Quasi-periodic Cocycles Research Articles

Continuity of the Lyapunov Exponent for Analytic Multi-frequency Quasiperiodic Cocycles

Abstract It is known that the Lyapunov exponent of analytic 1-frequency quasiperiodic cocycles is continuous in cocycle and, when the frequency is irrational, jointly in cocycle and frequency. In this paper, we extend a result of Bourgain to show the same continuity result for multifrequency quasiperiodic $M(2,\mathbb{C})$ cocycles. Our corollaries include applications to multifrequency Jacobi cocycles with periodic background potentials.

Analyticity of the Lyapunov exponents of random products of quasi-periodic cocycles

We show that the top Lyapunov exponent , with for each i, associated with a random product of quasi-periodic cocycles depends real analytically on the transition probabilities p whenever is simple. Moreover if the spectrum at p is simple (all Lyapunov exponents having multiplicity one ) then all Lyapunov exponents depend real analytically on p.

Spectral characteristics of Schrödinger operators generated by product systems

Motivated by the question of what spectral properties of dynamically defined Schrödinger operators may be preserved under periodic perturbations, we study ergodic Schrödinger operators defined over product dynamical systems in which one factor is periodic and the other factor is either a subshift over a finite alphabet or an irrational rotation of the circle. The scenario given by a periodic background potential corresponds to a separable structure in which the sampling function is the sum of two pieces, each of which depends only on a single factor of the product system. However, in each case that we study, our methods apply more generally to sampling functions that allow non-trivial dependencies between the product factors. In the case in which one factor is a Boshernitzan subshift, we prove that either the resulting operators are periodic or the resulting spectra must be Cantor sets. The main ingredient is a suitable stability result for the Boshernitzan condition under taking products. We also discuss the stability of purely singular continuous spectrum, which, given the zero-measure spectrum result, amounts to stability results for eigenvalue exclusion. In particular, we examine situations in which the existing criteria for the exclusion of eigenvalues are stable under periodic perturbations. As a highlight of this, we show that any simple Toeplitz subshift over a binary alphabet exhibits uniform absence of eigenvalues on the hull for any periodic perturbation whose period is commensurate with the coding sequence. This is new, even in the case in which the periodic background vanishes entirely. In the case of a full shift, we give an effective criterion to compute exactly the spectrum of a random Anderson model perturbed by a potential of period two, and we further show that the naive generalization of this criterion does not hold for period three. Next, we consider quasi-periodic potentials with potentials generated by trigonometric polynomials with periodic background. We show that the quasiperiodic cocycle induced by passing to blocks of period length is subcritical when the coupling constant is small and supercritical when the coupling constant is large. Thus, the spectral type is absolutely continuous for small coupling and pure point (for a.e. frequency and phase) when the coupling is large.

Almost reducibility of quasiperiodic [formula omitted]-cocycles in ultradifferentiable classes

Given a quasiperiodic cocycle in sl(2,R) sufficiently close to a constant, we prove that it is almost reducible in ultradifferentiable class under an adapted arithmetic condition on the frequency vector.

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

We establish a quantitative version of strong almost reducibility result for quasi-periodic cocycle close to a constant in the Gevrey class. We prove that, if the frequency is Diophantine, the long range operator has pure point spectrum with sub-exponentially decaying eigenfunctions for almost all phases; for the one dimensional quasi-periodic Schrödinger operators with small Gevrey potentials, the length of spectral gaps decays sub-exponentially with respect to its labelling; and the spectrum is an interval for discrete Schrödinger operators acting on with small separable potentials.

Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

We show that for a large class of ultra-differentiable potential, and every frequency α∈R∖Q, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either rotations reducible or has positive Lyapunov exponents. This partially answers the open problem by Fayad-Krikorian [26,40]. As spectral application, we prove the “Last's intersection spectrum conjecture” for Gervey potential, which answers an open problem raised by Jitomirskaya-Marx [36].

Strong Birkhoff ergodic theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles

<p style='text-indent:20px;'>In this paper, we first prove the strong Birkhoff Ergodic Theorem for subharmonic functions with the irrational shift on the Torus. Then, we apply it to the analytic quasi-periodic Jacobi cocycles and show that for suitable frequency and coupling number, if the Lyapunov exponent of these cocycles is positive at one point, then it is positive on an interval centered at this point and Hölder continuous in <inline-formula><tex-math id="M1">\begin{document}$ E $\end{document}</tex-math></inline-formula> on this interval. What's more, if the coupling number of the potential is large, then the Lyapunov exponent is always positive for all irrational frequencies and Hölder continuous in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula> for all finite Liouville frequencies. For the Schrödinger cocycles, a special case of the Jacobi ones, its Lyapunov exponent is also Hölder continuous in the frequency and the lengths of the intervals where the Hölder condition of the Lyapunov exponent holds only depend on the coupling number.</p>

Reducibility of ultra-differentiable quasiperiodic cocycles under an adapted arithmetic condition

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-R{\"u}ssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition.

