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

Karaliolios, Nikolaos

doi:10.1007/s00220-017-3034-3

Cited by 7 publications

(15 citation statements)

References 28 publications

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“…In particular, we obtain the following affirmative answer to [, problem 2] in our context. We remind that the conjugation operator associated to a cocycle

Φ = (α, A (\cdot))

is the operator acting on

C^{\infty} false(T^{d}, G false)

H false(\cdot false) \mapsto H false(\cdot + α false) . A false(\cdot false) H^{*} false(\cdot false) .

Theorem is thus the non‐linear analogue of the main theorem in . Corollary If the conjugation operator of an almost reducible cocycle (or quasi‐periodic flow) in

{double-struckT}^{d} \times G

G

a compact Lie group, is globally hypoelliptic, then

G

is a torus.…”

Section: Statement Of Resultsmentioning

confidence: 92%

“…The concept of almost reducibility, introduced by Eliasson, captures the fact that a generic cocycle is not reducible, that is, is not conjugate to a rigid rotation. This is a direct consequence of the main theorem of [5], which was subsequently refined and strengthened in [9,13]. One thus has to renounce in reducibility of all systems, but the next best thing was proved to be true: any cocycle, sufficiently close to a constant one, can be conjugated arbitrarily close to a constant one (see Definition 2.10 for a formal definition).…”

Section: Some Commentsmentioning

confidence: 99%

“…In this notation, the K.A.M. normal form of almost reducible cocycles (see [9,12,13] for a figure) is established as follows. For a cocycle in normal form, we abbreviate n i to i.…”

Section: Cocyclesmentioning

confidence: 99%

“…Proof The proof follows directly from [, § 5.1]. Assume without loss of generality that both sequences of conjugations

H_{i} false(\cdot false)

and

H_{i}^{'} false(\cdot false)

correspond to K.A.M.…”

Section: The Basic Lemmasmentioning

confidence: 99%

“…In this notation, the K.A.M. normal form of almost reducible cocycles (see for a figure) is established as follows. Lemma Under the assumptions of Theorem , there exists a conjugation

D false(\cdot false) \in C^{\infty} false(T^{d}, G false)

such that the K.A.M.…”

Section: Notationmentioning

confidence: 99%

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Fibered rotation vector and hypoellipticity for quasi‐periodic cocycles in compact Lie groups

Karaliolios¹

2020

Bull. London Math. Soc.

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Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi‐periodic cocycles in double-struckTd×G, G a compact Lie group, over a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in Td, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield–Wallach conjecture in PDEs.

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“…In particular, we obtain the following affirmative answer to [, problem 2] in our context. We remind that the conjugation operator associated to a cocycle

Φ = (α, A (\cdot))

is the operator acting on

C^{\infty} false(T^{d}, G false)

H false(\cdot false) \mapsto H false(\cdot + α false) . A false(\cdot false) H^{*} false(\cdot false) .

Theorem is thus the non‐linear analogue of the main theorem in . Corollary If the conjugation operator of an almost reducible cocycle (or quasi‐periodic flow) in

{double-struckT}^{d} \times G

G

a compact Lie group, is globally hypoelliptic, then

G

is a torus.…”

Section: Statement Of Resultsmentioning

confidence: 92%

Section: Some Commentsmentioning

confidence: 99%

“…In this notation, the K.A.M. normal form of almost reducible cocycles (see [9,12,13] for a figure) is established as follows. For a cocycle in normal form, we abbreviate n i to i.…”

Section: Cocyclesmentioning

confidence: 99%

“…Proof The proof follows directly from [, § 5.1]. Assume without loss of generality that both sequences of conjugations

H_{i} false(\cdot false)

and

H_{i}^{'} false(\cdot false)

correspond to K.A.M.…”

Section: The Basic Lemmasmentioning

confidence: 99%

“…In this notation, the K.A.M. normal form of almost reducible cocycles (see for a figure) is established as follows. Lemma Under the assumptions of Theorem , there exists a conjugation