Structural stability of C1 diffeomorphisms

“…We say that f satisfies the strong transversality condition if for every x, y ∈ N W (f ), W s (x) is transverse to W u (y). According to Mañé-Robinson's theorem [35], [47], a diffeomorphism f is structurally stable if and only if f is an A-diffeomorphism satisfying the strong transversality condition.…”

Section: Main Definitionsmentioning

confidence: 99%

On structurally stable diffeomorphisms with codimension one expanding attractors

Grines

Zhuzhoma

2004

Trans. Amer. Math. Soc.

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Abstract. We show that if a closed n-manifold M n (n ≥ 3) admits a structurally stable diffeomorphism f with an orientable expanding attractor Ω of codimension one, then M n is homotopy equivalent to the n-torus T n and is homeomorphic to T n for n = 4. Moreover, there are no nontrivial basic sets of f different from Ω. This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on T n , n ≥ 3.

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“…For example, in the diffeomorphisms theory the following result has been known. Theorem A [4,7,16]. Let f be a Cx diffeomorphism of a closed smooth manifold M. Then the following are equivalent:…”

Section: Introductionmentioning

confidence: 99%

“…Theorem A [4,7,16]. Let f be a Cx diffeomorphism of a closed smooth manifold M. Then the following are equivalent:…”

Section: Introductionmentioning

confidence: 99%

Infinitesimally Stable Endomorphisms

Ikeda¹

1994

Transactions of the American Mathematical Society

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Abstract.It is well known that infinitesimal stability of difleomorphisms is an open property. However, infinitesimal stability of endomorphisms is not an open property. So we consider the interior of the set of all infinitesimally stable endomorphisms. We prove that if / belongs to the interior of the set of all infinitesimally stable endomorphisms, then f is iî-stable. This means a generalization of Smale's Q-stability theorem for diffeomorphisms. Moreover, it is proved that for Anosov endomorphisms structural stability is equivalent to lying in the interior of the set of infinitesimally stable endomorphisms. IntroductionIn the theory of dynamical systems structural stability is one of important concepts. Moreover infinitesimal stability is closely related to structural stability. For example, in the diffeomorphisms theory the following result has been known. Theorem A [4,7,16]. Let f be a Cx diffeomorphism of a closed smooth manifold M. Then the following are equivalent:(a) / ¿s C1 structurally stable, (b) / satisfies Axiom A and the strong transversality condition; (c) / is infinitesimally stable. Now we shall attempt to give an outline of the development of infinitesimal stability without giving precise definitions. In the case of diffeomorphisms, Robbin [14], inspired by Moser [9], introduced the concept of infinitesimal stability to prove the structural stability theorem. At first, infinitesimal stability was a medium to prove structural stability; i.e., (b) => (c) => (a). Moreover Mané [4] generalized this concept and proved that (b) is equivalent to (c). In this paper the concept of infinitesimal stability of diffeomorphisms means Mane's generalized version. From the result of Mané [7], we now know that for C1 diffeomorphisms infinitesimal stability is equivalent to structural stability. The concept of infinitesimal stability of endomorphisms virtually appeared in [5]. Using the concept we obtained properties similar to those of infinitesimally stable diffeomorphisms [2,3]. In [3] the following question was stated:Question. Is infinitesimal stability of endomorphisms an open property?

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Structural stability of C1 diffeomorphisms

Cited by 162 publications

References 19 publications

C²DIFFEOMORPHISMS WITH THE INVERSE SHADOWING PROPERTY

C²DIFFEOMORPHISMS WITH THE INVERSE SHADOWING PROPERTY

On structurally stable diffeomorphisms with codimension one expanding attractors

Infinitesimally Stable Endomorphisms

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Structural stability of C1 diffeomorphisms

Cited by 162 publications

References 19 publications

C2DIFFEOMORPHISMS WITH THE INVERSE SHADOWING PROPERTY

C2DIFFEOMORPHISMS WITH THE INVERSE SHADOWING PROPERTY

On structurally stable diffeomorphisms with codimension one expanding attractors

Infinitesimally Stable Endomorphisms

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Product

Resources

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C²DIFFEOMORPHISMS WITH THE INVERSE SHADOWING PROPERTY

C²DIFFEOMORPHISMS WITH THE INVERSE SHADOWING PROPERTY