Anosov-Katok constructions for quasi-periodic $\mathrm{SL}(2,{\mathbb{R}})$ cocycles

Our first result concerns the regularity of Lyapunov exponent (LE):

as a function of the mapping $A(\cdot)$ defining the dynamics in the fibers. The LE is a central topic in the spectral theory of Schrödinger operators, since it relates with intergrated density of states through the Thouless formula. The LE also arise naturally in the study of smooth dynamics, and continuity of LE has been the object of important recent research, see [50] and the references therein.

In the analytic topology, the LE is always continuous with respect to both ${\alpha}$ and $A$ at any given cocycle $(\alpha,A)$, provided that $\alpha\in{\mathbb{T}}^{d}$ is rationally independent, see [18, 21, 35]. This holds true even for cocycles in higher dimensional groups $GL(d,{\mathbb{C}})$, with $\alpha\in{\mathbb{R}}\backslash{\mathbb{Q}}$ [9]. If $\alpha\in DC_{d}$²²2We recall that $\alpha$ is Diophantine, denoted by $\alpha\in{\rm DC}_{d}(\kappa^{\prime},\tau)$, if there exist $\kappa^{\prime}>0$ and $\tau>d-1$ such that (1.4) $${\rm DC}_{d}(\kappa^{\prime},\tau):=\left\{\alpha\in{\mathbb{T}}^{d}:\inf_{j\in{\mathbb{Z}}}\left|\langle n,\alpha\rangle-j\right|>\frac{\kappa^{\prime}}{|n|^{\tau}},\quad\forall\ n\in{\mathbb{Z}}^{d}\backslash\{0\}\right\}.$$ The set of Diophantine numbers is denoted by ${\rm DC}_{d}:=\bigcup_{\kappa^{\prime}>0,\,\tau>d-1}{\rm DC}_{d}(\kappa^{\prime},\tau)$., and the potential $V$ is small enough (the smallness of $V$ depending on $\alpha$), then $L(\alpha,S_{E}^{V})$ is $\frac{1}{2}$ Hölder continuous w.r.t $E$ [15]. Subsequently, Avila-Jitomirskaya in [7] generalized the $\frac{1}{2}$-Hölder continuity to the non-perturbative regime, lifting the dependence of the smallness of $V$ on $\alpha$. By the ART, see [2, 4], the LE is $\frac{1}{2}$-Hölder continuous in the whole subcritical regime [47]. This kind of $\frac{1}{2}$-Hölder continuity is sharp, since the LE is exactly $\frac{1}{2}$ Hölder continuous at the end of spectral gaps by [48]. On the other hand, if $(\alpha,A)$ is conjugated to constant $\mathrm{SO}(2,{\mathbb{R}})$ cocycle or $A\in\mathcal{UH}_{\alpha}$, then LE is Lipschitz. Therefore a natural question is whether there exist subcritical cocycles such that the optimal Hölder exponent of $L(\alpha,A)$ can be any fixed number between $\frac{1}{2}$ and $1$. In fact, we will show that this holds in a dense set in $\mathcal{AR}_{\alpha}\backslash\mathcal{UH}_{\alpha}$ for each admissible Hölder exponent.

Some arithmetic condition on $\alpha$ is needed for the theorem to be true. As pointed out by Avila-Jitomirskaya [7], the Lyapunov Exponent is discontinuous at rational $\alpha$, which implies then for generic $\alpha\in{\mathbb{R}}\backslash{\mathbb{Q}}$, LE cannot be Hölder continuous at any order. Recently, Avila-Last-Shamis-Zhou [11] showed that if $\alpha$ is very Liouvillean, then LE can even be not $\log^{t}$-Hölder continuous.

In the positive Lyapunov Exponent regime, Goldstein and Schlag proved that the LE is Hölder continuous (for one-frequency cocycles) or weak Hölder continuous (for multi-frequency cocycles) for Schrödinger operators whose frequency satisfies a strong Diophantine condition [30, 31].

Continuity of the LE also depends sensitively on the smoothness of $A(\cdot)$. In the $C^{0}$ topology, any non uniformly hyperbolic $\mathrm{SL}(2,{\mathbb{R}})$ cocycle can be approximated by cocycles with zero LE, see [16, 17], and thus the LE is not continuous. In the $C^{k}$ ($k\in{\mathbb{N}}$) topology, results are quite different. Wang-You, [51], showed that the LE can be discontinuous. On the other hand, Xu-Ge-Wang, [53], recently constructed a class of $C^{2+\epsilon}$ $\cos$-type potential, and for any $\frac{1}{2}\leq\kappa<1$ showed the existence of an energy, such that LE is exactly $\kappa$-Hölder continuous. For other results of regularity of LE in the smooth or in the Gervey topology, one can consult [22, 52, 45] and the references therein.

