Linearization of cohomology-free vector fields

Flaminio, Livio; Paternain, Miguel

doi:10.3934/dcds.2011.29.1031

Cited by 2 publications

(2 citation statements)

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“…Namely, Greenfield and Wallach in [13] as well as Katok in [14] conjectured: This conjecture is proved in several special situations: when M is the torus (see [18] and references therein); when dim(M) ≤ 3 [11], [18]; and when M is a homogeneous finite volume space G/D (where G is a connected Lie group and D a closed subgroup of G such that G/D has a finite G-invariant smooth measure) and ρ is a homogeneous flow [9]. Other advances in the direction of Conjecture 1 on general manifolds are [26], [10].…”

Section: Introduction and Resultmentioning

confidence: 99%

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Damjanović¹,

Tanis²,

Wang³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We define globally hypoelliptic smooth R k actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When k = 1, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous R k actions (k ≥ 2) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic R 2 actions on homogeneous spaces G/Γ, with at least one quasi-unipotent generator, where G = SL(n, R). We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.2010 Mathematics Subject Classification. Primary: 37C15, 37C85, 37D20.

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Section: Introduction and Resultmentioning

confidence: 99%

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Damjanović¹,

Tanis²,

Wang³

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…The best general result to date in the direction of a proof is the joint paper of the third author [RHRH06] where it is proved that every cohomology-free vector field has a factor smoothly conjugate to Diophantine linear flow on a torus of the dimension equal to the first Betti number of the manifold M. This result has been developed independently by several authors [For08], [Koc09], [Mat09] to give a complete proof of the conjecture in dimension 3 and by the first author in the joint paper [FP11] to prove that every cohomology-free flow can be embedded continuously as a linear flow in a possibly non-separated Abelian group.…”

Section: Introductionmentioning

confidence: 99%

Invariant distributions for homogeneous flows and affine transformations

Flaminio

Forni

Hertz³

2016

JMD

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We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases. on the first version of this paper which led to a significant improvement of the exposition and of the results.L.The goal of this section is to prove the following theorem.Theorem 2.1. Let M be a compact connected manifold, φ t , t ∈ R or t ∈ Z, an R-action (a flow) or a Z-action (a diffeomorphism) on M and assume φ t leaves invariant a foliation F with smooth leaves and continuous tangent bundle, e.g. the unstable foliation of a partially hyperbolic flow. Assume also that the action φ t expands the norm of the vectors tangent to F uniformly. Then there are infinitely many different φ t -minimal sets.

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Linearization of cohomology-free vector fields

Abstract: We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.

Cited by 2 publications

References 25 publications

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Invariant distributions for homogeneous flows and affine transformations

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