Accessible points rotate as prime ends in backward or forward time

Hernández-Corbato, Luis

doi:10.1112/jlms.12159

Cited by 4 publications

(7 citation statements)

References 13 publications

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“…In full generality, it is not true that when 𝜌(𝐷) is rational, there are periodic points in 𝜕𝐷 and for some examples, 𝜌(𝐷) is irrational and 𝜕𝐷 is not periodic point free. Anyway, the only result on this subject we need is the following classical lemma (as usual, a point 𝑧 ∈ 𝜕𝐷 is said to be accessible if there exists a simple arc 𝛾 ∶ [0, 1] → 𝐷 ∪ 𝜕𝐷 such that 𝛾([0, 1[) ⊂ 𝐷 and 𝛾(1) = 𝑧) whose proof, for instance, can be found in [9,Theorem 16].…”

Section: Prime Ends Compactification Of Open Disksmentioning

confidence: 99%

Mather's regions of instability for annulus diffeomorphisms

Addas‐Zanata,

Tal

2023

Bulletin of London Math Soc

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Let be a diffeomorphism of the closed annulus that preserves orientation and the boundary components, and be a lift of to its universal covering space. Assume that is a Birkhoff region of instability for , and the rotation set of is a nondegenerate interval. Then there exists an open ‐invariant essential annulus whose frontier intersects both boundary components of , and points and in , such that the positive (resp., negative) orbit of converges to a set contained in the upper (resp., lower) boundary component of and the positive (resp., negative) orbit of converges to a set contained in the lower (resp., upper) boundary component of . This extends a celebrated result originally proved by Mather in the context of area‐preserving twist diffeomorphisms.

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Section: Prime Ends Compactification Of Open Disksmentioning

confidence: 99%

Mather's regions of instability for annulus diffeomorphisms

Addas‐Zanata,

Tal

2023

Bulletin of London Math Soc

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“…In order to relate the value of prime ends rotation number with the actual dynamics, one has to understand the sets of accessible points of K. For this purpose we will need a recent result obtained by Hernández-Corbato [14], to be stated in Lemma 2.2. We start by recalling the context of his result; see [14, Even though the original statements in [14] did not include the adjective 'exterior', we have added it here, because later when we deal with cofrontiers we shall also consider the points that are accessible from the interior, and their respective interior prime ends rotation numbers (see Remark 2.3). Since any cofrontier separate S 2 into exactly two components, we may call either one of them the exterior and the other complementary domain is then called the interior.…”

Section: Surface Dynamics and Prime Ends Rotation Numbersmentioning

confidence: 99%

“…In order to relate the value of prime ends rotation number with the actual dynamics, one has to understand the sets of accessible points of

K

. For this purpose we will need a recent result obtained by Hernández‐Corbato [14], to be stated in Lemma 2.2. We start by recalling the context of his result; see [14, Sections 2–3] for more details.…”

Section: Preliminariesmentioning

confidence: 99%

“…For this purpose we will need a recent result obtained by Hernández‐Corbato [14], to be stated in Lemma 2.2. We start by recalling the context of his result; see [14, Sections 2–3] for more details. Let

K

be a non‐separating continuum in the interior of a closed disk

D

.…”

Section: Preliminariesmentioning

confidence: 99%

“…The lifted exterior prime end rotation number

{\overset{ρ}{true}}_{ex} false(\overset{f}{true}, K false) \in R

, is defined as the rotation number of the restriction of

\overset{\tilde{f}}{true}

L (U)

, and

{\overset{ρ}{true}}_{ex} false(\overset{f}{true}, K false)

modulo

double-struckZ

is the exterior prime ends rotation number

ρ_{ex} false(f, K false)

. Even though the original statements in [14] did not include the adjective ‘exterior’, we have added it here, because later when we deal with cofrontiers we shall also consider the points that are accessible from the interior, and their respective interior prime ends rotation numbers (see Remark 2.3). Since any cofrontier separate

S^{2}

into exactly two components, we may call either one of them the exterior and the other complementary domain is then called the interior.…”

Section: Preliminariesmentioning

confidence: 99%

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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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Parametrized family of pseudo‐arc attractors: Physical measures and prime end rotations

Činč

Oprocha

2022

Proceedings of London Math Soc

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The main goal of this paper is to study topological and measure‐theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure‐preserving map f$f$ generates the pseudo‐arc as inverse limit with f$f$ as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure‐preserving maps f$f$ the background Oxtoby–Ulam measures induced by Lebesgue measure for f$f$ on the interval are physical on the disc and in addition there is a dense set of maps f$f$ defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure‐preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo‐arc attractors with prime ends rotation numbers varying continuously in false[0,1/2false]$[0,1/2]$. It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.

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Accessible points rotate as prime ends in backward or forward time

Cited by 4 publications

References 13 publications

Mather's regions of instability for annulus diffeomorphisms

Mather's regions of instability for annulus diffeomorphisms

Prime ends dynamics in parametrised families of rotational attractors

Parametrized family of pseudo‐arc attractors: Physical measures and prime end rotations

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