michael, 1951] [Michael, 1951] [Curtis, 1980]

AR

[Curtis, 1980]

Let us recall that, by a Polish space, we understand as usually a complete and separable metric space and that X is an AR (ANR) if, for each Y and every closed A ⊂ Y , every continuous mapping f : A → X is extendable over Y (a neighbourhood of A in Y ). Furthermore, a metric space (X, d) is locally continuumconnected if, for each neighbourhood U of each point x ∈ X, there is a neighbourhood V ⊂ U of x such that each point of V can be connected with x by a subcontinuum (i.e. a compact, connected subset) of U . In locally compact (e.g. Euclidean) spaces (X, d), the local continuum-connectedness can be simply replaced by the local connectedness in Table 1. Since the ANRs and ARs are locally continuum-connected and the ARs are still connected, for (K(X), dH ) to be an ANR (AR), it is obviously enough to assume that so are (X, d), respectively. For more details concerning the ANRs and ARs, see e.g. [Andres& G´ orniewicz, 2003], [Andres & Väth, 2007], [Curtis, 1980]. It will be also convenient to recall some equivalent properties of (X, d) and (C(X), dH ) in Table 2. However, let us note that, unlike in Table 1, many properties are not induced here from spaces to hyper-

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spaces. For instance, because of a counter-example in [Kurihara et al., 2005], (C(Rn ), dH ) was shown there to be non-separable and, in particular, not Polish. More precisely, (C(Rn ), dH ) has according to [Kurihara et al., 2005, Proposition 7.2] uncountably many components (i.e. maximal connected subsets) and (K(Rn ), dH ) is the only separable (connected, closed and open) component of (C(Rn ), dH ). In particular, (C(Rn ), dH ) is disconnected. It is also not locally compact (cf. [Constantini et al., 2005]). This demonstrates that connectedness and local compactness are not induced. (X, d) compact complete relatively compact

(C(X), dH ) compact complete relatively compact

references [Beer, 1993], [Hu & Papageorgiu, 1997] [Beer, 1993], [Hu & Papageorgiu, 1997] [Beer, 1993]

On the other hand, according to [Banakh & Voytsitskyy, 2007, Corollary 3.8], a metric locally convex space (X, d) is normable if and only if its hyperspace (C(X), dH ) is an ANR. This implies that (C(Rn ), dH ) is an ANR and, in view of its just mentioned disconnectedness, it cannot be an AR. In particular, (C(Rn ), dH ) is only locally (path-)connected. Furthermore, e.g. the hyperspace (C(R∞ ), dH ), where R∞ is the countable product of lines, is not an ANR (despite (R∞ , |.|) is an AR) and, equivalently (see the comments to Diagram 1 in [Banakh & Voytsitskyy, 2007]), it is even not locally connected. Similarly, non-normable Fréchet spaces (X, d) do not induce, unlike Banach spaces, (C(X), dH ) to be locally connected. In [Banakh & Voytsitskyy, 2007], sufficient and necessary conditions were established, for both (C(X), dH ) and (B(X), dH ), to be ANR (AR). Thus, for instance, (B(X), dH ), where X ⊂ E is a convex subset of a normed linear space (E, ||.||) is an AR (cf. [Antosiewicz & Cellina, 1977]). Moreover, every component of (C(E n ), dH ), where (E, ||.||) is a Banach space, was shown in [Kurihara et al., 2005] to be a complete AR. For more details concerning the relationship between spaces and hyperspaces see e.g. [Antosiewicz & Cellina, 1977], [Beer, 1993], [Banakh & Voytsitskyy, 2007], [Constantini et al., 2005], [Curtis, 1980], [Hu & Papageorgiu, 1997], [Illanes & Nadler, 1999], [Kurihara et al., 2005], [Klein & Thompson, 1984], [Michael, 1951], [Nadler, 1978], [Nadler et al., 1975]–[Nadler et al., 1979], [Shori & West, 1975], [Wicks, 1991]. Now, we proceed to multivalued maps and the induced (in hyperspaces) hypermaps. In view of applications in the next section, we would like to have at least continuous hypermaps. There are counter-examples (see e.g. [Andres & Fiˇser, 2004]) that the upper semicontinuity of multivalued maps is sufficient for this aim. It is well-known (see e.g. [Andres& G´ orniewicz, 2003], [Hu & Papageorgiu, 1997]) that compact-valued upper semicontinuous maps φ : X → K(Y ) induce the (single-valued) hypermaps φ : K(X) → K(Y ) which are continuous only w.r.t. the upper-Vietoris topology, and subsequently the upper-Hausdorff topology, but not necessarily continuous w.r.t. the Hausdorff metric topology. On the other hand, if a compact-valued φ : X → K(Y ) is continuous w.r.t. a metric in X and the Hausdorff metric topology in K(Y ), then the induced hypermap is also continuous w.r.t. the Hausdorff metric topology. If (X, d1 ) and (Y, d2 ) are, in particular, compact metric spaces (by which (K(X), dH1 ) and (K(Y ), dH2 ) become compact as well, see Table 1), then φ : K(X) → K(Y ) with closed (= compact) values is continuous if and only if φ(clX (∪A∈K(X) A)) = clY (∪A∈K(X) φ(A)). Since continuous w.r.t. dH2 in C(Y ) multivalued maps φ : X → C(Y ) with closed but not necessarily compact values are only lower semicontinuous (for the definition, see below) in general (see e.g. [Andres& Górniewicz, 2003]), i.e. continuous only w.r.t. d1 in X and the lower-Vietoris topology in C(X), the induced hypermaps are obviously again not necessarily continuous w.r.t. dH1 and dH2 . Thus, in order to preserve continuity by induction from spaces to hyperspaces, its concept must be different here. Hence, a map φ : X → C(Y ) with closed values is said to be upper semicontinuous (u.s.c.) if, for every open U ⊂ Y , the set {x ∈ X|φ(x) ⊂ U } is open in X. It is said to be lower semicontinuous (l.s.c.) if, for every open U ⊂ Y , the set {x ∈ X|φ(x) ∩ U = ∅} is open in X. If it is both u.s.c. and l.s.c., then it is called continuous.

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Unlike for multivalued maps with noncompact values, for compact-valued multivalued maps, this continuity concept coincides with the continuity w.r.t. d1 in X and the Hausdorff metric topology induced by dH2 in K(Y ) (cf. e.g. [Andres& Górniewicz, 2003], [Hu & Papageorgiu, 1997]). Remark 2.

The following implications hold for continuous multivalued maps and the induced hypermaps (cf. [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3], [Hu & Papageorgiu, 1997]): φ : X → C(X) ⇒ φ : C(X) → C(X), Lipschitz φ : X → B(X) ⇒ φ : B(X) → B(X), φ : X → K(X) ⇒ φ : K(X) → K(X), where φ means that φ(A) :=

φ(x) = clX (

x∈A

φ(x)),

for all A ∈ C(X),

x∈A

but obviously φ : E → CCo (E) ⇒ φ : CCo (E) → CCo (E), where (E, ||.||) is a Banach space. Nevertheless, for special maps of the form φ0 : E n → CCo E n ), φ0 (x) := A x + C , · · × E and (E, ||.||) is a Banach space, we also where A is a real n × n−matrix, C ∈ CCo (E n ), E n = E × · have

n-times

φ0 : E n → CCo (E n ) ⇒ φ0 : CCo (E n ) → CCo (E n ). Let us note that although, in vector spaces, the linear combinations of convex sets are convex (see e.g. [Beer, 1993, Theorem 1.4.1]), they need not be closed (see e.g. [Aliprantis & Border, 1999, Example 5.3.]) which justifies to use φ0 instead of φ0 . On the other hand, in vector spaces, scalar multiples of closed sets are closed and the algebraic sum of a compact set and a closed set is closed (see e.g. [Aliprantis & Border, 1999, Lemma 5.2], [Dunford & Schwartz, 1958, Lemma V.2.3]). Thus, if C ∈ KCo (E n ), then the bar need not be used for φ0 , in order the last implication to be satisfied. Moreover, an affine function image of a convex set is convex (see e.g. [Beer, 1993, Theorem 1.4.1]). Since φ0 is Lipschitz continuous, in view of the above implications, we still get φ0 : E n → BCo (E n ) ⇒ φ0 : BCo (E n ) → BCo (E n ), φ0 : E n → KCo (E n ) ⇒ φ0 : KCo (E n ) → KCo (E n ), provided C ∈ BCo (E n ) and C ∈ KCo (E n ), respectively. If C ∈ KCo (E n ), then we can also simply write φ0 : E n → CCo (E n ) ⇒ φ0 : CCo (E n ) → CCo (E n ), φ0 : E n → BCo (E n ) ⇒ φ0 : BCo (E n ) → BCo (E n ). Now let us recall that a multivalued mapping φ : X → C(X) is said to be weakly contractive (cf. [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003]) if, for any x, y ∈ X, dH (φ(x), φ(y)) ≤ h(d(x, y)), where h : [0, ∞) → [0, ∞) is a continuous, nondecreasing function such that h(0) = 0 and 0 < h(t) < t, for t > 0. For h(t) := Lt, t ∈ [0, ∞), (⇒ L ∈ [0, 1)), the mapping φ is obviously a contraction. For weakly contractive single-valued maps φ : X → X, it is enough to replace dH by d. As pointed out in [Andres et al., 2005, Remark 1] and [Kirk & Sims, 2001, Remark on p. 8], the notion of a weak contraction can be weaken, namely that the function h need not be monotonic and it also suffices to take a right upper semicontinuous (in a single-valued sense) h. The equivalences for weakly contractive maps and hypermaps were proved in [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Proposition A 3.20], provided still limt→∞ t − h(t) = ∞, but this condition does not play any role.

Remark 3.

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7

The implications in Table 3 concerning the properties of hypermaps induced by those of maps were proved in [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3]. Because of the trivial reverse implications these properties are, in fact, equivalent. The last equivalences, for the map φ0 defined above, follow directly from the preceding ones, on the basis of the above arguments. map continuous φ : X → K(X) compact φ : X → K(X) (weakly) contractive φ : X → C(X) φ : X → B(X) φ : X → K(X) contraction φ0 : E n → CCo (E n ) φ0 : E n → BCo (E n ) φ0 : E n → KCo (E n )

hypermap continuous φ : K(X) → K(X) compact φ : K(X) → K(X) (weakly) contractive φ : C(X) → C(X) φ : B(X) → B(X) φ : K(X) → K(X) contraction φ0 : CCo (E n ) → CCo (E n ) φ0 : BCo (E n ) → BCo (E n ) φ0 : KCo (E n ) → KCo (E n )

references [Andres & Fiˇser, 2004], [Andres& G´ orniewicz, 2003, Prop. A 3.43] [Andres & Fiˇser, 2004], [Andres& G´ orniewicz, 2003, Prop. A 3.47] [Andres & Fiˇser, 2004], [Andres& G´ orniewicz, 2003, Prop. A 3.20] trivial consequences of the upper equivalences

As already pointed out, the bar can be omitted for φ0 in Table 3, provided C ∈ KCo (E n ) in the definition of φ0 . Remark 4.

