Iterated Function Systems Enriched with Symmetry

Leśniak, Krzysztof; Snigireva, Nina

doi:10.1007/s00365-021-09560-3

Cited by 2 publications

(5 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Figure 2 shows the lower transition attractor and Figure 3 shows the upper transition attractor for the IFS family of Example 1.1. The subject of transition attractors, in two guises, was introduced independently in [20] and [33]. Definition 1.2.…”

Section: Lip(f D)mentioning

confidence: 99%

See 1 more Smart Citation

Transition Phenomena for the Attractor of an Iterated Function System

Leśniak,

Snigireva,

Strobin

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Lip(f D)mentioning

confidence: 99%

“…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%

Transition Phenomena for the Attractor of an Iterated Function System

Leśniak,

Snigireva,

Strobin

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…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%

“…Figure 2 shows the lower transition attractor and figure 3 shows the upper transition attractor for the IFS family of example 1.1. The subject of transition attractors, in two guises, was introduced independently in [20,34].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Transition phenomena for the attractor of an iterated function system*

et al. 2022

View full text Add to dashboard Cite

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.

show abstract

Iterated Function Systems Enriched with Symmetry

Cited by 2 publications

References 26 publications

Transition Phenomena for the Attractor of an Iterated Function System

Transition Phenomena for the Attractor of an Iterated Function System

Transition phenomena for the attractor of an iterated function system*

Contact Info

Product

Resources

About