A Poincaré–Bendixson theorem for translation lines and applications to prime ends

Koropecki, Andres; Passeggi, Alejandro

doi:10.4171/cmh/457

Cited by 8 publications

(16 citation statements)

References 10 publications

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“…Note that the set

\tilde{ω} false(normalΓ false)

is a continuum which does not separate

S^{2}

. Following [18], we call a continuum

K

a rotational attractor if it is a topological attractor for

f

, and the corresponding external prime ends rotation number is non‐zero (modulo 1). A point

p \in K

is accessible if there is an arc

λ : [0, 1] \to D

, such that

λ ([0, 1)) \subset D ∖ K

and

λ (1) = p \in K

.…”

Section: Preliminariesmentioning

confidence: 99%

“…Remark Our application of Lemma 2.2 to sphere separating attractors is obtained by considering the filled (and thus sphere non‐separating) attractors (see [18]). Later, in Section 6 we shall discuss cofrontier attractors in the annulus, and use the lemma above for the exterior and interior prime end circles, to both of which the lemma can be applied by filling either one of the two complementary domains of the attractors.…”

Section: Preliminariesmentioning

confidence: 99%

“…In view of Theorem A, during the conference Surfaces in Luminy , held at Centre International de Rencontres Mathématiques, 3–7 October 2016, Frédéric Le Roux asked if the boundary of the rotational attractor

\tilde{ω} false(normalΓ false)

could be a Lakes of Wada continuum. Possible existence of such examples was conjectured in [18]. Clearly, in order to construct such an example, it is necessary to understand the corresponding exterior prime ends dynamics better.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Prime ends dynamics in parametrised families of rotational attractors

Boroński

Činč

Liu

2020

J. London Math. Soc.

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We provide several new examples in dynamics on the 2-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincaré-Bendixson Theorem obtained recently by Koropecki and Passeggi, we show the existence of homeomorphisms of S 2 with Lakes of Wada rotational attractors, with an arbitrarily large number of complementary domains, and with or without fixed points, that are arbitrarily close to the identity. This answers a question of Le Roux. Second, from reduced Arnold's family we construct a parametrised family of Birkhoff-like cofrontier attractors, where at least for uncountably many choices of the parameters, two distinct irrational prime ends rotation numbers are induced from the two complementary domains. This example complements the resolution of Walker's Conjecture by Koropecki, Le Calvez and Nassiri from 2015. Third, answering a question of Boyland, we show that there exists a non-transitive Birkhoff-like attracting cofrontier which is obtained from a BBM embedding of inverse limit of circles, such that the interior prime ends rotation number belongs to the interior of the rotation interval of the cofrontier dynamics. There exists another BBM embedding of the same attractor so that the two induced prime ends rotation numbers are exactly the two endpoints of the rotation interval.

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“…Note that the set

\tilde{ω} false(normalΓ false)

is a continuum which does not separate

S^{2}

. Following [18], we call a continuum

K

a rotational attractor if it is a topological attractor for

f

, and the corresponding external prime ends rotation number is non‐zero (modulo 1). A point

p \in K

is accessible if there is an arc

λ : [0, 1] \to D

, such that

λ ([0, 1)) \subset D ∖ K

and

λ (1) = p \in K

.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

\tilde{ω} false(normalΓ false)

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Prime ends dynamics in parametrised families of rotational attractors

Boroński

Činč

Liu

2020

J. London Math. Soc.

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show abstract

“…The Poincaré-Bendixson theorem is one of the most fundamental tools to capture the limit behaviors of orbits of flows and was applied to various phenomena (e.g. [5,10,17,19,[29][30][31][32]). In [6], Birkhoff introduced the concepts of ω-limit set and α-limit set of a point.…”

Section: Introductionmentioning

confidence: 99%

The $ω$-limit set of a flow with arbitrarily many singular points on a surface and surgeries to add singular points

Yokoyama¹

2022

Preprint

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The ω-limit set is one of the fundamental objects in dynamical systems. Using the ω-limit sets, the Poincaré-Bendixson theorem captures the limit behaviors of orbits of flows on surfaces. It was generalized in several ways and was applied to various phenomena. In this paper, we consider what kinds of the ω-limit sets do appear in the non-wandering flows on compact surfaces, and show that the ω-limit set of any non-closed orbit of a non-wandering flow with arbitrarily many singular points on a compact surface is either a subset of singular points or a locally dense Q-set. To show this, we demonstrate a similar result holds for locally dense orbits without non-wanderingness. Moreover, the ω-limit set of any non-closed orbits of a Hamiltonian flow with arbitrarily many singular points on a compact surface consists of singular points.

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“…Koropecki has recently generalized this theorem to any circloid. Furthermore, the periodic orbit can be chosen from

\partial X

provided the rotation interval is non‐degenerate .…”

Section: Introductionmentioning

confidence: 99%

Accessible points rotate as prime ends in backward or forward time

Hernández-Corbato

2018

J. London Math. Soc.

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Let f be a homeomorphism of A¯, the closed annulus, isotopic to the identity and let X be a closed f‐invariant subset of A¯ whose complement is homeomorphic to the half‐open annulus. The dynamics in the circle of prime ends of the complement of X has an associated rotation number, ρ. We prove that either the rotation number of all forward semi‐orbits of accessible points of X are well defined and equal to ρ or the rotation number of all backward semi‐orbits of accessible points of X is well defined and equal to ρ. The result generalizes to the open and half‐open annulus provided the semi‐orbits under consideration are relatively compact.

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A Poincaré–Bendixson theorem for translation lines and applications to prime ends

Cited by 8 publications

References 10 publications

Prime ends dynamics in parametrised families of rotational attractors

Prime ends dynamics in parametrised families of rotational attractors

The $ω$-limit set of a flow with arbitrarily many singular points on a surface and surgeries to add singular points

Accessible points rotate as prime ends in backward or forward time

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