Recurrence and genericity

Bonatti, Christian; Crovisier, Sylvain

doi:10.1016/s1631-073x(03)00203-6

Cited by 10 publications

(3 citation statements)

References 12 publications

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“…Take f ∈ Diff 1 ω (T 4 ) a diffeomorphisms sufficiently C 1 -close to f a,b such that the homoclinic class H(q f ) is the entire manifold T 4 , where q f is the continuation of the hyperbolic periodic point q. This is possible since C 1 -generically in Diff 1 ω (T 4 ) the entire manifold is the homoclinic class of any periodic point, see theorem 1.3 in [BC03].…”

Section: The Examplementioning

confidence: 99%

On the stable ergodicity of diffeomorphisms with dominated splitting

Obata

2019

Nonlinearity

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In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chainhyperbolicity. It is obtained one criterion for diffeomorphisms with dominated splitting and one criterion for weakly partially hyperbolic diffeomorphisms. As an application of one of these criteria, we obtain the C 1 -density of stable ergodicity inside a certain class of weakly partially hyperbolic diffeomorphisms.

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Section: The Examplementioning

confidence: 99%

On the stable ergodicity of diffeomorphisms with dominated splitting

Obata

2019

Nonlinearity

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“…LEMMA 2.1 ( [BC1]). Generically in Diff 1 (M), every homoclinic class is a chain recurrent class; Equivalently, every chain recurrent class containing a periodic point p coincide with the homoclinic class of p.…”

Section: Preliminariesmentioning

confidence: 99%

A Geometric Model of Mixing Lyapunov Exponents Inside Homoclinic Classes in Dimension Three

Li¹

2017

Tokyo J. Math.

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For C 1 diffeomorphisms of three dimensional closed manifolds, we provide a geometric model of mixing Lyapunov exponents inside a homoclinic class of a periodic saddle p with non-real eigenvalues. Suppose p has stable index two and the sum of the largest two Lyapunov exponents is greater than log(1 − δ), then δ-weak contracting eigenvalues are obtained by an arbitrarily small C 1 perturbation. Using this result, we give a sufficient condition for stabilizing a homoclinic tangency within a given C 1 perturbation range.

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“…(a) Every periodic point of f is hyperbolic and all their invariant manifolds are transverse (Kupka-Smale). (b) C f (p) = H f (p), where p is a hyperbolic periodic point ( [3]).…”

Section: Then Clearly ξ ⊂ H F (P) Since F Has the Shadowing Property ...mentioning

confidence: 99%