Because of counter-examples (see e.g. [Le´sniak, 2005, Examples 1 and 2]), compact hypermaps φ and φ0 (even φ and φ0 or Lipschitz φ and Lipschitz φ0 ), on the hyperspaces C(X), B(X) and CCo (E n ), BCo (E n ), need not imply the compactness of the related maps φ and φ0 , on the spaces X and E n . Nevertheless compact maps φ and φ0 imply there the compact induced hypermaps φ and φ0 . Remark 5.

Remark 6. In view of the above arguments, we can still add one more assertion. Let X ⊂ E n be a nonempty,

compact, convex subset of E n such that φ0 : X → KCo (X). Since φ0 is (Lipschitz-)continuous, so must be φ0 : KCo (X) → KCo (X) as well, and vice versa. Observe that, in this case, the Lipschitz constant L need not necessarily satisfy L ∈ [0, 1). Let us supply a brief information about the related space of probability measures and the Markov operators acting on it. Let (X, d) be a metric space. Let dM K be the Monge–Kantorovich metric on the set P (X) of probability Borel measures on X, i.e.

f (x)dμ − f (x)dν , dM K (μ, ν) := sup f ∈L (X,R)

X

X

where L (X, R) := {f : X → R||f (x) − f (y)| ≤ d(x, y), for all x, y ∈ X}. It is well-known (see e.g. [Hutchinson, 1981], [Barnsley, 2003, Chapter 9]) that the space of probability measures (P (X), dM K ) is compact if and only if so is (X, d). An operator M : P (X) → P (X) is called the Markov operator 3 if it satisfies (1) M (λ1 μ + λ2 ν) = λ1 M (μ) + λ2 M (ν), for all λ1 , λ2 ∈ [0, ∞) and μ, ν ∈ P (X), (2) M (μ(X)) = μ(X), for all μ ∈ P (X). If there is still a dual operator U : C 0 (X) → C 0 (X), where C 0 (X) denotes the space of continuous functions f : X → R endowed with the sup-norm on X, such that 3

In fact, the Markov operators act on the space of finite Borel measures on X, but we are interested in those transforming them into probability measures on the σ-algebra B(X) of Borel subsets of X (cf. e.g. [Lasota & Myjak, 1998], [Lasota & Myjak, 1999]).

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U f (x)dμ = operator. X

X

f (x)M dμ, for every f ∈ C 0 (X) and μ ∈ P (X), then M is called the Markov–Feller

It can be proved that every nonexpansive Markov operator is Markov–Feller. For more details and properties of Markov operators, see e.g. [Lasota & Myjak, 1998]–[Lasota & Myjak, 2002], [Myjak & Szarek, 2003]. Finally, we can define the hyperspaces (H1 (H2 (X)), dHH ) as certain classes of nonempty subsets of H2 (X), endowed with the Hausdorff metric dHH , induced by the metric dH , and (P (H(X)), dM KH ) as the space of probability measures on H(X), endowed with the Monge–Kantorovich metric dM KH defined as follows

sup f (x)dμ − f (x)dν , dM KH (μ, ν) := f ∈L (H(X),R)

H(X)

H(X)

where L (H(X), R) := {f : H(X) → R||f (x) − f (y)| ≤ dH (x, y), for all x, y ∈ H(X)}. Obviously, many properties of these new spaces can be directly induced from the supporting spaces. For instance, (P (K(X)), dM KH ) and (K(KCo (X)), dHH ) are compact if and only if so is (X, d). Similarly many properties of hypermaps on these new hyperspaces can be induced by those of maps on the supporting spaces. For instance, φ : K(K(X)) → K(K(X)) is compact continuous if and only if so is φ : X → K(X) or φ : K(X) → K(K(X)) and φ0 : K(KCo (E n )) → K(KCo (E n )) is a contraction if and only if so is φ0 : X → KCo (E n ) or φ0 : KCo (E n ) → K(KCo (E n )).

3. Fixed point theory in hyperspaces The fixed point theory in spaces is one of the mostly developed parts of nonlinear analysis. For the metric (Banach-like) theory, see e.g. the handbook [Kirk & Sims, 2001] and for the topological (Schauder-like) theory, see e.g. the handbook [Brown et al., 2005]. On the other hand, the results concerning the fixed point theory in hyperspaces are rather rare (cf. [Andres, 2001], [Andres & Väth, 2007], [De Blasi, 2006], [De Blasi, 2010], [De Blasi & Georgiev, 2001], [Hagopian, 1975], [Hu & Fang, 2007], [Hu & Huang, 1997], [Illanes & Nadler, 1999, Chapter VI], [Liu et al., 2007], [Nadler, 1987], [Rahmat & Noorami, 2006], [Ruiz del Portal & Salazar, 2001] and [Segal, 1962]). Everybody knows Banach’s (see e.g. [Granas & Dugundji, 2003, Theorem 1.1]) and Schauder’s (see e.g. [Granas & Dugundji, 2003, Theorem 3.2]) fixed point theorems. In applications, we will also need their generalizations. The following generalization is a particular case of the Boyd–Wong version of the Banach theorem (see e.g. [Kirk & Sims, 2001, Theorem 3.2 on pp. 7–8]). Lemma 1 [Boyd–Wong]. Let (X, d) be a complete metric space and f : X → X be a weakly contractive map. Then f has exactly one fixed point. The Covitz–Nadler multivalued version of the Banach theorem (see e.g. [Granas & Dugundji, 2003, Theorem 3.1 on p. 28], [Kirk & Sims, 2001, Theorem 5.1 on pp. 15–16]) reads as follows. Lemma 2 [Covitz–Nadler]. Let (X, d) be a complete metric space and φ : X → B(X) be a contraction. Then φ admits a fixed point, i.e. there exists x0 ∈ X such that x0 ∈ φ(x0 ). The following Granas version of the Lefschetz fixed point theorem (see e.g. [Granas & Dugundji, 2003, Theorem 4.3 on p. 425]) is a far reaching generalization of the Schauder theorem. Lemma 3 [Granas]. Let X be an ANR-space and f : X → X be a compact map. Then the generalized Lefschetz number Λ(f ) is defined and if Λ(f ) = 0, then f has a fixed point. In particular, if X is an AR-space, then Λ(f ) = 1, and subsequently a compact map f : X → X has a fixed point.

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Since the space of probability measures (P (X), dM K ) is compact if and only if so is (X, d), as a direct consequence of the Banach fixed point theorem, we can immediately give the following lemma. Lemma 4. Let (X, d) be a compact metric space and (P (X), dM K ) be the space of probability Borel measures

on X. If the Markov operator M : P (X) → P (X) is a contraction with a constant L ∈ [0, 1), i.e. dM K (M (μ), M (ν)) ≤ LdM K (μ, ν),

for all μ, ν ∈ P (X),

then there exists a unique fixed point μ0 ∈ P (X), μ0 = M (μ0 ), called the invariant measure w.r.t. M . The first statement for hypermaps is a slight improvement of its analogy in [Andres & Fiˇser, 2004] (see also [Andres& Górniewicz, 2003, Appendix 3]) in the sense of Remark 3. Proposition 1. Let (X, d) be a complete metric space and φ1 : X → C(X), φ2 : X → B(X), φ3 : X → K(X)

be (weak) contractions. Then each hypermap φ1 : C(X) → C(X), φ2 : B(X) → B(X), φ3 : K(X) → K(X) has exactly one fixed point xi , i = 1, 2, 3. Moreover, each of multivalued maps φ1 , φ2 , φ3 with bounded values possesses fixed points in x1 , x2 , x3 ⊂ X. Proof. By the above arguments, (C(X, dH )), (B(X, dH )), (K(X, dH )) are complete hyperspaces (see Table 1 and Table 2) and the induced (single-valued) hypermaps φ1 , φ2 , φ3 are self-maps. Moreover, φ1 , φ2 , φ3 are (weak) contractions (see Table 3). Hence, applying Lemma 1 resp. Banach’s theorem, they have exactly one fixed point xi , i = 1, 2, 3, representing, up to its boundary, positively invariant subset in X. Since x1 , x2 , x3 are closed subsets of a complete space X, they are also complete. Applying Lemma 2, the multivalued maps φ1 |x1 , φ2 |x2 , φ3 |x3 with bounded values possess fixed points in x1 , x2 , x3 ⊂ X.

By the same arguments, we can give the second metric statement. Proposition 2. Let (E, ||.||) be a Banach space. Assume that |||A ||| < 1, for the matrix norm of A , and

C1 ∈ C(E n ), C2 ∈ B(E n ), C3 ∈ K(E n ), at the affine maps φ01 : E n → CCo (E n ), φ02 : E n → BCo (E n ), φ03 : E n → KCo (E n ). Then each hypermap φ01 : CCo (E n ) → CCo (E n ), φ02 : BCo (E n ) → BCo (E n ), φ03 : KCo (E n ) → KCo (E n ) has exactly one fixed point xi , i = 1, 2, 3. Moreover, each of multivalued maps φ01 , φ02 , φ03 with bounded values admits a fixed point in xi ⊂ E n , i = 1, 2, 3. By the above arguments, (CCo (E n ), dH ), (BCo (E n ), dH ), (KCo (E n ), dH ) are complete hyperspaces. One can readily check that the multivalued maps φ01 , φ02 , φ03 are contractions. Thus, the induced (singlevalued) hypermaps φ01 , φ02 , φ03 , which are by the above arguments self-maps, must be contractions as well (see Table 3). Hence, applying the Banach fixed point theorem, they have exactly one fixed point xi , i = 1, 2, 3, representing, up to its boundary, a positively invariant subset in E n . Since x1 , x2 , x3 are closed subsets of a Banach space E n , they are also complete. Applying Lemma 2, the multivalued maps φ01 |x1 , φ02 |x2 , φ03 |x3 with closed bounded values possess fixed points in x1 , x2 , x3 ⊂ E n . Proof.

Remark 7. Propositions 1 and 2 can be naturally extended to suitable hyper-hyperspaces H1 (H2 (X)), when considering the multivalued (weak) contractions φ : H2 (X) → H1 (H2 (X)). On the other hand, if we consider multivalued (weak) contractions on the supporting space X, as in Propositions 1 and 2, then the unique fixed points of the induced hyper-hypermaps φ : H(H(X)) → H(H(X)) must be the same as those of hypermaps φ : H(X) → H(X). Remark 8. Condition |||A ||| < 1 in Proposition 2 is certainly not necessary. Let, for instance, E = R and C ∈ Rn . Then the map φ0 : Rn → Rn has a unique fixed point x0 ∈ Rn if and only if (A − I) is regular, i.e. 1 ∈ / σ(A ). The induced hypermap φ0 : K(Rn ) → K(Rn ) has exactly the same unique fixed point which can be explicitly calculated as a solution of the algebraic system (A − I)x = −C .