Random product of quasi-periodic cocycles

Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of $C^r$, $0\leq r \leq \infty$ (or analytic) $k+1$-tuples of quasi periodic cocycles taking values in $SL_2(\mathbb{R})$ such that the random product of them has positive Lyapunov exponent contains a $C^0$ open and $C^r$ dense subset which is formed by $C^0$ continuity point of the Lyapunov exponent For $k+1$-tuples of quasi periodic cocycles taking values in $GL_d(\mathbb{R})$ for $d>2$, we prove that if one of them is diagonal, then there exists a $C^r$ dense set of such $k+1$-tuples which has simples Lyapunov spectrum and are $C^0$ continuity point of the Lyapunov exponent.

Reducibility of Finitely Differentiable Quasi-Periodic Cocycles and Its Spectral Applications

In this paper, we prove the generic version of Cantor spectrum property for quasi-periodic Schrödinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency \(C^k\) long-range operators on \(\ell ^2({\mathbb Z}^d)\). These results are based on reducibility properties of finitely differentiable quasi-periodic \(SL(2,{\mathbb R})\) cocycles. More precisely, we prove that if the base frequency is Diophantine, then a \(C^k\) \(SL(2,{\mathbb R})\)-valued cocycle is reducible if it is close to a constant cocycle, sufficiently smooth and the rotation number of it is Diophantine or rational with respect to the frequency.

Rigidity of Reducibility of Finitely Differentiable Quasi-Periodic Cocycles on U(n)

We consider the reducibility problem of quasi-periodic cocycles $$(\alpha ,A)$$ on $${\mathbb {T}}^d\times U(n)$$ in $$C^k$$ class, with k large enough and $$\alpha $$ being a Diophantine vector. We show that if $$(\alpha ,A)$$ is conjugated to a constant cocycle $$(\alpha ,C)$$ via $$(\theta ,X)\mapsto (\theta , B(\theta )X)$$ , with a map $$B:{\mathbb {T}}^d\rightarrow U(n)$$ being measurable, then it can be conjugated to $$(\alpha ,C)$$ in $$C^{k^{'}} (k^{'}<k)$$ class for almost all C, provided that A is sufficiently close to constants. When $$d=1$$ , such a conclusion can even be extended to the global case: if $$(\alpha ,A)$$ is conjugated to a constant cocycle $$(\alpha ,C)$$ via measurable $$B:{\mathbb {T}}^d\rightarrow U(n)$$ , it can be conjugated to $$(\alpha ,C)$$ in $$C^{k^{'}} (k^{'}<k)$$ class, for almost all $$\alpha $$ and C.

Non-perturbative positivity and weak Holder continuity of Lyapunov exponent of analytic quasi-periodic Jacobi cocycles defined on a high dimension torus

When analytic quasi-periodic cocycles are defined on a high dimension torus, their Lyapunov exponents have perturbative positivity and continuity. In this article, we study a class of analytic quasi-periodic Jacobi cocycles defined on a two dimension torus. We show that in the non-perturbative large coupling regimes, the Lyapunov exponent is positive for any frequency and weak Holder continuous for the full-measured frequency.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/51/abstr.html

Erratum

Erratum

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 a Diophantine rotation. We then prove that if this rotation vector is Diophantine with respect to the rotation in $\mathbb{T}^{d}$, the cocycle is smoothly reducible, thus establishing a hypoellipticity property in the spirit of the Greenfield-Wallach conjecture in PDEs.

Positive Lyapunov exponent for a class of quasi-periodic cocycles

Young [ 17 ] proved the positivity of Lyapunov exponent in a large set of the energies for some quasi-periodic cocycles. Her result is also proved to be applicable for some quasi-periodic Schrodinger cocycles by Zhang [ 18 ]. However, her result cannot be applied to the Schrodinger cocycles with the potential \begin{document}$ v = \cos(4\pi x)+w( x) $\end{document} , where \begin{document}$ w\in C^2(\mathbb R/\mathbb Z,\mathbb R) $\end{document} is a small perturbation. In this paper, we will improve her result such that it can be applied to more cocycles.

Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles

An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We establish large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we obtain continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem for the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible everywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.

Sharp Hölder continuity of the Lyapunov exponent of finitely differentiable quasi-periodic cocycles

We show that if the base frequency is Diophantine, then the Lyapunov exponent of a $$C^{k}$$ quasi-periodic $$SL(2,{\mathbb {R}})$$ cocycle is 1 / 2-Holder continuous in the almost reducible regime. As a consequence, we show that if the frequency is Diophantine, and the potential is small, then the integrated density of states of the corresponding quasi-periodic Schrodinger operator is 1 / 2-Holder continuous.

The Set of Smooth Quasi-periodic Schrödinger Cocycles with Positive Lyapunov Exponent is Not Open

One knows that the set of quasi-periodic Schrodinger cocycles with positive Lyapunov exponent is open and dense in analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be arbitrarily approximated by ones with zero Lyapunov exponent in the space of $${\mathcal{C}^ l (1 \le l \le \infty)}$$ smooth quasi-periodic cocycles, which shows that the set of quasi-periodic Schrodinger cocycles with positive Lyapunov exponent is not open in smooth topology.

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

We continue our study of the local theory for quasiperiodic cocycles in {mathbb{T} ^{d} times G} , where {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)} . 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.

Uniform positivity of Lyapunov exponent for a class of smooth Schrödinger cocycles with weak Liouville frequencies

We prove uniform positivity of the Lyapunov exponent for quasiperiodic Schrodinger cocycles with C2 cos-type potentials, large coupling constants, and fixed weak Liouville frequencies.