Liapunov functions for closed relations

Wiandt, Tamas

doi:10.1080/10236190701809315

Cited by 7 publications

(6 citation statements)

References 4 publications

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“…Since ϕ = K, the compactness of X implies the existence of a finite subset of K whose intersection is a subset of η. Therefore, 13 (η) = h and the proof is complete.…”

Section: Theorem 414 If F and G Are Closed Relations On A Compact Hmentioning

confidence: 79%

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Conley decomposition for closed relations

McGehee

Wiandt

2006

Journal of Difference Equations and Applications

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“…Since ϕ = K, the compactness of X implies the existence of a finite subset of K whose intersection is a subset of η. Therefore, 13 (η) = h and the proof is complete.…”

Section: Theorem 414 If F and G Are Closed Relations On A Compact Hmentioning

confidence: 79%

“…Although not presented in this paper, Wiandt has established the existence of a Liapunov function for any closed relation on a second countable compact Hausdorff space [13], thus completing the extension of Conley's theorem to this very general setting.…”

Section: Introductionmentioning

confidence: 99%

Conley decomposition for closed relations

McGehee

Wiandt

2006

Journal of Difference Equations and Applications

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“…In this section we define the Conley attractor of an IFS, discuss the existence of attractorrepeller pairs (Theorem 12.5) and their relation to the dynamics of the IFS (Theorem 12.7), and explain Conley's "landscape picture" as it applies to an IFS (Theorem 12.9). Our presentation is a simplified version of the elegant work of McGehee and Wiandt [92,126] about iterated closed relations on compact Hausdorff spaces.…”

Section: Conley Attractorsmentioning

confidence: 99%

“…Finally Conley's "landscape picture" is described, as it applies to an invertible IFS F on a compact metric space. Suppose that A is an attractor of F. We have simplified the original statement of Theorem 12.9 in [126,Theorem 6.19] by noting that a compact metric space is second countable.…”

Section: Conley Attractorsmentioning

confidence: 99%

Developments in fractal geometry

Barnsley¹,

Vince²

2013

Bull. Math. Sci.

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Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.

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“…Here, the updating map f is set-valued, and in particular a function from X into the set of non-empty closed subsets of X. The general dynamical theory of set-valued (closed) updating maps has been studied under the terminology of closed relations by McGehee [19] and Akin [1] and pushed further by a variety of authors, for example [18], [22] and [8]. Specific instances of the dynamics of set-valued maps have been considered even earlier, for example in [16].…”

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confidence: 99%

2-manifolds and inverse limits of set-valued functions on intervals

Greenwood¹,

Suabedissen²

2017

Discrete &Amp; Continuous Dynamical Systems - A

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Suppose for each n ∈ N, fn : [0, 1] → 2 [0,1] is a function whose graph Γ(fn) = (x, y) ∈ [0, 1] 2 : y ∈ fn(x) is closed in [0, 1] 2 (here 2 [0,1] is the space of non-empty closed subsets of [0, 1]). We show that the generalized inverse limit lim ← − (fn) = (xn) ∈ [0, 1] N : ∀n ∈ N, xn ∈ fn(x n+1 ) of such a sequence of functions cannot be an arbitrary continuum, answering a longstanding open problem in the study of generalized inverse limits. In particular we show that if such an inverse limit is a 2-manifold then it is a torus and hence it is impossible to obtain a sphere.2010 Mathematics Subject Classification. Primary: 54C08, 54E45; Secondary: 54F15, 54F65.

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Liapunov functions for closed relations

Cited by 7 publications

References 4 publications

Conley decomposition for closed relations

Conley decomposition for closed relations

Developments in fractal geometry

2-manifolds and inverse limits of set-valued functions on intervals

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