The first topological statement in hyperspaces generalizes its analogy in [Andres & Fiˇser, 2004] (cf. also [Andres& Górniewicz, 2003, Appendix 3]). Proposition 3. Let (X, d) be a locally continuum-connected metric space and φ : X → K(X) be a compact

continuous mapping. Then the induced hypermap φ : K(X) → K(X) admits a fixed point.

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Proof. If (X, d) is still connected, then (K(X), dH ) is an AR (see Table 1). The induced hypermap φ : K(X) → K(X) is compact and continuous as well (see Table 3). Thus, applying Lemma 2, there is a fixed point. If (X, d) is disconnected then, unlike in the supporting space, in the hyperspace (K(X), dH ) which is an ANR (see Table 1), K(X) consists of a finite number of disjoint ARs, and subsequently Λ(φ) ≥ 1 (for more details, see [Andres & Väth, 2007]). Applying Lemma 2, φ admits also in this case a fixed point.

Since in (K(X), dH ), where (X, d) is locally continuum-connected, we get even N (φ) = Λ(φ) ≥ 1, where N (φ) denotes the Nielsen number for the lower estimate of fixed points of φ, we have in fact to our disposal a multiplicity result. The problem of calculation of N (φ) namely reduces there to a simple combinatorical situation on a finite set (see [Andres & Väth, 2007]).

Remark 9.

Remark 10. Propositions 1 and 3 can be extended, by means of degree arguments, to hyperspace continuation principles, where the induced hypermaps and hyperhomotopies need not be self-maps (see [Andres, 2004], [Andres, 2004], [Andres et al., 2005], [Ruiz del Portal & Salazar, 2001]). There also exists a fixed point theorem for condensing hypermaps (cf. [Le´sniak, 2005]), but the nontrivial induction of condesity seems to be a difficult task. On the other hand, there is no direct way for obtaining the hyperspace analogy of the Schauder fixed point theorem, because the hyperspaces of linear spaces have never a linear structure. Despite it, using the R˚ adström theorem (see [R˚ adstr¨ om, 1952], [De Blasi, 2010]) which allows us an embedding of the hyperspace as a positively semilinear subspace, some analogies of the Schauder theorem in terms of the Hausdorff topology were obtained in [De Blasi, 2006], [De Blasi, 2010], [De Blasi & Georgiev, 2001].

We already know from Section 2 that (B(X), dH ) is an AR, provided e.g. X ⊂ E is a convex subset of a normed space (E, ||.||) (cf. [Antosiewicz & Cellina, 1977]). In general, it is according to [Banakh & Voytsitskyy, 2007, Theorem 3.5] an AR if and only if a metric space (X, d) is uniformly locally chain equi-connected on each bounded subset of X and that each bounded subset of X lies in a bounded chain equi-connected subspace of X. For the definitions and more details, see [Banakh & Voytsitskyy, 2007]. Proposition 3 can be, therefore, also partially extended to multivalued compact continuous maps φ : X → B(X). Analogous criteria can be also found, in view of [Banakh & Voytsitskyy, 2007, Theorem 3.2], for multivalued compact continuous maps φ : X → C(X) and their inductions on (C(X), dH ). Remark 11.

Since (KCo (X), dH ) is, according to [Hu & Huang, 1997], [Liu et al., 2007] (cf. also [Nadler et al., 1975]– [Nadler et al., 1979]), a compact convex subset of (K(X), dH ), provided X ⊂ E n is a compact, convex subset of a Banach space E n , the following statement can be also regarded as a particular hyperspace version of the Schauder-type theorem. Proposition 4. Let (E n , ||.||) be a Banach space and X ⊂ E n be a nonempty, compact, convex, subset of

E n . Let C ∈ KCo (X) be at the affine map φ0 : X → KCo (X), defined in the foregoing section. Then KCo (X) the induced hypermap φ0 : → KCo (X) admits a fixed point x0 . If still |||A ||| < 1 holds, for the matrix norm of A at φ0 , then the fixed point x0 is unique. Moreover, the multivalued map φ0 : X → KCo (X) admits a fixed point in a convex, compact, positively invariant subset x0 ⊂ X ⊂ E n . Since (KCo (X), dH ) is convex and compact, it must be also a compact AR. By the above arguments (see Remark 5), the induced hypermap φ0 is a Lipschitz-continuous self-map, i.e. φ0 : KCo (X) → KCo (X), which is compact. Applying Lemma 2, there is a fixed point x0 ∈ KCo (X). For |||A ||| < 1, one can alternatively apply the Banach fixed point theorem to obtain the uniqueness of x0 . The point x0 ∈ KCo (X) is at the same time a convex, compact, positively invariant subset of X ⊂ E n such that φ0 |x0 : x0 → KCo (x0 ) is Lipschitz-continuous. Applying a suitable Kakutani-type fixed point theorem whose all assumptions are satisfied (see e.g. [Granas & Dugundji, 2003, Theorem 8.4 on pp. 168–169]), the multivalued map φ0 has a fixed point in x0 ⊂ X ⊂ E n . Proof.

Remark 12. Similarly as for metric statements, Proposition 3 can be naturally extended to the hyperspace K(K(X)), when considering the multivalued compact continuous maps φ : K(X) → K(K(X)),

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11

provided (X, d) is a locally continuum-connected metric space. Proposition 4 can be extended to the hyper-hyperspace K(KCo (X)), when considering the multivalued affine maps φ0 : KCo (X) → K(KCo (X)), provided X ⊂ E n is a nonempty, compact, convex subset of a Banach space E n .

4. Multivalued fractals and hyperfractals In our approach to fractals, we follow the classical ideas of J. E. Hutchinson [Hutchinson, 1981] and M. F. Barnsley [Barnsley, 2003] concerning the iterated function systems (IFSs) {(X, d), Fi : X → X, i = 1, 2, . . . n}, where (X, d) is a complete metric space and Fi , i = 1, 2, . . . n, are contractions. The prehistory of this approach can be already detected in the paper [Williams, 1971] of R. F. Williams. Replacing single-valued contractions Fi by multivalued ones, we speak about iterated multifunction systems (IMSs). This name was used for the first time in our paper [Andres& Górniewicz, 2001]. In [Andres, 2001], we also used for the first time the term multivalued fractals for the attractors of IMSs. Later on, in [Andres & Fiˇser, 2004], [Andres et al., 2005], [Andres& Górniewicz, 2003, Appendix 3], we extended this notion to fixed points in hyperspaces of the related Hutchinson–Barnsley operators determined by multivalued maps. We also distinguished there between metric and topological multivalued fractals, according to the applied metric (Banach-like) and topological (Schauder-like) fixed point theorems. Of course, because of identical images of sets, every set would be a topological fractal, but we always implicitly assumed that there are at least two maps in the generating systems under consideration. Let us note that this terminology seems to be nowadays standard (cf. e.g. [Boriceanu et al., 2010], [Colapinto & La Torre, 2008], [Chifu & Petru¸sel, 2008], [Fiˇser, 2004], [Kunze et al., 2007], [Kunze et al., 2008], [Mihail, 2008].) Multivalued fractals were considered for the first time in 2001 in [Andres, 2001], [Andres& G´ orniewicz, 2001] and, independently, by A. Petru¸sel and I. A. Rus in [Petru¸sel, 2001], [Petru¸sel & Rus, 2001]. At the same time, similar ideas were also implicitly present in the papers [Lasota & Myjak, 2000], [Lasota & Myjak, 2002] of A. Lasota and J. Myjak, where an alternative approach was used leading to the notion of semifractals (cf. also [Lasota & Myjak, 1998], [Lasota & Myjak, 1999], [Myjak & Szarek, 2003] and the references therein). In Introduction, we already indicated the relationship between hyperfractals, defined as fixed points in hyper-hyperspaces of the Hutchinson–Barnsley hyperoperators and determined by a rather general class of multivalued maps, to superfractals (cf. [Barnsley, 2006], [Barnsley et al., 2005]–[Barnsley et al., 2003], [Singh et al., 2009]) and to multivalued fractals. We also would like to clarify in this section the relationship between hyperfractals and invariant measures, defined on hyperspaces, of the Markov hyperoperators. Hence, both multivalued fractals and hyperfractals will be investigated here in terms of the fixed point theory from the foregoing section. We start with a theorem for metric multivalued fractals. Theorem 1. Let (X, d) be a complete metric space and F1i : X → C(X), F2i : X → B(X), F3i : X → K(X)

be, for all i = 1, 2, . . . , n, weak contractions. Then each Hutchinson–Barnsley operator ⎫ φ1 : C(X) → C(X), φ1 (A) := ni=1 clC(X) ( x∈A F1i (x)), ⎪ ⎪ ⎪ ⎬ n φ2 : B(X) → B(X), φ2 (A) := i=1 clB(X) ( x∈A F2i (x)), ⎪ ⎪ ⎪ n ⎭ φ3 : K(X) → K(X), φ3 (A) := i=1 x∈A F3i (x)

(2)

has exactly one fixed point Aj , j = 1, 2, 3, which is at the same time a positively invariant (for j = 1, 2, up to its boundary) set w.r.t. the related Hutchinson–Barnsley maps ⎫ F1 : X → C(X), F1 (x) := ni=1 F1i (x)), ⎪ ⎪ ⎬ n F2 : X → B(X), F2 (x) := i=1 F2i (x)), (3) ⎪ ⎪ n ⎭ F3 : X → K(X), F3 (x) := i=1 F3i (x)). Moreover, each of the maps F1 , F2 , F3 possesses fixed points in A1 , A2 , A3 ⊂ X, provided F1i have bounded values, for all i = 1, 2, . . . , n.

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Since a finite union of closed sets is closed, of bounded sets is bounded and of compact sets is compact, we can define the maps F1 , F2 , F3 as in (3). Since the operators φ1 , φ2 , φ3 in (2) can be equivalently defined as φ1 (A) := clC(X) ( F1 (x)), φ2 (A) := clB(X) ( F2 (x)), Proof.

x∈A

φ3 (A) :=

x∈A

F3 (x),

x∈A

they have the same properties as the induced (single-valued) hypermaps φ1 , φ2 , φ3 in Proposition 1. Hence, the application of Proposition 1 completes the proof. Remark 13. Because of a weaker notion of a weak contractivity (see Remark 3), the first part of Theorem 1 is slightly more general than its analogies in [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3]. It is also a generalization for φ3 , with multivalued contractions F3i , of the analogous results in [Andres& G´ orniewicz, 2001], [Kunze et al., 2007], [Petru¸sel, 2001], [Petru¸sel & Rus, 2001] and for φ3 , with singlevalued contractions F3i , of the classical results in [Hutchinson, 1981], [Barnsley, 2003]. Since the completeness of (X, d) implies the one of (C(X), dH ), (B(X), dH ), (K(X), dH ), and subsequently of (C(C(X)), dHH ), (B(B(X)), dHH ), (K(K(X)), dHH ), and since weak contractions are induced on these spaces, the following statement can be regarded as a corollary of Theorem 1 (cf. Remark 7). Corollary 1. Let (X, d) be a complete metric space and F1i : X → C(X), F2i : X → B(X), F3i : X → K(X)

be, for all i = 1, 2, . . . , n, weak contractions. Then each Hutchinson–Barnsley hyperoperator φ1 : C(C(X)) → C(C(X)), φ1 : B(B(X)) → B(B(X)),

φ1 (A) := φ2 (A) :=

φ3 : K(K(X)) → K(K((X)), φ3 (A) :=

n

clC(X) (

i=1 y∈A n i=1 y∈A n

F1i (x)),

x∈y

clB(X) (

F2i (x)),

x∈y

F3i (x)

i=1 y∈A x∈y

has exactly one fixed point Aj , j = 1, 2, 3, which is at the same time a positively invariant (for j = 1, 2, up to its boundary) set w.r.t. the related Hutchinson–Barnsley hypermaps F1 : C(X) → C(C(X)), F1 (y) := F2 : B(X) → B(B(X)), F2 (y) := F3 : K(X) → K(K(X)), F3 (y) :=

n i=1 n i=1 n

clC(X) (

F1i (x)),

x∈y

clB(X) (

F2i (x)), .

x∈y

F3i (x).

i=1 x∈y

Moreover, each of the hypermaps F1 , F2 , F3 possesses fixed points in A1 ⊂ C(X), A2 ⊂ B(X), A3 ⊂ K(X), provided F1i have bounded values, for all i = 1, 2, . . . , n. Corollary 1 is a generalization for φ3 , with special multivalued contractions F3i , of Theorem 8 in [Kunze et al., 2008]. Remark 14.

