Quasi-periodic Cocycles Research Articles

Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups

We study close-to-constants quasiperiodic cocycles in $\mathbb{T} ^{d} \times G$, where $d \in \mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We prove differentiable rigidity for such cocycles: if such a cocycle is measurably conjugate to a constant one satisfying a Diophantine condition with respect to the rotation, then it is $C^{\infty}$-conjugate to it, and the K.A.M. scheme actually produces a conjugation. We also derive a global differentiable rigidity theorem, assuming the convergence of the renormalization scheme for such dynamical systems.

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter's butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila's global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequency-dependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila's Global Theory to the present setting, self-duality via the Fourier transform, and a Johnson-type theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.

Large deviation type estimates for iterates of linear cocycles

We describe some methods used to derive large deviation type (LDT) estimates for quantities associated to random and quasi-periodic linear cocycles. We then explain how such LDT estimates can be used in an inductive scheme to prove continuity properties of the Lyapunov exponents as functions of the cocycle. This is a survey of recent work to appear in a research monograph.

Global aspects of the reducibility of quasiperiodic cocycles in semisimple compact Lie groups

In this m\'emoire 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 that $C^{\infty}$ reducible cocycles are dense in the $C^{\infty}$ topology, for a full measure set of frequencies. Moreover, we will show that every cocycle (or an appropriate iterate of it, if homotopy appears as an obstruction) is almost torus-reducible (i.e. can be conjugated arbitrarily close to cocycles taking values in an abelian subgroup of G). In the course of the proof we will firstly define two invariants of the dynamics, which we will call energy and degree and which give a preliminary distinction between (almost-)reducible and non-reducible cocycles. We will then take up the proof of the density theorem. We will show that an algorithm of renormalization converges to perturbations of simple models, indexed by the degree. Finally, we will analyze these perturbations using methods inspired by K.A.M. theory.

Continuity of the Lyapunov Exponents for Quasiperiodic Cocycles

Consider the Banach manifold of real analytic linear cocycles with values in the general linear group of any dimension and base dynamics given by a Diophantine translation on the circle. We prove a precise higher dimensional Avalanche Principle and use it in an inductive scheme to show that the Lyapunov spectrum blocks associated to a gap pattern in the Lyapunov spectrum of such a cocycle are locally Holder continuous. Moreover, we show that all Lyapunov exponents are continuous everywhere in this Banach manifold, irrespective of any gap pattern in their spectra. These results also hold for Diophantine translations on higher dimensional tori, albeit with a loss in the modulus of continuity of the Lyapunov spectrum blocks.

Examples of discontinuity of Lyapunov exponent in smooth quasiperiodic cocycles

We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_\omega, A)$ where $T_\omega$ is an irrational rotation $x\to x+ 2\pi\omega$ on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$. For any fixed $l=0, 1, 2, \cdots, \infty$ and any fixed $\omega$ of bounded-type, we construct $D_{l}\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$ such that the Lyapunov exponent is not continuous at $D_{l}$ in ${\cal C}^l$-topology. We also construct such examples in a smaller Schr\"odinger class.

Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications

In this paper, we prove that any analytic quasi-periodic cocycle close to constant is the Poincar\'{e} map of an analytic quasi-periodic linear system close to constant. With this local embedding theorem, we get fruitful new results. We show that the almost reducibility of an analytic quasi-periodic linear system is equivalent to the almost reducibility of its corresponding Poincar\'e cocycle. By the local embedding theorem and the equivalence, we transfer the recent local almost reducibility results of quasi-periodic linear systems \cite{HoY} to quasi-periodic cocycles, and the global reducibility results of quasi-periodic cocycles \cite{A,AFK} to quasi-periodic linear systems. Finally, we give a positive answer to a question of \cite{AFK} and use it to prove Anderson localization results for long-range quasi-periodic operator with Liouvillean frequency, which gives a new proof of \cite{AJ05,AJ08,BJ02}. The method developed in our paper can also be used to prove some nonlinear local embedding results.

Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

This paper is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H.Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density, of reducible cocycles near a constant. Some algebraic structure can also be preserved, by doubling the period if needed.

Regularity of the Rotation Number for the One-Dimensional Time-Continuous Schrödinger Equation

Starting from results already obtained for quasi-periodic co-cycles in \(SL(2, \mathbb R),\) we show that the rotation number of the one-dimensional time-continuous Schrodinger equation with Diophantine frequencies and a small analytic potential has the behavior of a \(\frac{1}{2}-\)Holder function. We give also a sub-exponential estimate of the length of the gaps which depends on its label given by the gap-labeling theorem.

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣σ) for some 0 < σ < 1.

