Quasi-Anosov diffeomorphisms of 3-manifolds

Abstract

Abstract. In 1969, Hirsch posed the following problem: given a diffeomorphism f : N → N and a compact invariant hyperbolic set Λ of f , describe the topology of Λ and the dynamics of f restricted to Λ. We solve the problem where Λ = M 3 is a closed 3-manifold: if M 3 is orientable, then it is a connected sum of tori and handles; otherwise it is a connected sum of tori and handles quotiented by involutions.The dynamics of the diffeomorphisms restricted to M 3 , called quasi-Anosov diffeomorphisms, is also classified: it is the connected sum of DA-diffeomorphisms, quotiented by commuting involutions.

Trivial centralizers for codimension-one attractors

“…A diffeomorphism f is quasi-Anosov if there exists a neighborhood U of f such that each g ∈ U is expansive. For quasi-Anosov diffeomorphisms we will show the next corollary follows from results in [5] and Theorem 1.1.…”

“…✷ Proof of Corollary 1.4. From [5] we know that for every quasi-Anosov diffeomorphism of a 3-manifold there is an open and dense set of points in the manifold contained in the basin of a codimension-one attractor or repeller.…”

Filtrated Pseudo-Orbit Shadowing Property and Approximately Shadowable Measures

“…Every Anosov diffeomorphism is quasi-Anosov, but an example of quasi-Anosov, non-Anosov diffeomorphism is constructed by [6] (for more information, see [7]). In this paper, by making use of a quasi-Anosov diffeomorphism, we construct a C 1 -open set of Diff(M), any element of which does not have the shadowing property but admits an approximately shadowable Lebesgue measure (see Corollary 1).…”

Sistemas quase-Anosov

“…As referências principais são [7,8,14,17,23,26,28,29] e outras que serviram na complementação da teoria foram [6,9,10,12,18,21,25]. …”