Since the union of convex sets need not be convex, Proposition 2 cannot be applied as Proposition 1 above. Despite this impossibility, Corollary 1 can be still specified as follows.

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13

Theorem 2. Let (E, ||.||) be a Banach space. Consider the affine maps

F1i : E m → CCo (E m ), F1i (x) := A1i x + C1i , C1i ∈ CCo (E m ), F2i : E m → BCo (E m ), F2i (x) := A2i x + C2i , C2i ∈ BCo (E m ), F3i : E m → KCo (E m ), F3i (x) := A3i x + C3i , C3i ∈ KCo (E m ), where Aji , j = 1, 2, 3, i = 1, 2, . . . , n, are real m × m-matrices. If |||Aij ||| < 1 holds for the matrix norms of Aji , j = 1, 2, 3, i = 1, 2, . . . , n, then each Hutchinson–Barnsley hyperoperator ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x∈y ⎪ i=1 y∈A ⎪ ⎪ ⎪ ⎪ m m B(BCo (E )) → B(BCo (E )), ⎪ ⎪ ⎬ n clBCo (E m ) ( F1i (x)), ⎪ ⎪ ⎪ ⎪ x∈y i=1 y∈A ⎪ ⎪ ⎪ ⎪ m m K(KCo (E )) → K(KCo (E )), ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ F3i (x) ⎪ ⎭

φ01 : C(CCo (E m )) → C(CCo (E m )), n φ01 (A) := clCCo (E m ) ( F1i (x)), φ02 : φ02 (A) := φ03 : φ03 (A) :=

(4)

i=1 y∈A x∈y

has exactly one fixed point Aj , j = 1, 2, 3. Since (E m , ||.||) is a Banach space, the hyperspaces (CCo (E m ), dH ), (BCo (E m ), dH ), (KCo (E m ), dH ), are, by the above arguments, complete as well as the hyper-hyperspaces (C(CCo (E m )), dHH ), (B(BCo (E m )), dHH ), (K(KCo (E m )), dHH ) (cf. Tables 1 and 2). Since the affine multivalued maps Fji are, for |||Aji ||| < 1, obviously contractions, so are the induced (single-valued) hypermaps clCCo (E n ) F1i , clBCo (E n ) F2i , F3i , i = 1, 2, . . . , n, (cf. Table 3). Furthermore, since a finite union of these contractions in (4) is a contraction (cf. [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3]), the hyperoperators in (4) must be also contractions (cf. Table 3). Thus, these hyperoperators in (4) have the same properties as the induced (single-valued) hypermaps φ1 , φ2 , φ3 in Proposition 1 which completes the proof. Proof.

Remark 15. The unique fixed points Aj , j = 1, 2, 3, can be only regarded as closed or bounded, closed or compact subsets of closed, convex or bounded, closed, convex or compact, convex subsets of E m which are positively invariant (for j = 1, 2, up to their boundaries) w.r.t. the related Hutchinson–Barnsley hypermaps F01 : C(E m ) → C(C(E n )), F01 (y) := ni=1 clC(E n ) ( x∈y F1i (x)), F02 : B(E m ) → B(B(E n )), F02 (y) := ni=1 clB(E n ) ( x∈y F2i (x)), . F03 : K(E m ) → K(K(E n )), F03 (y) := ni=1 x∈y F3i (x).

Moreover, each of the hypermaps F01 , F02 , F03 possesses fixed points in A1 ⊂ C(E m ), A2 ⊂ B(E m ), A3 ⊂ K(E m ), provided C1i ∈ BCo (E m ), for all i = 1, 2, . . . , n. Proposition 3 can be applied for obtaining the following topological result. Theorem 3. Let (X, d) be a locally continuum-connected metric space and Fi : X → K(X) be, for all

i = 1, 2, . . . , n, compact continuous mappings. Then the Hutchinson–Barnsley operator φ : K(X) → K(X), φ(A) :=

n i=1 x∈A

Fi (x)

(5)

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admits a fixed point which is at the same time a positively invariant set w.r.t. the Hutchinson–Barnsley map F : X → K(X), F (x) :=

n

Fi (x).

(6)

i=1

Since a finite union of compact sets is compact, we can define the map F as in (6). Since the operator φ in (5) can be equivalently defined as F (x), φ(A) :=

Proof.

x∈A

it has the same properties as the induced (single-valued) hypermap φ in Proposition 3. Hence, the application of Proposition 3 completes the proof. Since (X, d) can be disconnected, it generalizes its analogies in [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3]. In view of Remark 9, we can even obtain in an extremely simple way the lower estimate of the number of fixed points of φ in (5). On the other hand, since e.g. (C(R), dH ) is, according to [Banakh & Voytsitskyy, 2007], only an ANR, but not an AR, it is a difficult task to find sufficient conditions in order (C(C(X)), dHH ) or (B(B(X)), dHH ) to be ARs. Thus, it seems to be also difficult to extended Theorem 3 to the Hutchinson-Barnsley operators on (C(X), dH ) and (B(X), dH ). Remark 16.

Since the compactness of (X, d) implies the one of (K(X), dH ), and subsequently of (K(K(X)), dHH ), and since a continuity is induced on these spaces, the following statement can be regarded as a corollary of Theorem 3 (cf. Remark 7). Corollary 2. Let (X, d) be a locally continuum-connected metric space and Fi : X → K(X) be, for all

i = 1, 2, . . . , n, compact continuous mappings. Then the Huchinson-Barnsley hyperoperator φ : K(K(X)) → K(K(X)), φ(A) :=

n

Fi (x)

i=1 y∈A x∈y

admits a fixed point which is at the same time a positively invariant set w.r.t. the Hutchinson–Barnsley hypermap F : K(X) → K(K(X)), F (y) :=

n

Fi (x).

i=1 x∈y

Despite the impossible application of Proposition 4, Corollary 2 can be still specified as follows. Theorem 4. Let (E, ||.||) be a Banach space and X ∈ E m be a nonempty, convex, compact subset of E m .

Consider the affine maps F0i := X → KCo (X), F0i (x) := Ai x + Ci , where Ai are real m × m-matrices and Ci ∈ KCo (X), for all i = 1, 2, . . . , n. The Hutchinson–Barnsley hyperoperator φ0 : K(KCo (X)) → K(KCo (X)), φ0 (A) :=

n

F0i (x)

(7)

i=1 y∈A x∈y

admits always a fixed point A0 . If still |||Ai ||| < 1 holds for the matrix norms of Ai , i = 1, 2, . . . , n, then the fixed point A0 is unique. Since X ⊂ E m is convex and compact, so is by the above arguments (cf. [Hu & Huang, 1997], [Liu et al., 2007], [Nadler et al., 1975]–[Nadler et al., 1977]) the hyperspace (KCo (X), dH ), by which Proof.

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15

the hyper-hyperspace (K(KCo (X)), dHH ) is a compact AR (cf. Table 1). Furthermore, since the multivalued affine maps F0i are evidently (Lipschitz-) continuous, so are the induced (single-valued) hypermaps F0i : KCo (X) → KCo (X) (cf. Remark 6). Moreover, since a finite union of these hypermaps is also (Lipschitz-) continuous (cf. [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3]), the induced Hutchinson–Barnsley hyperoperator in (7) must be (Lipschitz-) continuous as well (cf. Table 3). Thus, the hyperoperator in (7) has the same properties as the induced (single-valued) hypermap φ in Proposition 3 which completes the first (topological) part of the proof. For |||Ai ||| < 1, the multivalued affine maps F0i are obviously contractions, for all i = 1, 2, . . . , n, and so are the induced (single-valued) hypermaps (cf. Table 3). Furthermore, since a finite union of these contractions in (7) is a contraction (cf. [Andres & Fiˇser, 2004], [Andres& Górniewicz, 2003, Appendix 3]), so must also be the hyperoperator φ0 in (7) (cf. Table 3) which has in this way the same properties as the induced (single-valued) hypermap φ3 in Proposition 1. This completes the second (metric) part of the proof. Remark 17. Similarly as in Remark 14, the fixed points A0 can be only regarded as compact subsets of compact, convex subsets of X ⊂ E m which are positively invariant w.r.t. the related Hutchinson–Barnsley hypermap

F0 : K(X) → K(K(X)), F0 (y) :=

n

Foi (x).

i=1 x∈y

Moreover, for |||Ai ||| < 1, i = 1, 2, . . . , n, the hypermap F0 possesses fixed points in A0 ⊂ K(X). Observe that, unlike in Theorem 3, here the matrices Ai , i = 1, 2, . . . , n, can be without restrictions. Now, let (X, d) be a compact metric space and (P (X), dM K ) be the space of probability Borel measures on X. Let, for all i = 1, 2, . . . , n, Fi : X → X be contractions with factors Li ∈ [0, 1) and Pi : X → [0, 1] be the associated continuous probability functions such that ni=1 Pi (x) = 1, for all x ∈ X. We can define the Markov–Feller operators as follows (cf. e.g. [Lasota & Myjak, 1998]–[Lasota & Myjak, 2002],[Myjak & Szarek, 2003]): n

Pi (x)dμ(x), (8) M : P (X) → P (X), M (μ)(A) := i=1

Fi−1 (A)

for all μ ∈ P (X) and A ∈ B(X), where B(X) denotes the σ-algebra of Borel subsets of X. In a particular case of constant probabilities Pi (x) ≡ Pi , i = 1, 2, . . . , n, the formula (8) obviously simplifies into M (μ)(A) :=

n

Pi μ(Fi−1 (A)), μ ∈ P (X), A ∈ B(X).