Next, we move to spectral applications. In contrast with 1D random Schrödinger operator, one of the most remarkable phenomena exhibited by quasiperiodic Schrödinger operators is the existence of absolutely continuous (ac) spectrum. If $\alpha\in\mathrm{DC}_{d}$ and $V$ is small enough, then Dinaburg-Sinai in [23] proved that $H_{V,\alpha,\theta}$ has ac spectrum. Eliasson in [24] proved the stronger result that $H_{V,\alpha,\theta}$ has purely ac spectrum for every $\theta$. Avila’s ART [2, 4, 5] ensures that if the potential is subcritical, then the corresponding operator has purely ac spectrum. From the physicist’s point of view, ac spectrum corresponds to a phase where the material is a conductor, and the corresponding eigenstate is extended. Indeed, one can define the inverse participation ratio (IPR) [26] as

where $u_{m}$ is the $m-th$ eigenstate. If $\Gamma=-\lim_{L\rightarrow\infty}\frac{\ln({\rm IPR})}{\ln L}=1$, then the phase is extended. Thus, conductivity of the material is reflected in the growth of its eigenstate.

For $\alpha\in\mathrm{DC}_{d}$ and $V$ small enough, Eliasson [24] proved that if the energy lies in the end of spectral gaps, then its extended eigenstates have linear growth. On the other hand, for any $E\in\Sigma_{V,\alpha}$, the spectrum of the operator, the extended eigenstates at most have sub linear growth, i.e. $|u_{E}(n)|\leq o(n)$. This kind of behavior can be generalized to the whole subcritical regime [2, 4]. Thus it is interesting to ask whether this kind of sub-linear growth was optimal. In this paper, we prove that for a dense subset of subcritical potentials, the extended eigenstates of the associated operator have optimal sub-linear growth.

Theorem 1.2 shows that there exists extended eigenfunctions with optimal sub-linear growth. We remark howevert, that the universal hierarchical structure of any extended quasi-periodic eigenfunctions for almost Mathieu operators was recently obtained in [33]

The study of growth of extended eigenstates is not only interesting from the physicist’s point of view, but also important from the mathematical point of view. By (1.2), the study of the growth of the extended eigenstates is equivalent to study the growth of the Schrödinger cocycle. On one hand, the growth of Schrödinger cocycle is crucial for proving existence of ac spectrum. In the subcritical regime, for almost every $E\in\Sigma_{V,\alpha}$, the cocycles are uniformly bounded, see [24], which is sufficient for establishing the existence of ac spectrum by subordinacy theory [29]. However, whether the spectral measure is purely ac or not really depends on the growth of $\|S_{E}^{V}(\theta;j)\|$ on a Lebesgue zero measure set of energies by [3]. On the other hand, the growth of Schrödinger cocycle is important in proving the regularity of the spectral measure, see [8]. Indeed, using Jitomiskaya-Last’s inequality, see [37], Avila-Jitomirskaya in [8] showed that

where $P_{L}=\sum_{n=1}^{L}(S_{E}^{V})^{*}(\theta+\alpha;2n-1)S_{E}^{V}(\theta+\alpha;2n-1)$ satisfies $\det P_{L}=1/4\epsilon^{2}$. Therefore, by Theorem 1.2, it seems interesting to study high order Hölder continuity of spectral measure for a dense set of subcritical energies.

Our third result concerns the localization property of the quasi-periodic long-range operator:

where $V_{k}\in{\mathbb{C}}$ is the Fourier coefficient of $V\in C^{\omega}({\mathbb{T}}^{d},{\mathbb{R}})$. This operator has received a lot of attention, see [7, 13, 20] since it is the Aubry dual of the quasi-periodic Schrödinger operator defined in eq. (1.1). If $V(\theta)=2\cos\theta$, then it reduces to the extensively studied almost Mathieu operator:

