Contractive Iterated Function Systems Enriched with Nonexpansive Maps

Strobin, Filip

doi:10.1007/s00025-021-01451-0

Cited by 7 publications

(6 citation statements)

References 17 publications

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“…If the monoid M({f (i,1) : i ∈ J}) is compact, then the lower transition attractor A • of F t is compact.Proof. The statement follows from Theorem 4.2 (iv) and from[31, Theorems 4.1]. If either of the following two conditions hold, then the monoid M({f (i,1) : i ∈ J}) is compact.…”

mentioning

confidence: 90%

“…The notion of a semiattractor comes into play in [20,31], where functions that are not contrations are added to an IFS consisting of contractions. This allows for the use of standard methods for computer drawing of the attractor of contractive IFS.…”

Section: Theorem 41 ([26]mentioning

confidence: 99%

“…A generalization of the Hutchinson theorem is the following (see [31] and the references therein): Theorem 5.1. If an IFS F on (X, d) satisfies sup f ∈F Lip(f, d) < 1 and is compact, then it admits a Hutchinson attractor.…”

Section: The Existence Of a Unique Upper Transition Attractormentioning

confidence: 99%

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Transition Phenomena for the Attractor of an Iterated Function System

Leśniak,

Snigireva,

Strobin

et al. 2022

Preprint

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Iterated function systems (IFSs) and their attractors have been central to the theory of fractal geometry almost from its inception. And contractivity of the functions in the IFS has been central to the theory of iterated functions systems. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Recently, however, there has been an interest in what occurs to the attractor at the boundary between contractvity and expansion of the IFS. That is the subject of this paper. For a family Ft of IFSs depending on a real parameter t > 0, the existence and properties of two types of transition attractors, called the lower transition attractor A• and the upper transition attractor A • , are investigated. A main theorem states that, for a wide class of IFS families, there is a threshold t 0 such that the IFS Ft has a unique attractor At for t < t 0 and no attractor for t > t 0 . At the threshold t 0 , there is an Ft 0 -invariant set A • such that A • = limt→t 0 At.

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mentioning

confidence: 90%

Section: Theorem 41 ([26]mentioning

confidence: 99%

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Transition Phenomena for the Attractor of an Iterated Function System

Leśniak,

Snigireva,

Strobin

et al. 2022

Preprint

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“…Iterated function systems with uncountable many maps are not very common in the mathematical literature, but they do appear, see e.g. [Str21]. However, such systems are abundant in the fractal art community, whose goal is to generate and color interesting attractors.…”

Section: Contentsmentioning

confidence: 99%

Linear first order differential operators and their Hutchinson-invariant sets

Alexandersson¹,

Hemmingsson²,

Shapiro³

2022

Preprint

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In this paper, we initiate the study of a new interrelation between linear ordinary differential operators and complex dynamics which we discuss in details in the simplest case of operators of order 1. Namely, assuming that such an operator T has polynomial coefficients, we interpret it as a continuous family of Hutchinson operators acting on the space of positive powers of linear forms. Using this interpretation of T , we introduce its continuously Hutchinson invariant subsets of the complex plane and investigate a variety of their properties. In particular, we prove that for any T with non-constant coefficients, there exists a unique minimal under inclusion invariant set M T CH . Further, we completely characterize the class of operators T for which M T CH is compact and find it explicitly for several special types of operators. In particular, we present strong evidence that the boundary of M T CH is piecewise analytic in contrast with the boundaries of classical invariant sets occurring in complex dynamics.

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“…In a particular case, where an isometry is added to a contractive IFS, such a system is related to another contractive IFS. This allows one to use the standard methods for computer drawing of attractors of contractive IFSs, as is done in example 4.2 below; see [20,32]. However, in general, when an IFS contains noncontractive maps or maps with parabolic fixed points, specific methods of drawing attractors need to be used; see [26].…”

Section: The Lower Transition Attractor and Semiattractormentioning

confidence: 99%

Transition phenomena for the attractor of an iterated function system*

et al. 2022

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Iterated function systems (IFSs) and their attractors have been central in fractal geometry. If the functions in the IFS are contractions, then the IFS is guaranteed to have a unique attractor. Two natural questions concerning contractivity arise. First, whether an IFS needs to be contractive to admit an attractor? Second, what occurs to the attractor at the boundary between contractivity and expansion of an IFS? The first question is addressed in the paper by providing examples of highly noncontractive IFSs with attractors. The second question leads to the study of two types of transition phenomena associated with an IFS family that depend on a real parameter. These are called lower and upper transition attractors. Their existence and properties are the main topic of this paper. Lower transition attractors are related to the semiattractors, introduced by Lasota and Myjak in 1990s. Upper transition attractors are related to the problem of continuous dependence of an attractor upon the IFS. A main result states that, for a wide class of IFS families, there is a threshold such that the IFSs in the one-parameter family have an attractor for parameters below the threshold and they have no attractor for parameters above the threshold. At the threshold there exists a unique upper transition attractor.

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Contractive Iterated Function Systems Enriched with Nonexpansive Maps

Cited by 7 publications

References 17 publications

Transition Phenomena for the Attractor of an Iterated Function System

Transition Phenomena for the Attractor of an Iterated Function System

Linear first order differential operators and their Hutchinson-invariant sets

Transition phenomena for the attractor of an iterated function system*

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