(9)

i=1

It can be proved (see e.g. [Barnsley, 2003, Theorem 6.1 on p.351]) that, under the above assumptions, M defined by (9) is a contraction, i.e. dM K (M (μ), M (ν)) ≤ LdM K (μ, ν), for all μ, ν ∈ P (X), where (1 >)L := maxi=1,2,...,n {Li }. Thus, applying Lemma 4, there exists a unique fixed point μ0 ∈ P (X), μ0 = M (μ0 ), called the invariant measure w.r.t. M . In this light, for compact metric spaces, a particular case of contractions in Corollary 1 can be extended as follows. Theorem 5. Let (X, d) be a compact metric space and Fi : X → K(X) be Lipschitz-continuous multivalued

. . . , n (like e.g. F maps with factors Li ≥ 0, for i = 1, 2, 3i , i = 1, 2, . . . , n, in Theorem 2). Let Pi ∈ (0, 1] be the associated probabilities such that ni=1 Pi = 1 and ni=1 Li Pi < 1. Then the Markov hyperoperator M : P (K(X)) → P (K(X)), M (μ)(A) :=

n i=1

Pi μ(Fi−1 (A)),

(10)

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for all μ ∈ P (K(X)) and A ∈ B(K(X)), where B(K(X)) denotes the σ-algebra of Borel subsets of K(X), has exactly one fixed point μ0 ∈ P (K(X)) such that supp(μ0 ) := {x ∈ K(X)|μ0 (B(x, r)) > 0, for every r > 0} is the smallest positively invariant set w.r.t. the Hutchinson–Barnsley hypermap F :=

n

Fi : K(X) → K(K(X)).

(11)

i=1

In view of Table 1, (K(X), d) and (K(K(X)), dH ) are compact. Moreover, by the above arguments, Lipschitz-continuos multivalued maps induce (single-valued) Lipschitz hypermaps with the same factors nL Li ≥ 0, i = 1, 2, . . . , n. Therefore, if i Pi < 1, then the Markov hyperoperator M in (10), on i=1 the hyperspace (P (K(X)), dM KH ), has exactly one fixed point μ0 ∈ P (K(X)) such that supp(μ0 ) is a semiattractor of the hyperIFS {(K(X), dH ), Fi : K(X) → K(X), i = 1, 2, . . . , n} (see [Lasota & Myjak, 1998, Theorem 3.1], [Myjak & Szarek, 2003, Fact 3.2 and Corollary 5.6]). At the same time, it is the smallest positively invariant set w.r.t. the hypermap F in (11) (see e.g. [Lasota & Myjak, 1998, Theorem 2.1], [Myjak & Szarek, 2003, Theorem 5.2]). Proof.

Observe that the factors Li in Theorem 5 can be greater than 1, for some i = 1, 2, . . . , n. On the other hand, for non-unique positively invariant sets A ⊂ K(X) w.r.t. F in (11), we only have that supp(μ0 ) ⊂ A, but not the equality, as for a uniqueness. Nevertheless, the relationship between invariant measures and topological hyperfractals can be clarified in this way. More precisely, we know from Corollary 2 that if a compact X is still locally connected, then there is always a positively invariant set A ⊂ K(X) w.r.t. F in (11). Now, we also know that, under the assumptions of Theorem 5, supp(μ0 ) ⊂ A. Moreover, in the case of uniqueness, we have that supp(μ0 ) = A. Remark 18.

In order to avoid the handicap mentioned in Remark 18, we can give the following corollary of Theorem 5 which already concerns a unique positively invariant set w.r.t. F in (11). Corollary 3. Let (X, d) be a compact space and Fi : X → K(X) be, for all i = 1, 2, . . . , n, weak contrac-

tions. Moreover, let at least one Fi , say F1 , be a contraction with factor L1 < 1. Let Pi ∈ (0, 1] be the associated probabilities such that ni=1 Pi = 1. Let (P (K(X)), dM KH ) be the hyperspace of probability Borel measures on (K(X), dH ). Then the Markov–Feller hyperoperator M : P (K(X)) → P (K(X)), which takes the same form as in (10), has exactly one fixed point μ0 ∈ P (K(X)), called the invariant measure w.r.t. the hyperoperator M in (10), such that supp(μ0 ) = A3 , where A3 comes from Corollary 1. Since compact-valued weak contractions are, by definition, nonexpansive, the induced (singlevalued) maps contractive (see Table 3), and subsequently nonexpansive. Thus, we always n n must be weakly a contraction with a constant L1 < 1, so must be the have that i=1 Li Pi ≤ i=1 Pi = 1. Since F1 is still induced (single-valued) map (cf. Table 3), by which ni=1 Li Pi < 1. Applying Theorem 5, there is a unique invariant measure μ0 of the related Markov hyperoperator M whose support is the smallest positively invariant set w.r.t. the Hutchinson–Barnsley hypermap F in (11). Since this set is according to Corollary 1 unique, we have that supp(μ0 ) = A3 , where A3 comes from Corollary 1, as claimed. Proof.

Corollary 3 is a generalization of its analogy in [Kunze et al., 2007], [Kunze et al., 2008], where all Fi , i = 1, 2, . . . n, were strict special contractions. In this case, the proof can be done in a more straightforward way by means of Lemma 4 and multivalued contractions with compact values need not be of a special type as in [Kunze et al., 2007], [Kunze et al., 2008]. Remark 19.

Remark 20. In [Lasota & Myjak, 1998], [Myjak & Szarek, 2003] (cf. also the references therein), the authors

considered, for the invariant measures, the supporting Polish space (X, d), the hyperspace (C(X), dH ) and the space (P (X), dM K ) of probability Borel measures on X. Although completeness is implied by (X, d) to (C(X), dH ) (cf. Table 2), separability is not preserved in this way, by the arguments explained in Section 2. Thus, we could not directly extend Theorem 5 and Corollary 3, when just replacing a compact (X, d) by a Polish space. Moreover, since the supports of invariant measures in [Lasota & Myjak, 1998], [Myjak & Szarek, 2003] can be noncompact, it follows that, in Theorem 5 and Corollary 3, a compact (X, d) cannot

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17

be replaced by a Polish space, when considering the hyperspace (K(X), dH ). On the other hand, Theorem 5 and Corollary 3 can be generalized, when replacing (K(K(X)), dHH ) by (C(K(X)), dHH ), where (X, d) is a Polish space, and x∈A F by clC(K(X)) x∈A F , for each A ∈ (C(K(X)), dHH ), where F is defined in (11). In Corollary 3, A3 is still replaced by A1 from Corollary 1. The metric part of Theorem 4 can be generalized in a probabilistic way as follows. Theorem 6. Let (E m , ||.||) be a real Banach space and X ⊂ E m be a nonempty, convex, compact subset of

E m . Consider the affine maps

F0i : X → KCo (X), F01 (x) := Ai x + Ci , where Ai are real m × m-matrices and Ci ∈ KCo (X), for all i = 1, 2, . . . , n. Let Pi ∈ [0, 1], i = 1, 2, . . . , n, be the associated probabilities such that n

Pi = 1

i=1

and P (KCo (X), dM KH ) be the hyperspace of probability Borel measures on (KCo (X), dH ). If |||Ai ||| < 1 holds, for the matrix norms of Ai , for all i = 1, 2, . . . , n, then the Markov–Feller hyperoperator M0 : P (KCo (X)) → P (KCo (X)), M0 (μ)(A) :=

n

−1 Pi μ(F0i (A)),

(12)

i=1

for all μ ∈ P (KCo (X)) and A ∈ B(KCo (X)), has exactly one fixed point μ0 ∈ P (KCo (X)), called the invariant measure w.r.t. the hyperoperator M0 . Since X ⊂ E m is compact, so is by the above arguments (cf. [Hu & Huang, 1997]) (KCo (X), dH ), and subsequently (P (KCo (X)), dM KH ). Furthermore, because of |||Ai ||| < 1, the affine multivalued maps F0i are obviously contractions, and so are (cf. Table 3) the induced (single-valued) hypermaps F0i : KCo (X) → KCo (X), for all i = 1, 2, . . . , n. Thus, the Markov–Feller hyperoperator defined in (9) is, by the above arguments, a contraction as well. The application of Lemma 4, therefore, completes the proof. Proof.

Remark 21.

For regular matrices Ai , i = 1, 2, . . . , n, the formula (12) takes the more explicit form. M0 : P (KCo (X)) → P (KCo (X)), M0 (μ)(A) :=

n

Pi μ[(A − Ci )Ai−1 ],

(13)

i=1

for all μ ∈ P (KCo (X)) and A ∈ B(KCo (X)), where B(KCo (X)) denotes the σ-algebra of Borel subsets of KCo (X). Besides this advantage, the only improvement of Corollary 3 consists in fact that the space of probability Borel measures and the invariant measure μ0 from it are on (KCo (X), dH ), i.e. on convex, compact subsets of X. This fact will be successfully used, similarly as Theorem 2 and the second part of Theorem 4, in the next section. Otherwise, Theorem 6 can be only regarded as a consequence of Corollary 3. On the other hand, we still have that supp(μ0 ) = A0 , where A0 comes from Theorem 4, provided Pi > 0, for all i = 1, 2, . . . , n. We conclude this section by indicating the relationship of the obtained results in terms of fractals. [terminological] Fixed points in Theorems 1 and 3 are called fractals, while the other fixed points are called hyperfractals. To distinguish them still by means of the applied fixed point theorems, we speak about fixed points in Theorems 1, 2 and, for |||Ai ||| < 1, i = 1, 2, . . . , n, in Theorem 4 and Corollary 1 as metric, while about those in Theorems 3, 4 and Corollary 2 as topological. Thus, the unique fixed points in Corollary 1 represent metric hyperfractals whose “shadows” (called the underlying fractals in [Barnsley, 2006]) on the supporting space (X, d) coincide with respective metric multivalued fractals represented by fixed points in Theorem 1. The fixed points in Corollary 2 represent topological hyperfractals whose “shadows” on (X, d) coincide with topological multivalued fractals represented by the fixed points in Theorem 3. The topological hyperfractal supp(μ0 ) in Theorem 5 can be also called a hyper-semifractal, Remark 22.

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in the lines of [Lasota & Myjak, 1998], [Myjak & Szarek, 2003]. Furthermore, the support of the unique invariant measure in Corollary 3 coincides, for Pi > 0, i = 1, 2, . . . , n, with a metric hyperfractal in a particular case of Corollary 1 and the support of the invariant measure in Theorem 6 coincides, for Pi > 0 and |||Ai ||| < 1, i = 1, 2, . . . , n, with a metric hyperfractal in Theorem 4. The metric hyperfractals in Theorem 2 as well as, for |||Ai ||| < 1, i = 1, 2, . . . , n, in Theorem 4, and the topological hyperfractals in Theorem 4 are rather exceptional (cf. Remarks 14 and 16), but their “shadows” on the supporting spaces coincide with special metric multivalued fractals in Theorem 1 and topological multivalued fractals in Theorem 3, respectively.

