On classification of higher rank Anosov actions on compact manifold

Damjanović, Danijela; Xu, Di

doi:10.1007/s11856-020-2038-4

Cited by 7 publications

(5 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Global rigidity results-in which all Anosov actions on tori and nilmanifolds are shown to be smoothly conjugate to affine actionshave been established in [53,80,81,[109][110][111]178] with the most complete result being [179]. Under strong dynamical hypotheses, a number of these results including [53,109,110] establish global rigidity results without any assumption on the underlying manifold. [66,123,183].…”

Section: Actions Of Higher-rank Abelian Groupsmentioning

confidence: 99%

Entropy, Lyapunov exponents, and rigidity of group actions

Brown

Alvarez

Malicet

et al. 2019

View full text Add to dashboard Cite

show abstract

Section: Actions Of Higher-rank Abelian Groupsmentioning

confidence: 99%

Entropy, Lyapunov exponents, and rigidity of group actions

Brown

Alvarez

Malicet

et al. 2019

View full text Add to dashboard Cite

show abstract

“…Moreover, for Anosov diffeomorphisms, if the centralizer contains a Z 2 subgroup that does not factor onto a virtually Z-action, Katok and Spatzier conjectured that f is then smoothly conjugate to a hyperbolic (infra)nilmanifold automorphism, and in particular it has a full rank centralizer smoothly conjugate to a group of automorphisms. We refer to [26], [87] and references therein for the history and most recent results in the direction of this conjecture.…”

Section: Lebesgue Disintegration and Large Centralizermentioning

confidence: 99%

“…It was conjectured by Katok and Spatzier that any higher rank Anosov action on a compact manifold is essentially algebraic, i.e. smoothly conjugate to an affine action on a nilmanifold, up to a finite cover of M and up to a finite index subgroup in G. (For more on the Katok-Spatzier global rigidity conjecture, see for example [26] and references therein). This conjecture was proved for Anosov actions on nilmanifolds by F. Rodriguez Hertz and Wang [76] (for the statement on T d , see Theorem 11 in Section 3.8).…”

mentioning

confidence: 99%

Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms

Damjanović¹,

Wilkinson²,

Xu³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

We discover a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. In particular, for smooth, ergodic perturbations of certain algebraic systems -including the discretized geodesic flows over hyperbolic manifolds and certain toral automorphisms with simple spectrum and exactly one eigenvalue on the unit circle, the smooth centralizer is either virtually Z or contains a smooth flow.At the heart of this work are two very different rigidity phenomena. The first was discovered in [2, 3]: for a class of volume-preserving partially hyperbolic systems including those studied here, the disintegration of volume along the center foliation is either equivalent to Lebesgue or atomic. The second phenomenon is the rigidity associated to several commuting partially hyperbolic diffeomorphisms with very different hyperbolic behavior transverse to a common center foliation [25].We introduce a variety of techniques in the study of higher rank, abelian partially hyperbolic actions: most importantly, we demonstrate a novel geometric approach to building new partially hyperbolic elements in hyperbolic Weyl chambers using Pesin theory and leafwise conjugacy, while we also treat measure rigidity for circle extensions of Anosov diffeomorphisms and apply normal form theory to upgrade regularity of the centralizer.

show abstract

“…For higher rank Anosov actions on compact manifolds, the rigidity problem has been widely studied (cf. [11,15,21,23,29], etc.). For local rigidity of certain higher rank partially hyperbolic abelian actions, see [5,7,8,33] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

2022

Self Cite

View full text Add to dashboard Cite

Let $$\mathcal {F}$$ F and $$\mathcal {K}$$ K be commuting $$C^\infty $$ C ∞ diffeomorphisms of the cylinder $$\mathbb {T}\times \mathbb {R}$$ T × R that are, respectively, close to $$\mathcal {F}_0 (x, y)=(x+\omega (y), y)$$ F 0 ( x , y ) = ( x + ω ( y ) , y ) and $$T_\alpha (x, y)=(x+\alpha , y)$$ T α ( x , y ) = ( x + α , y ) , where $$\omega (y)$$ ω ( y ) is non-degenerate and $$\alpha $$ α is Diophantine. Using the KAM iterative scheme for the group action we show that $$\mathcal {F}$$ F and $$\mathcal {K}$$ K are simultaneously $$C^\infty $$ C ∞ -linearizable if $$\mathcal {F}$$ F has the intersection property (including the exact symplectic maps) and $$\mathcal {K}$$ K satisfies a semi-conjugacy condition. We also provide examples showing necessity of these conditions. As a consequence, we get local rigidity of certain class of $$\mathbb {Z}^2$$ Z 2 -actions on the cylinder, generated by commuting twist maps.

show abstract

On classification of higher rank Anosov actions on compact manifold

Cited by 7 publications

References 41 publications

Entropy, Lyapunov exponents, and rigidity of group actions

Entropy, Lyapunov exponents, and rigidity of group actions

Pathology and asymmetry: centralizer rigidity for partially hyperbolic diffeomorphisms

On simultaneous linearization of certain commuting nearly integrable diffeomorphisms of the cylinder

Contact Info

Product

Resources

About