On manifolds supporting distributionally uniquely ergodic diffeomorphisms

Ávila, Artur; Fayad, Bassam; Kocsard, Alejandro

doi:10.4310/jdg/1421415561

Cited by 6 publications

(15 citation statements)

References 5 publications

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“…The efficiency of the Anosov-Katok construction in producing realizations of exotic dynamics makes it a good candidate for producing counter-examples to the conjecture. To our best knowledge, there have been two recent such attempts, both of them in spaces of quasi-periodic skew-product diffeomorphism spaces, [AFK15] and [Kar14]. Both attempts fail for slightly different reasons, but in the present article we will establish the reason why such attempts should not be expected to produce counter-examples to the conjecture.…”

Section: Introduction 1generalities and Statement Of The Resultsmentioning

confidence: 85%

Cohomological rigidity and the Anosov-Katok construction

Karaliolios

2017

Preprint

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We provide a general argument for the failure of Anosov-Katok-like constructions (as in [AFK15] and [Kar14]) to produce Cohomologically Rigid diffeomorphisms in manifolds other than tori. A C ∞ smooth diffeomorphism f of a compact manifold M is Cohomologically Rigid iff the equation, known as Linear Cohomological one,As an application, we show that no Cohomologically Rigid diffeomorphisms exist in the Almost Reducibility regime for quasi-periodic cocycles in homogeneous spaces of compact type, even though the Linear Cohomological equation over a generic such system admits a solution for a dense subset of functions ϕ. We thus confirm a conjecture by M. Herman and A. Katok in that context and provide some insight in the mechanism obstructing the construction of counterexamples.

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Section: Introduction 1generalities and Statement Of The Resultsmentioning

confidence: 85%

Cohomological rigidity and the Anosov-Katok construction

Karaliolios

2017

Preprint

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“…Remark 4. Group actions whose distributional kernel contains only an ergodic invariant probability measure are labelled distributionally uniquely ergodic (DUE) in [2], where diffeomorphisms with this property were obtained on manifolds different than tori. This indicates that DUE condition is significantly weaker than GH.…”

Section: Direct Consequence Of Proposition 21 Ismentioning

confidence: 99%

“…Fix a finite measurable set E ⊂ Z. For each z ∈ E, let f z = h, where h ∈ C ∞ c (R) with support inside the interval [1,2] satisfying…”

Section: Sl(2 R) × Sl(2 R)mentioning

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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“…Further, and certainly non-exhaustive, literature in the subject includes the works of Chavaudret [Cha11,Cha12,Cha13], also in collaboration with St. Marmi [CM12] and with Stolovich [CS] HP13], Zhou [YZ13], and the paper of Avila-FayadKocsard [AFK12], which triggered this finer study that we took up in our recent papers.…”

Section: Given Any Two Cocycles (αmentioning

confidence: 99%

Continuous Spectrum or Measurable Reducibility for Quasiperiodic Cocycles in $${\mathbb{T} ^{d} \times SU(2)}$$ T d × S U ( 2 )

Karaliolios

2017

Commun. Math. Phys.

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Abstract:We continue our study of the local theory for quasiperiodic cocycles in, over a rotation satisfying a Diophantine condition and satisfying a closeness-to-constants condition, by proving a dichotomy between measurable reducibility (and therefore pure point spectrum), and purely continuous spectrum in the space orthogonal to. Subsequently, we describe the equivalence classes of cocycles under smooth conjugacy, as a function of the parameters defining their K.A.M. normal form. Finally, we derive a complete classification of the dynamics of one-frequency (d = 1) cocycles over a recurrent Diophantine rotation.

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On manifolds supporting distributionally uniquely ergodic diffeomorphisms

Cited by 6 publications

References 5 publications

Cohomological rigidity and the Anosov-Katok construction

Cohomological rigidity and the Anosov-Katok construction

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Continuous Spectrum or Measurable Reducibility for Quasiperiodic Cocycles in $${\mathbb{T} ^{d} \times SU(2)}$$ T d × S U ( 2 )

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