5. Dimension of self-similar hyperfractals In this section, we will calculate the Hausdorff dimension of some self-similar hyperattractors from Corollary 1, Theorem 2 and Theorem 4. It immediately follows from the proofs of Theorem 2 and the second part of Theorem 4, based on the application of the Banach contraction principle, that the sets A1 ∈ C(CCo (E m )), A2 ∈ B(BCo (E m )) and A3 ∈ K(KCo (E m )) in Theorem 2 and A0 ∈ K(KCo (X)) in the second part of Theorem 4 are hyperattractors. It also follows from the complete version of Lemma 1 (see e.g. [Kirk & Sims, 2001, Theorem 3.2 on pp. 7–8]) that the sets A1 ∈ C(C(X)), A2 ∈ B(B(X)), A3 ∈ K(K(X)) in Corollary 1, proved by means of Lemma 1, are hyperattractors as well. For instance, for A3 ∈ K(K(X)) from Corollary 1, it means that lim dHH (φk3 (A), A3 ) = 0,

k→∞

for any A ∈ K(K(X)).

On the other hand, the related Collage theorem, i.e. dHH (φk3 (A), A3 ) ≤

Lk dH (A, φ3 (A)), 1−L H

for any A ∈ K(K(X)),

holds only for contractions F3i with factors Li ∈ [0, 1), i = 1, 2, . . . , n, where L := maxi=1,2,...,n {Li }. Otherwise, e.g. for nonexpansive maps, there can still exist iteration (e.g. Ishikawa’s, Mann’s, etc., cf. [Kirk & Sims, 2001], [Liu et al., 2007]) methods such that one can always reach points which are approximately fixed, but these points need not be near fixed points even if they exist. Moreover, the assumptions concerning the spaces are usually not induced to suitable hyperspaces. It is well-known (see e.g. [Hutchinson, 1981], [Peitgen et al., 1992], [Schief, 1996]) that the (hyper)attractor of the (hyper)IFS containing similitudes is self-similar. In particular, the hyperattractor of a hyperIFS induced in a hyperspace by the IMS containing similitudes must be self-similar, too. Let us recall that by a similitude F : X → Y , where (X, d1 ) and (Y, d2 ) are complete metric spaces, we mean that (cf. [Hutchinson, 1981]) d2 (F (x), F (y)) = rd1 (x, y),

for all x, y ∈ X,

for some r ∈ [0, 1). Let us also recall (cf. [Barnsley, 2003], [Hutchinson, 1981], [Peitgen et al., 1992], [Schief, 1996]) that, in a complete metric space (X, d), for every s ≥ 0, δ > 0 and S ∈ X, we can define the s-dimensional Hausdorff measure of S by H s (S) := lim Hδs (S), δ→0

(14)

∞ s where Hδs (S) := inf{ ∞ i=1 (diam Si ) |S ⊂ i=1 Si , diam Si ≤ δ}, and the Hausdorff dimension dimH (S) of S by dimH (S) := inf{s ≥ 0|H s (S) = 0} = sup{s ≥ 0|H s (S) = ∞}.

(15)

For the calculation of the Hausdorff dimension D = dimH (S) of S ⊂ X by means of contraction factors of the (hyper)IFSs, the crucial role is played by the Moran–Hutchinson formula (MHF) (see [Moran, 1946], [Hutchinson, 1981], [Schief, 1996] and notice that the paper [Moran, 1946] was already published in 1946!). Hence, let (X, d) be a complete metric space and Fi : X → X be similitudes with contraction factors

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19

ri ∈ [0, 1), i = 1, 2, . . . , n. Then, for D = dimH (S), where S ∈ K(X) is the associated attractor, the MHF holds, i.e. n

riD = 1,

(16)

i=1

provided the strong open set condition (SOSC) is satisfied, namely that there exists an open subset V ⊂ X such that (i) ni=1 Fi (V ) ⊂ V , (ii) Fi (V ) ∩ Fj (V ) = ∅, for i = j; i, j = 1, 2, . . . , n, and (iii) V ∩ S = ∅. Let us note that in Rm only conditions (i) and (ii), called the open set condition (OSC), are enough for the validity of (16) (cf. [Moran, 1946], [Hutchinson, 1981]). For the calculation of the Hausdorff dimensions, we will also apply the following two propositions. Proposition 5. (cf. [Schief, 1996]) Assume that, for all i = 1, 2, . . . , n, Fi : X → X are similitudes with contraction factors ri ∈ [0, 1) in a complete metric space (X, d) and Si = Fi (S), where S ⊂ X is the associated attractor, are pair-wise disjoint. Then H D (S) > 0, where H D (S) is defined in (14) and D satisfies (16). This, in particular, implies (cf. (15)) that D = dimH (S). Proposition 6. (cf. [Castro & Reyes, 1997]) Assume that (X, d) and (Y, d ) are metric spaces, S ⊂ X and

f : S → Y satisfies the inequalities

a · d(x, y) ≤ d (f (x), f (y)) ≤ b · d(x, y),

for all x, y ∈ S,

with suitable constants a > 0 and b > 0. Then as H s (E) ≤ H s (f (E)) ≤ bs H s (E) holds, for every s ≥ 0. For applications of Propositions 5 and 6 in hyperspaces, the convex analysis is essential. Hence, let us recall at least the basic properties of convex sets we need in the sequel. For more details, we recommend the monographs [Aliprantis & Border, 1999], [Beer, 1993] [Hiriart-Urruty & Lemaréchal, 2001], [Rockafellar, 1997]. Let (E, ||.||) be a real Banach space and A ⊂ E, B ⊂ E its subsets. Defining as usually (cf. e.g. [Aliprantis & Border, 1999], [Andres& G´ orniewicz, 2003]) A + B := {x x = a + b, a ∈ A, b ∈ B}, c · A := {x x = c · a, a ∈ A}, c ∈ R, we can say the following. If A and B are convex subsets of E, then A + B and c · A are convex (cf. e.g. [Beer, 1993], [Dunford & Schwartz, 1958]). adström, 1952]) Moreover, if A, B, C ∈ (KCo (E), dH ), then (cf. [R˚ dH (A + C, B + C) = dH (A, B).

(17)

Defining still the convex hull conv(A) of A ∈ K(E) as (see e.g. [Dunford & Schwartz, 1958, Chapter V.2]) conv(A) := {x ∈ E|x =

p i=1

αi ai ,

n+1

αi = 1, αi ≥ 0, ai ∈ A, i = 1, . . . , p},

i=1

p = 1, 2, . . . , it is obviously the smallest convex set containing A ⊂ E. Let us note that, in Rm , we can simply fix p = m + 1. Moreover, for any A, B ⊂ E, it holds (see e.g. [Dunford & Schwartz, 1958, Lemma V.2.4]): (i) conv(A + B) = conv(A) + conv(B), (ii) conv(c · A) = c · conv(A), c ∈ R. We can also give the following lemma which is probably known but, for the sake of completeness, we decided to prove it here.

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Lemma 5. For A, B, C ∈ K(E), we have that

dH (A, B) ≥ dH (conv(A), conv(B))

(18)

dH (conv(A + C), conv(B + C)) = dH (conv(A), conv(B)).

(19)

and

Proof.

At first, we shall prove the inequality (18). Letting x ∈ conv(A), we have p

x=

αi ai ,

for all i = 1, . . . , p.

i=1

It follows from the compactness of A, B ∈ K(E) that, for every ai ∈ A, we can find bi ∈ B such that Defining y ∈ conv(B) as y =

||ai − bi || ≤ dH (A, B).

p

i=1 αi bi ,

||x − y|| = ||

p

it holds: αi ai −

i=1

≤

p i=1

(20)

p

αi bi || ≤

i=1

p

αi ||ai − bi ||

i=1

αi max ||ai − bi || ≤ max ||ai − bi ||. i=1,...,p

i=1,...,p

In view of (20), this already implies (18). Now, we prove (19). It follows from the property (i) of the convex hull and (17) that, for any A, B, C ∈ K(E), dH (conv(A + C), conv(B + C)) = = dH (conv(A) + conv(C), conv(B) + conv(C)) = dH (conv(A), conv(B)), i.e. (19) holds as claimed.

We are ready to give the main results of this section. They concern the particular cases of multivalued maps of the form Fi : E → K(E m ), Fi (x) = ri Qi x + Ci ,

i = 1, 2, . . . , n,

(21)

where (E, ||.||) is a real Banach space and, for each i = 1, 2, . . . , n, Ci ∈ K(E m ), ri are reals, Qi are orthonormal m × m-matrices, i.e. Qi · QiT = I or equivalently QiT = Qi−1 , whose columns and rows are orthonormal unit vectors. The first theorem in this section deals with the hyperattractors in (K(Rm ), dH ) and (KCo (Rm ), dH ). Theorem 7. Consider the maps Fi , i = 1, 2, . . . , n, of the form (21) in Rm (i.e. E = R) such that ri ∈ [0, 12 ),

for all i = 1, 2, . . . , n. Assume that there exist a subset U ∈ K(K(Rm )) of K(Rm ) and a positive number dmin H such that φ(U ) :=

n

Fi (x) ⊂ U

(22)

i=1 y∈A x∈y

> 0, i = j; i, j = 1, 2, . . . , n, for all A, B ∈ U . Then and dH (conv(Fi (A)), conv(Fj (B))) ≥ dmin H D = dimH (A0 ) from (16) is the Hausdorff dimension in K(Rm ) of the associated hyperattractor A0 ∈ K(K(Rm )), i.e. the unique fixed point of the Hutchinson–Barnsley hyperoperator φ : K(K(Rm )) → K(K(Rm )). If Fi are still convex-valued, i.e. Fi : Rm → KCo (Rm ), i = 1, 2, . . . , n, then the same conclusion holds, provided ri ∈ [0, 1), i = 1, . . . , n.

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21

We start with the special case that Fi are still convex-valued and ri ∈ [0, 1), for each i = 1, 2, . . . , n. Then the Hutchinson–Barnsley hyperoperator φ : K(KCo (Rm )) → K(KCo (Rm )) defined in (22) has, according to Theorem 2, a unique fixed point A0 ∈ K(KCo (Rm )) which is at the same time the attractor of the related hyperIFS in KCo (Rm ). In order to calculate dimH (A0 ), where A0 ∈ K(KCo (Rm )), by means of Proposition 5, we need to show that Fi : KCo (Rm ) → KCo (Rm ), i = 1, 2, . . . , n, are similitudes. Since A, B ⊂ KCo (Rm ) are convex sets, so must be, for each i = 1, 2, . . . , n, ri Qi A and ri Qi B (cf. e.g. [Be, Theorem 1.4.1]). Furthermore, since it is well-known (see e.g. [Hutchinson, 1981]) that, in Euclidean spaces (Rm , |.|), Qi acts as an isometry, we obtain in view of (17) that Proof.

dH (Fi (A), Fi (B)) = dH (ri Qi A + Ci , ri Qi B + Ci ) = ri dH (Qi A, Qi B) = ri max{ sup d(a, Qi B), sup d(b, Qi A)} a∈Qi A

b∈Qi B

inf |a − b|, sup

= ri max{ sup

a∈Qi A b∈Qi B

inf |a − b|}

b∈Qi B a∈Qi A

= ri max{sup inf |Qi a − Qi b|, sup inf |Qi a − Qi b|} a∈A b∈B

b∈B a∈A

= ri max{sup inf |a − b|, sup inf |a − b|} = ri dH (A, B), b∈B a∈A

a∈A b∈B

i.e. Fi , i = 1, 2, . . . , n, are similitudes, as required. Since obviously A0 ⊂ U ∈ K(K(Rm )), where by the hypothesis (cf. (18)) Fi (U ) ∩ Fj (U ) = ∅, i = j; i, j = 1, 2, . . . , n, we get that Fi (A0 ) ∩ Fj (A0 ) = ∅,

i = j; i, j = 1, 2, . . . , n.