For the almost Mathieu operator, there is a sharp phase transition line $\lambda=e^{\beta(\alpha)}$ ³³3 Here, $\beta(\alpha):=\limsup_{n\rightarrow\infty}\frac{\ln q_{n+1}}{q_{n}},$ where $\frac{p_{n}}{q_{n}}$ is the continued fraction best rationnal approximants of $\alpha\in{\mathbb{R}}\backslash{\mathbb{Q}}$. from singular continuous spectrum to Anderson localization (pure point spectrum with exponentially decaying eigenfunctions), see [13, 38]. In the transition line $\lambda=e^{\beta(\alpha)}$, for frequencies in a dense set, $H_{\lambda,\alpha,\theta}$ displays pure point spectrum, see [10], but the eigenfunction does not decay exponentially, see [38]. As pointed out in [10], the insights gained from the critical parameters often shed light on the creation, dissipation, and the mechanism behind the phases of non critical parameters as well. Thus it is interesting to ask whether there exists real power law decay eigenfunction, i.e. if the eigenfunctions can decay polynomially. In this paper, we establish the following result.

Just as Theorem 1.2, Theorem 1.3 also holds if the dual Schrödigner operator (1.1) lies in the subcritical regime. However, under the assumption that $\alpha\in\mathrm{DC}_{d}$, the phenomenon exhibited by Theorem 1.3 was not expected for Schrödigner operators (1.1). Indeed, if $V(\theta)$ is an even function, then for a $G_{\delta}$ dense set of $\theta$, $H_{V,\alpha,\theta}$ has no eigenvalues, see [40]. It is widely believed for a.e. $\theta$, $H_{V,\alpha,\theta}$ has Anderson localization in the positive LE regime. In fact, $H_{\lambda,\alpha,\theta}$ even exhibits a sharp transition in the phase $\theta$ between singular continuous spectrum and Anderson localization [39].

If $d=1$, Bourgain-Jitomirskaya in [20] proved that for any fixed $\alpha\in DC$, $\mathcal{L}_{\lambda V,\alpha,\phi}$ has Anderson localization for sufficiently small $\lambda$ and a.e. $\phi$. If $\alpha\in{\mathbb{R}}\backslash{\mathbb{Q}}$, not necessary Diophantine, then for dense set of small potentials $V$ and a.e. $\phi$, $\mathcal{L}_{V,\alpha,\phi}$ has point spectrum with exponentially decaying eigenfunctions, see [56], while it is still open whether $\mathcal{L}_{V,\alpha,\phi}$ has Anderson localization. If $d\geq 2$, Jitomirskaya-Kachkovskiy in [36] proved that for fixed $\alpha\in DC_{d}$, $L_{\lambda V,\alpha,\theta}$ has pure point spectrum for sufficiently small $\lambda$ and $a.e.$ $\theta$. Recently, Ge-You-Zhou [32] further proved that under the same assumption, $L_{\lambda V,\alpha,\theta}$ has exponentially dynamical localization.

Recall that a cocycle $(\alpha,A)\in{\mathbb{T}}^{d}\times C^{\omega}({\mathbb{T}}^{d},\mathrm{SL}(2,{\mathbb{R}}))$ is $C^{s}$ reducible if there exists $B\in C^{s}(2{\mathbb{T}}^{d},\mathrm{SL}(2,{\mathbb{R}}))$ such that

then by Aubry duality, Theorem 1.3 is an immediately corollary of the following reducibility result.

We point out that this result is interesting in itself, from the dynamical systems point of view. The study of the reducibility of cocycles is related with the linearization of circle diffeomorphisms. Arnol’d in [1] proved that if an analytic diffeomorphism $f\in\mathrm{Diff}^{\omega}({\mathbb{T}})$ is close to a rotation $T_{\rho}$, where $\rho$ is the rotation number of $f$, and $\rho\in DC$, then $f$ is analytically linearizable. One of the great achievements of Herman-Yoccoz, see [34, 54], is the proof of the fact that the sharp arithmetic condition for $C^{\infty}$ linearizability of $f\in\mathrm{Diff}^{\infty}({\mathbb{T}})$, without any smallness condition imposed a priori on $f$, is that its rotation number be Diophantine. In the Liouvillean regime, Herman [34] (see also [44]) proved that for any $s\in{\mathbb{N}}^{+}$ and any Liouvillean number $\rho$, the set of $f\in\mathrm{Diff}^{\infty}({\mathbb{T}})$ with rotation number $\rho$ which is $C^{s}$ but not $C^{s+1}$ linearizable is locally dense. For a more precise description of the theory, we refer the reader for the recent survey of Eliasson-Fayad-Krikorian [25] on circle diffeomorphisms.

