Axiom A
diffeomorphisms derived from Anosov flows

Bonatti, Christian; Guelman, Nancy

doi:10.3934/jmd.2010.4.1

Cited by 25 publications

(23 citation statements)

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“…To overcome this issue, we have based our proof of theorem B on techniques coming from [2]. In that work they provide many tools for the analysis of the germ of a hyperbolic flow in the neighbourhood of a boundary component of an immersed Birkhoff section.…”

Section: Transitiveness and Open Book Decompositionsmentioning

confidence: 99%

“…This notion of well-positionedness will be called tameness and was taken from [2]. Under this assumption, the germ of the flow in a neighbourhood of each γ ∈ Γ i can be described using the information associated with the section (first return and homological coordinates), and theorem 3.9 can be reduced to a local problem near the sets Γ i .…”

Section: The Homeomorphism Hmentioning

confidence: 99%

“…The rest of the section is organized as follows: In subsection 3.1 we summarize all the information from [2] needed for proving theorem 3.9. In subsection 3.2 we state and prove a local version of 3.9.…”

Section: The Homeomorphism Hmentioning

confidence: 99%

“…It is thus relevant to understand if all of these topological Anosov toy models effectively correspond with a hyperbolic dynamic. 2 The general problem of the existence of smooth models for a given expansive dynamic has been treated in other contexts, from where it is relevant to remark the case of pseudo-Anosov homeomorphisms on closed surfaces treated in [18]. (See also [20] and [8]).…”

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confidence: 99%

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Hyperbolic models for transitive topological Anosov flows in dimension three

Shannon

2021

Preprint

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Section: Transitiveness and Open Book Decompositionsmentioning

confidence: 99%

Section: The Homeomorphism Hmentioning

confidence: 99%

Section: The Homeomorphism Hmentioning

confidence: 99%

mentioning

confidence: 99%

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Hyperbolic models for transitive topological Anosov flows in dimension three

Shannon

2021

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“…Notice, that in [4], [19] structurally stable 3-diffeomorphisms with onedimensional attractor-repeller dynamics were constructed, but the constructed basic sets were not canonically embedded in surfaces in that examples. Let M 2 be a closed surface and ψ : M 2 → M 2 be an Ω-stable diffeomorphism.…”

Section: A Rmentioning

confidence: 99%

There are no structural stable Axiom A 3-diffeomorphisms with dynamics "one-dimensional surfaced attractor-repeller"

Pochinka¹

2021

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In this paper, we study the structural stability of three-dimensional diffeomorphisms with source-sink dynamics. Here the role of source and sink is played by one-dimensional hyperbolic repeller and attractor. It is well known that in the case when the repeller and the attractor are solenoids (not embedded in the surface), the diffeomorphism is not structurally stable. The author proves that in the case when the attractor and the repeller are canonically embedded in a surface, the diffeomorphism is also not structurally stable.1 All the manifolds considered in the paper are assumed to be orientable and all diffeomorphisms are assumed to preserve orientation.2 A set A is called an attractor of a diffeomorphism f if it has a compact neighborhood) is called a basin of the attractor A. A repeller is defined as the attractor for f −1 . By a dimension of the attractor (repeller) we mean its topological dimension.

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Research Announcement: Partially Hyperbolic Diffeomorphisms Homotopic to the Identity on 3-Manifolds

et al. 2020

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We study partially hyperbolic diffeomorphisms homotopic to the identity in 3-manifolds. Under a general minimality condition, we show a dichotomy for the dynamics of the (branching) foliations in the universal cover. This allows us to give a full classification in certain settings: partially hyperbolic diffeomorphisms homotopic to the identity on Seifert fibered manifolds (proving a conjecture of Pujals [BW05] in this setting), and dynamically coherent partially hyperbolic diffeomorphisms on hyperbolic 3-manifolds (proving a classification conjecture of Hertz-Hertz-Ures [CRRU15] in this setting). In both cases, up to iterates we prove that the diffeomorphism is leaf conjugate to the time one map of an Anosov flow. Several other results of independent interest are obtained along the way.

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Axiom A diffeomorphisms derived from Anosov flows

Cited by 25 publications

References 0 publications

Hyperbolic models for transitive topological Anosov flows in dimension three

Hyperbolic models for transitive topological Anosov flows in dimension three

There are no structural stable Axiom A 3-diffeomorphisms with dynamics "one-dimensional surfaced attractor-repeller"

Research Announcement: Partially Hyperbolic Diffeomorphisms Homotopic to the Identity on 3-Manifolds

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