Cohomology free systems and the first Betti number

Hertz, Federico Rodriguez; Hertz, Jana Rodriguez

doi:10.3934/dcds.2006.15.193

Cited by 11 publications

(17 citation statements)

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“…In support of Katok's conjecture Federico and Jana Rodriguez Hertz have recently proved in [RHRH05] that any (compact, connected) manifold M admitting a cohomology free vector field fibres over a torus of dimension equal to the first Betti number of M .…”

Section: Main Resultmentioning

confidence: 93%

On the cohomological equation for nilflows

Flaminio¹,

Forni²

2007

Journal of Modern Dynamics

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Let X be a vector field on a compact connected manifold M. An important question in dynamical systems is to know when a function g:M -> R is a coboundary for the flow generated by X, i.e. when there exists a function f: M->R such that Xf=g. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions D_n such that any sufficiently smooth function g is a coboundary iff it belongs to the kernel of all the distributions D_n.Comment: 27 page

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Section: Main Resultmentioning

confidence: 93%

On the cohomological equation for nilflows

Flaminio¹,

Forni²

2007

Journal of Modern Dynamics

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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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“…For a very long time this was the only known obstruction for the existence of cohomologically rigid vector fields, until F. and J. Rodríguez-Hertz produced a breakthrough in [21], finding additional restrictions for the topology of the supporting manifold.…”

Section: Topological Obstructionsmentioning

confidence: 98%

“…It is important to notice that some authors [21,29,15] use the term "cohomology-free" or "parameter rigid" instead of our terminology. The reader can find some examples of cohomologically rigid actions in [12,16,28,24].…”

Section: Remark 22mentioning

confidence: 99%

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Cohomologically rigid vector fields: the Katok conjecture in dimension 3

Kocsard

2009

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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A smooth vector field X on a closed orientable d-manifold M is said to be cohomologically rigid when given anywhere L X is the Lie derivative in the X direction. In 1984, Anatole Katok conjectured that every cohomologically rigid vector field should be smoothly conjugated to a Diophantine vector field on the d-torus T d . In this work the validity of the Katok conjecture for 3-manifolds is proved. RésuméUn champ de vecteurs X sur une variété M compacte orientable de dimension d est dit cohomologiquement rigide si pour touteoù L X désigne la dérivée de Lie dans la direction de X. En 1984, Anatole Katok a conjecturé que tout champ de vecteurs cohomologiquement rigide devrait être conjugué par un difféomorphisme lisse à un champ linéaire diophantien sur le tore T d . Dans ce travail nous démontrons la conjecture de Katok pour les variétés de dimension trois.

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Cohomology free systems and the first Betti number

Abstract: Abstract. We prove that a cohomology free flow on a manifold M fibers over a diophantine translation on T β 1 where β1 is the first Betti number of M .

Cited by 11 publications

References 0 publications

On the cohomological equation for nilflows

On the cohomological equation for nilflows

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Cohomologically rigid vector fields: the Katok conjecture in dimension 3

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