Mathematics of Fractals

Yamaguti, Masaya; Hata, Masayoshi; Kigami, Jun

doi:10.1090/mmono/167

Cited by 63 publications

(60 citation statements)

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“…Some topological properties of the attractors of IFSs were studied in [9], [13], [19] and [27]. Also, general aspects of topology can be found in [5].…”

Section: Connectedness Of the Attactors Of (Iifs) Formed By Contractimentioning

confidence: 99%

Attractors of Infinite Iterated Function Systems Containing Con

Dumitru¹

2013

Annals of the Alexandru Ioan Cuza University - Mathematics

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Abstract. We consider a complete ε-chainable metric space (X, d) and an infinite iterated function system (IIFS) formed by an infinite family of (ε, φ)-functions on X. The aim of this paper is to prove the existence and uniqueness of the attractors of such infinite iterated systems (IIFS) and to give some sufficient conditions for these attractors to be connected. Similar results are obtained in the case when the IIFS is formed by an infinite family of uniformly ε-locally strong Meir-Keeler functions.Mathematics Subject Classification 2010: 28A80.

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“…Some topological properties of the attractors of IFSs were studied in [9], [13], [19] and [27]. Also, general aspects of topology can be found in [5].…”

Section: Connectedness Of the Attactors Of (Iifs) Formed By Contractimentioning

confidence: 99%

Attractors of Infinite Iterated Function Systems Containing Con

Dumitru¹

2013

Annals of the Alexandru Ioan Cuza University - Mathematics

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“…These are special cases of discrete dynamical systems which arise from a class of Iterated Function Systems; see [YHK97]. Our starting point is two features of these systems which we proceed to outline.…”

Section: Probabilities On Path Spacementioning

confidence: 99%

“…The map π is continuous and onto; see [Hut81] and [YHK97]. We will use it to lift the elements associated to the IFS, up from X to Ω, which is endowed with the endomorphism given by the shift r Ω .…”

Section: Lifting the Ifs Case To The Endomorphism Case Now Consider mentioning

confidence: 99%

Iterated function systems, Ruelle operators, and invariant projective measures

2006

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Abstract. We introduce a Fourier-based harmonic analysis for a class of discrete dynamical systems which arise from Iterated Function Systems. Our starting point is the following pair of special features of these systems. (1) We assume that a measurable space X comes with a finite-to-one endomorphism r : X → X which is onto but not one-to-one. (2) In the case of affine Iterated Function Systems (IFSs) in R d , this harmonic analysis arises naturally as a spectral duality defined from a given pair of finite subsets B, L in R d of the same cardinality which generate complex Hadamard matrices.Our harmonic analysis for these iterated function systems (IFS) (X, µ) is based on a Markov process on certain paths. The probabilities are determined by a weight function W on X. From W we define a transition operator R W acting on functions on X, and a corresponding class H of continuous R Wharmonic functions. The properties of the functions in H are analyzed, and they determine the spectral theory of L 2 (µ). For affine IFSs we establish orthogonal bases in L 2 (µ). These bases are generated by paths with infinite repetition of finite words. We use this in the last section to analyze tiles in R d .

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“…De Rham [14] deduced the properties of F directly from the functional equation (13) governing F , but his presentation is quite terse, as is quite clear from the more comprehensive account of it in Yamaguti, Hata and Kigami [63].…”

Section: Singular Invariant Distributions Of the Hellinger Typementioning

confidence: 99%

The nature of the steady state in models of optimal growth under uncertainty

Mitra

Montrucchio

Privileggi³

2003

Economic Theory

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Summary. We study a one-sector stochastic optimal growth model with a representative agent. Utility is logarithmic and the production function is of the Cobb-Douglas form with capital exponent . Production is a¤ected by a multiplicative shock taking one of two values with positive probabilities p and 1 p. It is well known that for this economy, optimal paths converge to a unique steady state, which is an invariant distribution. We are concerned with properties of this distribution. By using the theory of Iterated Function Systems, we are able to characterize such a distribution in terms of singularity versus absolute continuity as parameters and p change. We establish mutual singularity of the invariant distributions as p varies between 0 and 1 whenever < 1=2. More delicate is the case > 1=2. Singularity with respect to Lebesgue

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Mathematics of Fractals

Cited by 63 publications

References 0 publications

Attractors of Infinite Iterated Function Systems Containing Con

Attractors of Infinite Iterated Function Systems Containing Con

Iterated function systems, Ruelle operators, and invariant projective measures

The nature of the steady state in models of optimal growth under uncertainty

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