The construction of global attractors

Barge, Marcy; Martin, Joe

doi:10.1090/s0002-9939-1990-1023342-1

Cited by 43 publications

(33 citation statements)

References 6 publications

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“…If k = 0 then t = s which is impossible, and when k ≥ 1 then −t ≤ 0 < 2πk − s which is again a contradiction, since −t = 2πk − s. The following theorem summarizes a result of Barge and Martin from [2] that allows to present any arc-like continuum as a global attractor of a planar homeomorphism. …”

Section: Next Observe That If α Has Self-intersections Then There Arementioning

confidence: 95%

“…circles). 2 A continuum is indecomposable if it cannot be expressed as the union of two proper subcontinua. It is hereditarily indecomposable if each subcontinuum is indecomposable.…”

Section: Continuamentioning

confidence: 99%

“…Because it seems that the proof of this fact has never been published, for completeness sake we recall the procedure from [2], and adopt it to the case of degree 1 circle maps. Proof In our construction, as usual, we will be using polar coordinates, with the angular projection π θ given by π θ (r, θ) = θ .…”

Section: E H Restricted To K ← Agrees With the Shift Homeomorphism mentioning

confidence: 99%

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Rotational chaos and strange attractors on the 2-torus

Boroński

Oprocha

2014

Math. Z.

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We construct a 2-torus homeomorphism h homotopic to the identity with an attracting R. H. Bing's pseudocircle C such that the rotation set of h|C is not a unique vector. The only known examples of such an attractor were Birkhoff's attractors, arising for dissipative maps with a twist. Our construction relies heavily on the Barge-Martin method for constructing attractors as inverse limits of graphs.

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Section: Next Observe That If α Has Self-intersections Then There Arementioning

confidence: 95%

“…circles). 2 A continuum is indecomposable if it cannot be expressed as the union of two proper subcontinua. It is hereditarily indecomposable if each subcontinuum is indecomposable.…”

Section: Continuamentioning

confidence: 99%

Section: E H Restricted To K ← Agrees With the Shift Homeomorphism mentioning

confidence: 99%

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Rotational chaos and strange attractors on the 2-torus

Boroński

Oprocha

2014

Math. Z.

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“…On the one hand side, inverse limits of graphs are often used to construct attractors of dynamical systems on manifolds, where the dynamics on the attractor is conjugate to the shift homeomorphism (see e.g. [4,6,7,19,38,43]). On the other hand, famously there are situations where the converse is true: a dynamical system on a manifold can be locally translated to the dynamics of a shift homeomorphism on an inverse system of graphs (see e.g.…”

Section: Introductionmentioning

confidence: 99%

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

Boroński

Oprocha

2015

J Dyn Diff Equat

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We prove that if f : G → G is a map on a topological graph G such that the inverse limit lim ← − (G, f ) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if G = [0, 1], then h( f ) ∈ {0, ∞}, and combined with a result of Ito shows that certain dynamical systems on compact finitedimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

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“…In 1990, M. Barge and J. Martin [BM90] proved that the shift map on the inverse limit space ([0, 1], f ), for any map f : [0, 1] → [0, 1], can be realized as a global attractor in the plane. In 1960, M. Brown [Bro60] proved that the inverse limit space of any near homeomorphism (Definition 1.2) of a compact metric space is homeomorphic to the original space.…”

Section: Introductionmentioning

confidence: 99%