Persistent Nonhyperbolic Transitive Diffeomorphisms

Bonatti, Christian; Díaz, Lorenzo J.

doi:10.2307/2118647

Cited by 175 publications

(191 citation statements)

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“…Let f t : N → N , t ∈ S 1 , be a smooth family of volumepreserving diffeomorphisms on some compact manifold N , such that f t = id for t in some interval I ⊂ S 1 , and f t is partially hyperbolic for t in another interval J ⊂ S 1 . Such families may be obtained, for instance, using the construction of partially hyperbolic diffeomorphisms isotopic to the identity in [7]. Then f : S 1 ×N → S 1 ×N , f (t, x) = (t, f t (x)) is a volume-preserving diffeomorphism for which D ⊃ S 1 × J and Z ⊃ S 1 × I.…”

Section: Dichotomy For Volume-preserving Diffeomorphismsmentioning

confidence: 99%

The Lyapunov exponents of generic volume-preserving and symplectic maps

2005

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We show that the integrated Lyapunov exponents of C 1 volume-preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding Oseledets splitting is trivial (all Lyapunov exponents are equal to zero) or else dominated (uniform hyperbolicity in the projective bundle) almost everywhere.We deduce a sharp dichotomy for generic volume-preserving diffeomorphisms on any compact manifold: almost every orbit either is projectively hyperbolic or has all Lyapunov exponents equal to zero.Similarly, for a residual subset of all C 1 symplectic diffeomorphisms on any compact manifold, either the diffeomorphism is Anosov or almost every point has zero as a Lyapunov exponent, with multiplicity at least 2.Finally, given any set S ⊂ GL(d) satisfying an accessibility condition, for a residual subset of all continuous S-valued cocycles over any measure-preserving homeomorphism of a compact space, the Oseledets splitting is either dominated or trivial. The condition on S is satisfied for most common matrix groups and also for matrices that arise from discrete Schrödinger operators.

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Section: Dichotomy For Volume-preserving Diffeomorphismsmentioning

confidence: 99%

The Lyapunov exponents of generic volume-preserving and symplectic maps

2005

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“…) such that every g ∈ T is topologically transitive but not Anosov (see also [1] for a generalization). It is easy to see that every g ∈ T has the weak shadowing property but does not satisfy Axiom A (and the no-cycle condition).…”

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confidence: 99%

Diffeomorphisms with weak shadowing

Sakai¹

2001

Fund. Math.

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Abstract. The weak shadowing property is really weaker than the shadowing property. It is proved that every element of the C 1 interior of the set of all diffeomorphisms on a C ∞ closed surface having the weak shadowing property satisfies Axiom A and the no-cycle condition (this result does not generalize to higher dimensions), and that the non-wandering set of a diffeomorphism f belonging to the C 1 interior is finite if and only if f is Morse-Smale.The notion of pseudo-orbits very often appears in several branches of the modern theory of dynamical systems; especially, the shadowing property usually plays an important role in stability theory. The weak shadowing property (which is really weaker than the shadowing property) was introduced in [6]. Corless and Pilyugin (see [6, p. 30]) proved that the weak shadowing property is generic in the set of all homeomorphisms on a C ∞ closed manifold endowed with C 0 topology (see [7] for further results). Every diffeomorphism having the shadowing property has the weak shadowing property but the converse is not true. Indeed, an irrational rotation on the unit circle has the weak shadowing property but does not have the shadowing property.The purpose of this paper is to investigate the dynamics of a diffeomorphism belonging to the C 1 interior of the set of all diffeomorphisms on a C ∞ closed surface with the weak shadowing property. The motivation is the following remarkable example on the 2-torus constructed by Plamenevskaya [8] (see also [7]). We describe it briefly before stating our results. Example ([8]). There exists a diffeomorphism g on the 2-torus T 2 satisfying Axiom A and the no-cycle condition such that g has the weak shad-

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“…Bonatti and Diaz [3] have shown that there is an open set of transitive diffeomorphisms near F 0 = f × id (f is an Anosov diffeomorphism and id is the identity map of any manifold) as well as near the time-1 map of a topologically transitive Anosov flow. This result was used in [4] to construct examples of partially hyperbolic systems with minimal unstable foliation (i.e., every unstable leaf is dense in the manifold itself).…”

Section: 2mentioning

confidence: 99%