Local Rigidity For Anosov Automorphisms

Gogolev, Andrey; Kalinin, Boris; Sadovskaya, Victoria; Canosa, Rafael de la Llave

doi:10.4310/mrl.2011.v18.n5.a4

Cited by 27 publications

(46 citation statements)

References 22 publications

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“…) are close to the identity. Observe that h • h 0 is a C 1 conjugacy between h 0 • f • h −1 0 and A, then it naturally satisfies the smooth conjugacy hypothesis from [9], therefore we obtain that h 0 • f • h −1 0 is C ∞ conjugate to A. Now because h 0 and h • h −1 0 are C ∞ then h is C ∞ as we wanted to show.…”

Section: Preliminariesmentioning

confidence: 74%

Rigidity for partially hyperbolic diffeomorphisms

Varão¹

2017

Ergod. Th. Dynam. Sys.

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In this work we completely classify C ∞ conjugacy for conservative partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain C ∞ conjugacy to its linear Anosov automorphism. On a recent work Avila, Viana and Wilkinson proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate sceneario for our context since it is not possible to obtain an analogous result of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular a metric condition on the center foliation implies ergodicity and C ∞ center foliation.

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Section: Preliminariesmentioning

confidence: 74%

Rigidity for partially hyperbolic diffeomorphisms

Varão¹

2017

Ergod. Th. Dynam. Sys.

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“…Hence the cocycle A = L|E L i is conformal in some norm. In [GKS11], conformality of B at the periodic points together with the assumption that the distributions E L i and E f i are either one-or two-dimensional allows us to conclude that, by [KS10, Theorem 1.3], the cocycle B is conformal. In higher dimensions, conformality at the periodic points does not imply conformality [KS10, Proposition 1.2].…”

Section: An Application: Smooth Conjugacy To a Small Perturbation Formentioning

confidence: 99%

Cohomology of fiber bunched cocycles over hyperbolic systems

Sadovskaya¹

2014

Ergod. Th. Dynam. Sys.

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Abstract. We consider Hölder continuous fiber bunched GL(d, R)-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of the diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question when an Anosov diffeomorphism is smoothly conjugate to a C 1 -small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of GL(d, R), for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle. InroductionCocycles and their cohomology arise naturally in the theory of group actions and play an important role in dynamics. In this paper we study cohomology of Hölder continuous group-valued cocycles over hyperbolic dynamical systems. Our motivation comes in part from questions in local and global rigidity for hyperbolic systems and actions, where the derivative and the Jacobian provide important examples of cocycles. We state our results for the case of an Anosov diffeomorphism, but they also hold for cocycles over hyperbolic sets and symbolic dynamical systems.

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“…Thus the Gibbs expanding state which we will use in our considerations will be the volume, for all the one dimensional subfoliations. In order to show that the conjugacy or the center holonomy preserves various one dimensional subfoliations, and in order to bootstrap for better regularity, we will use some previous results of Gogolev and others ( [24,25,26]).…”

Section: 2mentioning

confidence: 99%

Lyapunov exponents and rigidity of Anosov automorphisms and skew products

Saghin

Yang

2019

Advances in Mathematics

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In this paper we obtain local rigidity results for linear Anosov diffeomorphisms in terms of Lyapunov exponents. More specifically, we show that given an irreducible linear hyperbolic automorphism L with simple real eigenvalues with distinct absolute values, any small perturbation preserving the volume and with the same Lyapunov exponents is smoothly conjugate to L.We also obtain rigidity results for skew products over Anosov diffeomorphisms. Given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an Anosov automorphism of the 2-torus, we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable Lyapunov exponents, the center foliation is smooth.

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Local Rigidity For Anosov Automorphisms

Cited by 27 publications

References 22 publications

Rigidity for partially hyperbolic diffeomorphisms

Rigidity for partially hyperbolic diffeomorphisms

Cohomology of fiber bunched cocycles over hyperbolic systems

Lyapunov exponents and rigidity of Anosov automorphisms and skew products

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