The application of Proposition 5 completes the proof in a special case that Ci ∈ KCo (Rm ), i = 1, 2, . . . , n. Now, we proceed to the general case. First of all, under the above assumptions, the unique fixed point A0 ∈ K(K(Rm )) of φ in (22) exists, according to Corollary 1, and by the above arguments it is a hyperattractor of a hyperIFS induced in K(Rm ) by the IMS consisting of Fi , i = 1, 2, . . . , n. For the calculation of D = dimH (A0 ), Proposition 6 will be applied. Hence, let us make the following notations (called the address system in [Hutchinson, 1981], [Barnsley, 2003]) Ai1 i2 ...ij ... := Fi1 ◦ Fi2 ◦ · · · ◦ Fij ◦ · · · (L), for any L ∈ K(Rm ), and ri1 i2 ···ij := ri1 · ri2 · · · rij , r :=

Qi1 i2 ...ij := Qi1 · Qi2 · · · Qij , max {ri },

i=1,2,...m

where the indices ik refer to any i ∈ {1, 2, . . . , n}, k = 1, 2, . . . In view of this notation, for any j ∈ N, we can write: Ai1 i2 ...ij ... = Ci1 + ri1 Qi1 (Ci2 + ri2 Qi2 (Ci3 + · · · + rij−1 Qij−1 (Cij + rij Qij (M )))), where M := Cij+1 + rij+1 Qij+1 (Cij+2 + rij+2 Qij+2 (. . . )), Ai1 i2 ...ij ... = Ci1 + ri1 Qi1 (Ci2 + ri2 Qi2 (Ci3 + · · · + rij−1 Qij−1 (Cij + Qri (N )))), j

where N := Cij+1 + rij+1 Qij+1 (Cij+2 + rij+2 Qij+2 (. . . )), and M, N ∈ U , ij = ij . Hence,

Ai1 i2 ...ij ... = Ci1 + ri1 Qi1 Ci2 + ri1 i2 Qi1 i2 Ci3 + · · · + ri1 i2 ...ij−1 Qi1 i2 ...ij−1 Fij (M ),

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Ai1 i2 ...ij ... = Ci1 + ri1 Qi1 Ci2 + ri1 i2 Qi1 i2 Ci3 + · · · + ri1 i2 ...ij−1 Qi1 i2 ...ij−1 Fij (N ). Since ri Qi : K(Rm ) → K(Rm ) were already proved, for each i = 1, 2, . . . , n, to be similitudes, it follows from the inequality (18) in Lemma 5 that dH (Ai1 i2 ...ij ... , Ai1 i2 ...ij ... ) ≥ ≥ dH (conv(ri1 i2 ...ij−1 Qi1 i2 ...ij−1 Fij (M )), conv(ri1 i2 ...ij−1 Qi1 i2 ...ij−1 Fij (N ))) = dH (ri1 i2 ...ij−1 Qi1 i2 ...ij−1 conv(Fij (M )), ri1 i2 ...ij−1 Qi1 i2 ...ij−1 conv(Fij (N ))) ≥ ri1 i2 ...ij−1 · dmin H . On the other hand, dH (Ai1 i2 ...ij ... , Ai1 i2 ...ij ... ) ≤ ri1 i2 ...ij−1 · diam(U )). In order to apply Proposition 6, we need to construct the auxiliary IFS with prescribed contraction factors whose attractor is the Cantor set. More concretely, for rj ∈ [0, 12 ), j = 1, 2, . . . , n, let us consider the IFS {[0, 1]k , gj , j = 1, 2, . . . n, 2k > n}, where gj (x) := rj · x +

1 · (j)k2 , 2

and (j)k2 means that the integer j is written in a binary number system and completed by zeros to be a k-dimensional vector. Let us denote by B0 ∈ K([0, 1]k ) the related attractor of the IFS. It is self-similar and the open set condition is satisfied on the open cube (0, 1)k . The dimension D of the attractor can be calculated from the prescribed contraction factors by means of (16) and 0 < H D (B0 ) < ∞, for the Hausdorff measure H D (B0 ) of B0 ∈ K([0, 1]k ). Denoting Bi1 i2 ...ij ... := gi1 ◦ gi2 ◦ · · · ◦ gij ◦ . . . (x), √ for any x ∈ [0, 1]k , from diam([0, 1]k ) = k (B0 ⊂ [0, 1]k ) and d(gi (x), gj (y)) ≥ we can see that

1 − r, 2

i = j; i, j = 1, 2, . . . , m,

√ 1 − r · ri1 i2 ...ij−1 ≤ d(Bi1 i2 ...ij−1 ij ... , Bi1 i2 ...ij−1 ij ... ) ≤ k · ri1 i2 ...ij−1 . 2

This already implies dmin H √ d(Bi1 i2 ...ij−1 ij ... , Bi1 i2 ...ij−1 ij ... ) ≤ dH (Ai1 i2 ...ij−1 ij ... , Ai1 i2 ...ij−1 ij ... ) k diam(U ) dH (Bi1 i2 ...ij−1 ij ... , Bi1 i2 ...ij−1 ij ... . ≤ 1 2 −r Applying Proposition 6, for f : B0 → A0 ,

f (Bi1 i2 ...ij−1 ij ... ) = Ai1 i2 ...ij−1 ij ... ,

the Hausdorff dimension of A0 is really D, because 0 < H D (B0 ) < ∞ (cf. (15)).

Since Qi in (21) acts as an isometry either in Rm as above or trivially in E (i.e. m = 1 and Qi = 1), but not in a general E m , we could not formulate Theorem 7 in a more general way in E m . On the other hand, we can still give the following theorem dealing with the hyperattractors in (K(E), dH ) and (KCo (E), dH ). Its proof will be omitted, because it is quite analogous as in Theorem 7.

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23

Theorem 8. Consider the maps Fi , i = 1, 2, . . . , n, of the form (21) in E (i.e. m = 1 and Qi = 1, i = 1, 2, . . . n) such that ri ∈ [0, 12 ), for all i = 1, 2, . . . , n. Assume that there exist a subset U ∈ K(K(E)) of K(E) and a positive number dmin H such that (22) holds and

dH (conv(Fi (A)), conv(Fj (B))) ≥ dmin H > 0,

i = j; i, j = 1, 2, . . . , n,

for all A, B ∈ U . Then D = dimH (A0 ) from (16) is the Hausdorff dimension in K(E) of the associated hyperattractor A0 ∈ K(K(E)), i.e. the unique fixed point of the Hutchinson–Barnsley hyperoperator φ : K(K(E)) → K(K(E)) in (22). If Fi are still convex-valued, i.e. Fi : E → KCo (E), i = 1, 2, . . . , n, then the same conclusion holds, provided ri ∈ [0, 1), i = 1, 2, . . . , n. For Ci ∈ / KCo (Rm ) and Ci ∈ / KCo (E), ri in Theorem 7 and Theorem 8 can be in particular √ 2 1 cases greater than 2 . For instance, if n = 2 and r1 , r2 ∈ [0, 2 ), then one can use as an auxiliary tool, instead of the Cantor-like IFS, the von Koch-like IFS whose attractor is disconnected. Remark 23.

Remark 24. Stimulated by modelling in quantitative linguistics, we constructed in [Andres& Rypka, 2011] the Cantor-like hyperfractals, by means of 2m − 1 induced maps by multivalued contractions with the 1 ≤ 12 , whose values consisted of sets of maximally m points in [0, 1]. Their same contraction factor r ≤ m Hausdorff dimension was log(2m − 1) , D= log 1r 1

i.e. we took r := (2m − 1)− D . Let us note that neither Theorem 7 (m = 1) nor Theorem 8 (E = R) can be applied there. Restricting ourselves to compact subsets of Rm or E, as in Remark 24, we are able to calculate in the following two theorems the Hausdorff dimension of the supports of some invariant measures from Corollary 3 and Theorem 6. Theorem 9. Consider the maps Fi , i = 1, 2, . . . , n, of the form (21) in Rm (i.e. E = R) such that ri ∈ [0, 1),

for all i = 1, 2, . . . , n, and assume that there exists a compact subset X ⊂ Rm such that Ci ∈ K(X) and Fi |X : X → K(X), i = 1, 2, . . . , n. Assume, furthermore, that there also exists a constant dmin H > 0 such that dH (conv(Fi |X (A)), conv(Fj |X (B))) ≥ dmin H > 0,

i = j; i, j = 1, 2, . . . , n,

for all A, B ∈ K(X). Then D from (16) is the Hausdorff dimension of the associated hyperattractor A0 ∈ K(K(X)) as well as of supp(μ0 ), i.e. D = dimH (A0 ) = dimH (supp(μ0 )), where μ0 ∈ P (K(X)) is the unique invariant measure w.r.t. the Markov–Feller hyperoperator M : P (K(X)) → P (K(X)) defined in (10). If Fi |X are still convex-valued, i.e. Fi |X : X → KCo (X), i = 1, 2, . . . , n, then the same conclusion holds, provided ri ∈ [0, 1), i = 1, 2, . . . , n. Proof. Under the above assumptions, according to Corollary 1, there exists a unique fixed point A0 ∈ K(K(X)) of φ in (22), where U := K(X). If still Fi |X : X → KCo (X), u = 1, 2, . . . , n, then φ : K(KCo (X)) → K(KCo (X)) defined in (22) has, according to the second part of Theorem 2, a unique fixed point A0 ∈ K(KCo (X)). One can easily check that the deterministic part, i.e. that D from (16) satisfies D = dimH (A0 ), where A0 ∈ K(K(X)) respectively A0 ∈ K(KCo (X)) are just mentioned unique fixed points, can be proved quite analogously as Theorem 7. In fact, it is a direct consequence of Theorem 7, when taking U := K(X). Furthermore, there exists, according to Corollary 3 (resp. Theorem 6), whose all assumptions are satisfied, a unique invariant measure μ0 ∈ P (K(X)) of M in (10) such that supp(μ0 ) = A0 ∈ K(K(X)) (resp. μ0 ∈ P (KCo (X)) of M0 in (12) such that supp(μ0 ) = A0 ∈ K(KCo (X))). Therefore, D from (16) also satisfies D = dimH (supp(μ0 )), where μ0 ∈ P (K(X)) (resp. μ0 ∈ K(KCo (X))), which completes the proof.

