On the invariant distributions of $$C^2$$ circle diffeomorphisms of irrational rotation number

Navas, Andrés; Triestino, Michele

doi:10.1007/s00209-012-1071-3

Cited by 15 publications

(16 citation statements)

References 4 publications

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“…In the same spirit of the preceding remark, there is an alternative end of proof for Lemma 3.9. Namely, after having detected the element h ∈ G of irrational rotation number, we may invoke a theorem of Herman from [20], according to which the sequence of diffeomorphisms h q i converges to the identity in the C 1 topology (see [32] for a much shorter proof of this fact). Now, using this sequence and Remark 2.9, we may conclude as above that there is a sequence of elements g n ∈ G for which lim n→∞ g n (x 0 ) > 1, which is a contradiction.…”

Section: Proof Of the Main Theorem For Minimal Actionsmentioning

confidence: 99%

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

Deroin¹,

Kleptsyn

Navas

2017

Invent. math.

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We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due toÉ. Ghys, G.

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Section: Proof Of the Main Theorem For Minimal Actionsmentioning

confidence: 99%

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

Deroin¹,

Kleptsyn

Navas

2017

Invent. math.

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“…One of the motivations for studying local flows for non‐locally discrete groups (see for instance ) was to extend the method of Katok and Herman to more general actions. Indeed, the group generated by a minimal circle diffeomorphism

f

is the most natural example of a non‐discrete group (and thus non locally discrete): if

(q_{n})

is the sequence of denominators of the rational approximations of the rotation number of

f

, then the sequence

f^{q_{n}}

tends to the identity in the

C^{1}

topology (see [, Chapter VII] and also ).…”

Section: Foreword and Resultsmentioning

confidence: 99%

Groups with infinitely many ends acting analytically on the circle

Alvarez

Filimonov

Kleptsyn

et al. 2019

Journal of Topology

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This article is inspired by two milestones in the study of non‐minimal group actions on the circle: Duminy's theorem about the number of ends of semi‐exceptional leaves, and Ghys' freeness result in real‐analytic regularity. Our first result concerns groups of real‐analytic diffeomorphisms with infinitely many ends: if the action is non‐expanding, then the group is virtually free. The second result is a Duminy type theorem for minimal codimension‐one foliations: either non‐expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.

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“…Write T := S 1 τ (x)dµ(x) and let pn qn n∈N be the approximation of θ by rational numbers given by the continued fractions algorithm. From the corollary in [NT13], which is a version of Denjoy-Koksma inequality (Corollary C in [AK11]), we obtain the following…”

Section: 21mentioning

confidence: 99%

On the centralizer of vector fields: criteria of triviality and genericity results

Obata

Santiago

2020

Math. Z.

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In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold M d has a "small" centralizer. In the C 1 case, we give two criteria, one of which is C 1 -generic, which guarantees that the centralizer of a C 1 -generic vector field is indeed small, namely collinear. The other criterion states that a C 1 separating flow has a collinear C 1 -centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as quasi-triviality. In particular, the C 1 -centralizer of a C 1 -generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a C 1 -generic vector field, which includes C 1 -generic Axiom A (or sectional Axiom A) vector fields and C 1 -generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is trivial (as small as it can be), and we show that in higher regularity, collinearity and triviality of the C d -centralizer are equivalent properties for a generic vector field in the C d topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity C 2 , the C 1 -centralizer is trivial. 9 4. Quasi-triviality 13 4.1. Collinear does not imply quasi-trivial 13 4.2. The case of hyperbolic zeros 13 5. The study of invariant functions and trivial centralizers 18 5.1. First integrals and trivial C 1 -centralizers 19 5.2. Some results in higher regularity 20 6. The generic case 23 Appendix A. The separating property is not generic 40 References

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On the invariant distributions of $$C^2$$ circle diffeomorphisms of irrational rotation number

Cited by 15 publications

References 4 publications

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

Groups with infinitely many ends acting analytically on the circle

On the centralizer of vector fields: criteria of triviality and genericity results

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