Almost reducibility for finitely differentiable -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 times differentiable functions, with k great enough. Almost reducibility is obtained by analytic approximation after a loss of differentiability which only depends on the frequency and on the constant part. As in the analytic case, if the fibred rotation number of these cocycles is diophantine or rational with respect to the frequency, they are in fact reducible. This extends Eliasson's theorem on Schrödinger cocycles to the differentiable case.

Reducibility of quasiperiodic cocycles under a Brjuno-Rüssmann arithmetical condition

The arithmetics of the frequency and of the rotation number play afundamental role in the study of reducibility of analyticquasiperiodic cocycles which are sufficiently close to a constant.In this paper we show how to generalize previous works byL.H. Eliasson which deal with the diophantine case so as to implementa Brjuno-Rüssmann arithmetical condition both on the frequency and onthe rotation number. Our approach adapts the Pöschel-Rüssmann KAMmethod, which was previously used in the problem of linearization ofvector fields, to the problem of reducing cocycles.

Reducibility Results for Quasiperiodic Cocycles With Liouvillean Frequency

We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics. Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic \({GL(d,\mathbb R)}\) cocycles which are close to constants. It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes.

Continuity of the Lyapunov exponent for analytic quasi-periodic cocycles with singularities

We prove that the Lyapunov exponent of quasi-periodic cocyles with singularities behaves continuously over the analytic category. We thereby generalize earlier results, where singularities were either excluded completely or constrained by additional hypotheses. Applications are one-parameter families of analytic Jacobi operators, such as extended Harper's model describing crystals subject to external magnetic fields.

Local rigidity of reducibility of analytic quasi-periodic cocycles on [formula omitted

In this paper, we consider the reducibility problem of an analytic d -dimensional quasi-periodic cocycle on G L ( n , C ) . We prove that, for any Diophantine vector α and diagonal matrix C satisfying some arithmetic conditions, if a cocycle ( α , A ) can be conjugated to the constant cocycle ( α , C ) by some ( 0 , B ) with B ∈ L 2 ( T d , G L ( n , C ) ) and A is sufficiently close to some diagonal matrix, it is analytically reducible. Moreover B is actually analytic if it is continuous.

Reducibility of quasiperiodic cocycles in linear Lie groups

Let G be a linear Lie group. We define the G-reducibility of a continuous or discrete cocycle modulo N. We show that a G-valued continuous or discrete cocycle which is GL(n,ℂ)-reducible is in fact G-reducible modulo two if G=GL(n,ℝ),SL(n,ℝ),Sp(n,ℝ) or O(n) and modulo one if G=U(n) .

Continuity of the Lyapunov exponent for analytic quasiperiodic cocycles

AbstractIt is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasiperiodic cocycles. In this paper we show that it is continuous in the analytic category. Our corollaries include continuity of the Lyapunov exponent associated with general quasiperiodic Jacobi matrices or orthogonal polynomials on the unit circle, in various parameters, and applications to the study of quantum dynamics.

Stable one-dimensional quasi-periodic cocycles on unitary group

In this paper, we study the stable one-dimensional quasi-periodic C∞ cocycles on U(N). We prove that any such cocycle on a generic irrational rotation is a limit point of reducible cocycles. The proof is based on Krikorian’s renormalization scheme and a local result of him.

Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$

In this paper, we consider the analytic reducibility problem of ananalytic $d-$dimensional quasi-periodic cocycle $(\alpha,\ A)$ on$U(n)$ where $ \alpha$ is a Diophantine vector. We prove that, ifthe cocycle is conjugated to a constant cocycle $(\alpha,\ C)$ by ameasurable conjugacy $(0,\ B)$, then for almost all $C$ it isanalytically conjugated to $(\alpha,\ C)$ provided that $A$ issufficiently close to some constant. Moreover $B$ is actuallyanalytic if it is continuous.

Hölder Continuity of the Rotation Number for Quasi-Periodic Co-Cycles in $${SL(2, \mathbb R)}$$

We prove two results on the rotation number of the skew-product system \({(\omega ,A):(\theta ,y)\in\mathbb T^d\times\mathbb R^2\mapsto (\theta +\omega ,A(\theta)y)\in\mathbb T^d\times\mathbb R^2,}\) where ω is Diophantine and \({A(\theta)\in SL(2, \mathbb R)}\) is homotopic to the identity. On the one hand, we prove that this function has the behavior of a \({\frac{1}{2}-}\) Holder function. On the other, we show that the length of the gaps has a sub-exponential estimate which depends on its label given by the gap-labeling theorem. We give also an estimate of the complement of the spectrum. These results are obtained by studying the reducibility of the quasi-periodic co-cycle (ω , A).