Since the following theorem can be proved quite analogously as Theorem 9, its proof will be omitted. In fact, the deterministic part is a direct consequence of Theorem 8, when taking U := K(X), where X ⊂ E

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is a compact subset of E. The probabilistic part then again follows by the application of Corollary 3 resp. Theorem 6. Theorem 10. Consider the maps Fi , i = 1, 2, . . . , n, of the form (21) in E (i.e. m = 1 and Qi = 1, i = 1, 2, . . . , n) such that ri ∈ [0, 12 ), for all i = 1, 2, . . . , n, and assume that there exists a compact subset X ⊂ E such that Ci ∈ K(X) and Fi |X : X → K(X), i = 1, 2, . . . , n. Assume, furthermore, that there also exists a constant dmin H > 0 such that

dH (conv(Fi |X (A)), conv(Fj |X (B))) ≥ dmin H > 0,

i = j; i, j = 1, 2, . . . n,

for all A, B ∈ K(X). Then D from (16) is the Hausdorff dimension of the associated hyperattractor A0 ∈ K(K(X)) as well as of supp(μ0 ), i.e. D = dimH (A0 ) = dimH (supp(μ0 )), where μ0 ∈ P (K(X)) is the unique invariant measure w.r.t. the Markov–Feller hyperoperator M : P (K(X)) → P (K(X)) defined in (10). If Fi |X are still convex-valued, i.e. Fi |X : X → KCo (X), i = 1, 2, . . . , n, then the same conclusion holds, provided ri ∈ [0, 1), i = 1, 2, . . . , n. We conclude this section by an illustrative example. Example 5.1. Consider the IMS with probabilities

{[0, 1]2 , Fi : [0, 1]2 → KCo ([0, 1]2 ), i = 1, 2, 3, 4, 5, P1 = P5 = 0.2, P2 = P3 = 0.19, P4 = 0.21}, where

x F1 y x F2 y x F3 y x F4 y x F5 y

1 := :=

3

0 1 3

0

1 :=

3

0 1

:=

3

0 1

:=

4

0

⎫ x 0 ⎪ ⎪ + , ⎪ 1 ⎪ y 0 ⎪ 3 ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ 0 x ⎪ ⎪ + , 2 1 ⎪ y ⎪ ⎪ 3 3 ⎪ ⎪ 2 ⎪ ⎬ 0 x 3 , + 1 y 0 ⎪ 3 ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ 0 x 3 ⎪ ⎪ + , 1 ⎪ 2 y ⎪ ⎪ 3 3 ⎪ ⎪ 17 19 ⎪ ⎪ ⎪ , 0 x ⎪ 48 48 ⎪ . + ⎭ 1 17 19 y , 4 48 48 0

(23)

Applying the second part of Theorem 4, the Hutchinson–Barnsley hyperoperator φ : K(KCo ([0, 1]2 )) → K(KCo ([0, 1]2 )), where φ(A) :=

5

Fi (x),

i=1 y∈A x∈y

has a unique fixed point A0 ∈ K(KCo ([0, 1]2 )) which is in K([0, 1]2 ) the hyperattractor of the hyperIFS induced by our IMS. Applying still Theorem 6 (cf. also Remark 21), the Markov–Feller hyperoperator M : P (KCo ([0, 1]2 )) → P (KCo ([0, 1]2 )), where M (μ(A)) :=

5

Pi μ(Fi−1 (A))

i=1

has, for each A ∈ B(KCo ([0, 1]2 )), a unique invariant measure μ0 ∈ P (KCo ([0, 1]2 )) such that supp(μ0) = A0 .

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Fig. 1.

25

Attractor of the IMS from Example 1.

Since one can easily check that all the assumptions of Theorem 9 (m = 2, X := [0, 1]2 ) are satisfied, we arrive after its application at dimH (supp(μ0 )) = dimH (A0 ) = D, where D satisfies the equation (cf. (16)): 4

D D 1 1 + = 1, 3 4

. i.e. dimH (supp(μ0 )) = dimH (A0 ) = 1.4024. The “shadow” of the invariant measure μ0 ∈ P (KCo ([0, 1]2 )), whose embedding is in Figure 3, is plotted in Figure 4. Let us note that the attractor of our IMS in Figure 1 coincides with the “shadow” of the hyperattractor A0 ⊂ K([0, 1]2 ) whose embedding is in Figure 2. According to the R˚ adstr¨ om theorem (see [R˚ adström, 1952], [De Blasi, 2010], [De Blasi & Georgiev, 2001]), (KCo (Rm ), dH ) can be embedded into the positive convex cone in a real Banach space. This embedding can be an isometric isomorphism. Moreover, (KCo (Rm ), dH ) is for m ≥ 2 homeomorphic to the punctured Hilbert cube [0, 1]∞ \point ≈ [0, 1]∞ × (0, 1] (see [Nadler et al., 1979]). Furthermore, since (K(Rm ), dH ) is separable (see Section 2), it follows from the proof of Urysohn’s metrization theorem (see e.g. [Aliprantis & Border, 1999]) that it can be isometrically embedded into the Hilbert cube [0, 1]∞ . Since the hyperfractals in Theorems 7, 8 and the supports of invariant measures in Theorems 9, 10 have finite dimensions D, it is a question whether or not they can be embedded by means of the R˚ adström theorem or the Urysohn theorem into Rk , where k ∈ N is a suitable number such that k ≥ D. The simulations indicate that it might be possible (see e.g. the hyperattractor A0 ∈ K(K([0, 1]2 )) of the induced hyperIFS and the invariant measure μ0 ∈ P (K([0, 1]2 )) of the Markov–Feller hyperoperator M in Example 1 which are embedded from (K(R2 ), dH ) into R2 in Figure 2 and Figure 3). Remark 25.

6. Concluding remarks The Hausdorff dimension calculated by means of the Moran–Hutchinson formula (16) is sometimes also called the self-similarity dimension (cf. [Hutchinson, 1981]). If some of the generating hypermaps is not a similitude, then one should therefore use another sort of dimension. For hyperfractals, the most appropriate dimension seems to be then the correlation dimension (see e.g. [Chin et al., 1997], [Peitgen et al., 1992]).

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Fig. 2.

Fig. 3.

Embedding of the hyperattractor from Example 1.

Embedding of the invariant measure from Example 1 (its support is light).

Example 6.1. As an example of such class of multivalued maps, we can give those involved in the IMS

{[0, 1]2 , Fi : [0, 1]2 → K([0, 1]2 ), i = 1, 2, 3}, namely

where

1 0 x x 0 := 2 1 + , F1 y y 0 0 2 1 0 0 x x := 2 1 + 1 , F2 y y 0 2 2 1 1 1 x x 3, 2 0 + 2 , := F3 1 1 y y 0 , 0 3 2

x x y y 1 1 1 1 , · x, , · y := , , , . 3 2 3 2 3 2 3 2

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Fig. 4.

27

The “shadow” of the invariant measure from Example 1 (its support is light).

Fig. 5.

“Fat” Sierpi´ nski’s triangle.

Applying Theorem 1, the IMS has an attractor usually called (because of its positive Lebesgue measure, cf. [Farmer, 1985], [Umberger & Farmer, 1985]) the “fat” Sierpi´ nski triangle (see Figure 5) which is in our terminology a metric multivalued fractal. Applying still Corollary 1, there is also a hyperattractor A0 ∈ K(K([0, 1]2 )) of the induced hyperIFS {K([0, 1]2 ), Fi : K([0, 1]2 ) → K([0, 1]2 ), i = 1, 2, 3} (see Figure 6, where the 2D-projection of its embedding is plotted). Moreover, considering the induced hyperIFS with probabilities {[0, 1]2 , Fi : K([0, 1]2 ) → K([0, 1]2 ), i = 1, 2, 3, P1 = P3 = 0.3, P2 = 0.4}, there is according to Corollary 3 a unique invariant measure μ0 ∈ P (K([0, 1]2 )) such that supp(μ0 ) = A0 (see Figure 7, where the 2D-projection of its embedding is plotted). Observe that its light part coincides with Sierpi´ nski’s hypertriangle in Figure 6 and that the light part of the “shadow” in Figure 8 coincides with the underlying multivalued fractal in Figure 5. On the other hand, since F3 : K([0, 1]2 ) → K([0, 1]2 ) is not a similitude, neither Theorem 9 nor Theorem 10 can be applied here for the calculation of dimH (A0 ) and dimH (supp(μ0 )). Nevertheless, the correlation

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Fig. 6.

Fig. 7.

Sierpi´ nski’s hypertriangle.

Embedding of the invariant measure from Example 2 (its support is light).

dimension dimCor (A0 ) = dimCor (supp(μ0 )) can be approximately computed by means of the chaos game 3 generation of 5000 sets (i.e. points of the hyperattractor A0 ), to be 1.55 (< log log 2 ). Let us note that the fast algorithm for the calculation of the correlation dimension which can be applied also in hyperspaces can be found in [Theiler, 1987]. The numerical techniques for multivalued fractals, including shadowing, were developed in [Andres et al., 2005], [Fiˇser, 2004] and [Glavan & Gut¸u, 2004]. For hyperfractals, the numerical analysis in [Barnsley, 2003] can be suitably adopted. According to Hambly’s theorem (cf. [Barnsley, 2006, p. 397]), one can calculate by means of the generalized Moran–Hutchinson formula involving probabilities the almost sure Hausdorff dimension of 1-variable fractal sets. These fractals are generated by the IMS with probabilities {Rm , Fi : Rm → Kif in (Rm ), Pi , i = 1, 2, . . . , n}, where Kif in (Rm ) := {A ⊂ Rm |A consists of a maximally finite number if in of points in Rm } and Fi are comprised by if in single-valued similitudes, i = 1, 2, . . . , n. In [Barnsley et al., 2011], a similar result was obtained for V -variable fractals. On the other hand, the calculation of a nontrivial fractal dimension, or at least its estimate, for our multivalued fractals, where the values of generating multivalued maps do not necessarily consist of a finite number of points, seems to be a difficult task. Nevertheless, if multivalued fractals in Rm are “fat” in the sense of Grebogi et al. [Grebogi et al., 1985], [OtT, 1994, Chapter 3.9] (i.e. if, for every point x in the

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Fig. 8.

29

The “shadow” of the invariant measure from Example 2 (its support is light).

set and every ε > 0, the ball Oε (x) contains a nonzero Lebesgue measure of points in the set, as well as outside the set), then the exterior dimension defined in [Grebogi et al., 1985], [OtT, 1994] can be computed. Although multivalued fractals in Figures 1 and 5 are “fat” in the sense of [Farmer, 1985], [Umberger & Farmer, 1985], they do not satisfy this definition. Obviously, their box-counting dimension (cf. [OtT, 1994], [Peitgen et al., 1992]) is trivially 2. As pointed out in [Umberger & Farmer, 1985], Mandelbrot called such objects “dusts with a positive volume”. The simulations in Figures 5 and 8 seem also to justify the possibility to investigate, for our IMSs with probabilities, the invariant measures on probability measure spaces.

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