On Anosov diffeomorphisms with asymptotically conformal periodic data

Kalinin, Boris; Sadovskaya, Victoria

doi:10.1017/s0143385708000357

Cited by 23 publications

(24 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In general, this is not the case for systems with higherdimensional distributions [dlL92, dlL02]. Positive results were established for certain classes of diffeomorphisms that are conformal on the full stable and unstable distributions, [dlL02,KS03,dlL04,KS09]. In a different direction, local rigidity was proved in [G08] for an irreducible Anosov toral automorphism L : T d → T d with real eigenvalues of distinct moduli, as well as for some nonlinear systems with similar structure.…”

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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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.

Self Cite

View full text Add to dashboard Cite

show abstract

“…[ KS03,Sad05,KS09] show that the existence of a conformal structure on the stable and the unstable foliations for an Anosov systems implies also some global properties of the manifold.…”

Section: Existence Of Invariant Conformal Structures On the Stable Anmentioning

confidence: 99%

“…Published after this paper was originally submitted, [Kal08] contains a result that removes the localization assumption for all matrix groups (more precisely, that the localization assumption is implied by the periodic-orbit obstruction). See also [KS09].…”

Section: Introductionmentioning

confidence: 99%

Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems

Llave¹,

Windsor²

2009

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

Link to this article: http://journals.cambridge.org/abstract_S014338570900039XHow to cite this article: RAFAEL DE LA LLAVE and ALISTAIR WINDSOR (2010). Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems.Abstract. The celebrated Livšic theorem [A. N. Livšic, Certain properties of the homology of Y -systems, Mat. Zametki 10 (1971), 555-564; A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296-1320] states that given a manifold M, a Lie group G, a transitive Anosov diffeomorphism f on M and a Hölder function η : M → G whose range is sufficiently close to the identity, it is sufficient for the existence of φ : M → G satisfying η(x) = φ( f (x))φ(x) −1 that a condition-obviously necessary-on the cocycle generated by η restricted to periodic orbits is satisfied. In this paper we present a new proof of the main result. These methods allow us to treat cocycles taking values in the group of diffeomorphisms of a compact manifold. This has applications to rigidity theory. The localization procedure we develop can be applied to obtain some new results on the existence of conformal structures for Anosov systems.

show abstract

“…Combining then Main Theorem and the local rigidity result for automorphisms with complex eigenvalues by Kalinin and Sadovskaya [KS09] yields the following corollary.…”

Section: Introductionmentioning

confidence: 73%

“…Deformation rigidity and then local (and global) rigidity was first discovered by de la Llave, Marco and Moriyón [MM87,dlL87] for hyperbolic automorphisms of the 2-torus and, since then, was generalized to certain classes of hyperbolic conformal automorphisms, i.e., automorphisms whose spectrum is confined to a circle of radius greater than 1 and a circle of radius less than 1 in C; see [dlL04,KS09] and references therein. Weaker form of local rigidity, which only yields C 1+Hölder regularity of the conjugating homeomorphism, was established for a rather large class of hyperbolic automorphisms of higher dimensional tori in [GKS11].…”

Section: Introductionmentioning

confidence: 99%

Bootstrap for Local Rigidity of Anosov Automorphisms on the 3-Torus

Gogolev

2017

Commun. Math. Phys.

View full text Add to dashboard Cite

We establish a strong form of local rigidity for hyperbolic automorphisms of the 3-torus with real spectrum. Namely, let L : T 3 → T 3 be a hyperbolic automorphism of the 3-torus with real spectrum and let f be a C 1 small perturbation of L. Then f is smoothly (C ∞ ) conjugate to L if and only if obstructions to C 1 conjugacy given by the eigenvalues at periodic points of f vanish. By combining our result and a local rigidity result of Kalinin and Sadovskaya for conformal automorphisms [KS09] this completes the local rigidity program for hyperbolic automorphisms in dimension 3. Our work extends de la Llave-Marco-Moriyón 2-dimensional local rigidity theory [MM87, dlL87, dlL92].

show abstract

On Anosov diffeomorphisms with asymptotically conformal periodic data

Cited by 23 publications

References 19 publications

Cohomology of fiber bunched cocycles over hyperbolic systems

Cohomology of fiber bunched cocycles over hyperbolic systems

Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems

Bootstrap for Local Rigidity of Anosov Automorphisms on the 3-Torus

Contact Info

Product

Resources

About