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

@article{Karaliolios2015ContinuousSO,
title={Continuous Spectrum or Measurable Reducibility for Quasiperiodic Cocycles in \$\$\{\mathbb\{T\} ^\{d\} \times SU(2)\}\$\$Td×SU(2)},
author={Nikolaos Karaliolios},
journal={Communications in Mathematical Physics},
year={2015},
volume={358},
pages={741-766}
}
• Nikolaos Karaliolios
• Published 30 November 2015
• Physics, Mathematics
• Communications in Mathematical Physics
AbstractWe continue our study of the local theory for quasiperiodic cocycles in $${\mathbb{T} ^{d} \times G}$$Td×G , where $${G=SU(2)}$$G=SU(2) , 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 $${L^{2}(\mathbb{T} ^{d}) \hookrightarrow L^{2}(\mathbb{T} ^{d} \times G)}$$L2(Td)↪L2(Td×G… Expand
3 Citations

#### Figures from this paper

Fibered rotation vector and hypoellipticity for quasi‐periodic cocycles in compact Lie groups
Using weak solutions to the conjugation equation, we define a fibered rotation vector for almost reducible quasi-periodic cocycles in $\mathbb{T}^{d} \times G$, $G$ a compact Lie group, over aExpand
Ju l 2 01 9 Invariant Distributions and local theory of quasiperiodic cocycles in T d × SU ( 2 )
We study the linear cohomological equation in the smooth category over quasi-periodic cocycles in T × SU(2). We prove that, under a full measure condition on the rotation in T, for a generic cocycleExpand
Self-duality triggered dynamical transition
• Physics, Mathematics
• 2020
A basic result about the dynamics of spinless quantum systems is that the Maryland model exhibits dynamical localization in any dimension. Here we implement mathematical spectral theory and numericalExpand

#### References

SHOWING 1-10 OF 40 REFERENCES
Invariant Distributions and local theory of quasiperiodic cocycles in $\mathbb{T} ^{d} \times SU(2)$}
We study the linear cohomological equation in the smooth category over quasi-periodic cocycles in $\mathbb{T} ^{d} \times SU(2)$. We prove that, under a full measure condition on the rotation inExpand
Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups
We study close-to-constants quasiperiodic cocycles in \begin{document} $\mathbb{T} ^{d} \times G$ \end{document} , where \begin{document} $d \in \mathbb{N} ^{*}$ \end{document} and \begin{document}Expand
On the degree of cocycles with values in the groupSU(2)
AbstractIn this paper are presented some properties of smooth cocycles over irrational rotations on the circle with values in the groupSU(2). It is proved that the degree of anyC2-cocycle (the notionExpand
Rigidity of the Reducibility of Gevrey Quasi-periodic Cocycles on U(n)
• Mathematics
• 2013
We consider the reducibility problem of cocycles $(\alpha,A)$ on $\T^d\times U(n)$ in Gevrey classes, where $\alpha$ is a Diophantine vector. We prove that, if a Gevrey cocycle is conjugated to aExpand
On cocycles with values in the group SU(2)
In this paper we introduce the notion of degree for $C^1$-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a $C^1$-cocycle $\phi:S^1\to SU(2)$ overExpand
The absolute continuous spectrum of skew products of compact Lie groups
Let $X$ and $G$ be compact Lie groups, $F_1:X\to X$ the time-one map of a $C^\infty$ measure-preserving flow, $\phi:X\to G$ a continuous function and $\pi$ a finite-dimensional irreducible unitaryExpand
Local Rigidity of Reducibility of Analytic Quasi-periodic Cocycles on U(n)
• Mathematics
• 2009
In this paper, we consider the analytic reducibility problem of an analytic $d-$dimensional quasi-periodic cocycle $(\alpha,\ A)$ on $U(n)$ where $\alpha$ is a Diophantine vector. We prove that,Expand
Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups
In this PhD thesis we study quasiperiodic cocycles in semi-simple compact Lie groups. For the greatest part of our study, we will focus ourselves to one-frequency cocyles. We will prove thatExpand
Rigidity results for quasiperiodic SL(2, R) -cocycles
• Mathematics
• 2010
In this paper we introduce a new technique that allows us to investigate reducibility properties of smooth SL(2, R)-cocycles over irrational rotations of the circle beyond the usual DiophantineExpand
Almost reducibility for finitely differentiable SL(2,R)-valued quasi-periodic cocycles
Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k timesExpand