Concerning the reducibility of $\mathrm{SL}(2,{\mathbb{R}})$ cocycles, if the base frequency $\alpha\in DC_{d}$, and the cocycle $(\alpha,A)$ is sufficiently close to constants (the closeness depends on $\alpha$), Eliasson [24] proved that if the fibered rotation number is Diophantine w.r.t $\alpha$, then $(\alpha,A)$ is analytically reducible. By global to local reduction, reducibility for a full measure set of frequencies holds in the non-perturbative regime, see [48], and in the subcritical regime [4]. Motivated by the results in circle diffeomorphism, then it is natural to study the density of $C^{s}$ but not $C^{s+1}$ reducible cocycles. Theorem 1.4 provides a positive answer to this question. The corresponding theorem in the case of $C^{\infty}$ smooth $SU(2)$ cocycles, which in some sense stands in midway between circle diffemorphisms and $\mathrm{SL}(2,{\mathbb{R}})$ cocycles, was proved by the first author in [41].

The proof of these results are based on the fast approximation by conjugation method introduced by Anosov and Katok in [14], where they constructed mixing diffeomorphisms of the unit disc arbitrarily close to Liouvillean rotations. We refer the reader to §3 for a more detailed description of the method, but in a nutshell it consists of the following idea.

The transient dynamics of a Liouvillean rotation is, for all practical reasons, periodic: if $\rho$ is Liouville, then there exist sequences $p_{n},q_{n}\in{\mathbb{N}}^{*}$ such that

which means that a diffeomorphism conjugate to the rotation by $\rho$ is practically periodic with period $q_{n}$, in scales of iteration comparable with arbitrarily big powers of $q_{n}$. It is tempting, therefore, to try to study diffeomorphisms with rotation number equal to $\rho$ by approximating them from the outside, i.e. with diffeomorphisms that have rational rotation numbers, and to boot, are conjugate to them.

Since the dynamics of a periodic rotation are determined by a finite number of iterations, they are easier to tamper with, and since $|q_{n+1}|\gg|q_{n}|$, it is also possible to modify the dynamics at the scale $q_{n+1}$ in a way that preserves what was constructed in the previous scale $q_{n}$.

The limit object, satisfying the mixing property was thus constructed as follows. A diffeomorphism $f_{n}$ is constructed such that

for some smooth conjugation $H_{n}$, satisfying some finitary version of mixing at a scale of iteration $q_{n}$. A $q_{n}$-periodic conjugation $h_{n}$ is constructed, such that the diffeomorphism

satisfying some improved finitary version of mixing at a scale of iteration $q_{n+1}\gg q_{n}$. This can be achieved by chosing $p_{n+1}/q_{n+1}$ very close to $p_{n}/q_{n}$, so that the $C^{0}$ norm of $h_{n}$ can be allowed to explode. This will ensure that the limit diffeomorphism $f=\lim f_{n}$ is smooth, but not measurably conjugate to $T_{\rho}$. The Liouvillean character of $\rho$ is necessary in order to assure the convergence of $f_{n}$ despite the divergence of $H_{n}$.

For more information on the method, results and references we point the reader to [28]. This approximation-by-conjugation method has been useful in producing examples of dynamics incompatible with quasi-periodicity, in the vicinity of quasi-periodic dynamics. It is in some sense the counterpart of the KAM method: KAM tends to prove rigidity in the Diophantine world, while Anosov-Katok is used in order to prove non-rigidity in the Liouvillean world. In the context of cocycles, the concept of reducibility, obtained notably via KAM, allows for studying the rigidity results [12] when the fibered rotation number is Diophantine with respect to the frequency, while Anosov-Katok’s construction will be an efficient method to study wild dynamics when the fibered rotation number is Liouville with respect to the frequency.

The fibered Anosov-Katok construction was introduced by the first author in [41]. In this context, the rotation in the basis is fixed, and the only freedom is in the choice of the mapping in the fibers. The rotation $\alpha$ in the basis can be chosen to be Diophantine, and the role of periodic rotations in the classical constructions is taken by resonant cocycles, i.e. cocycles whose rotation number is $k\alpha$, where $k\in{\mathbb{Z}}$. The rest of the construction remains the same.

From an almost reducibility point of view, the construction consists in engineering the parameters of the KAM normal form, introduced in [42], and further exploited in [41, 43] in order to study a variety of dynamical phenomena present in the almost reducibility regime for cocycles in $SU(2)$. Using the KAM schemes of [22, 47], we further develop these techniques in order to adapt them to $SL(2,{\mathbb{R}})$ cocycles and in the context of the analytic category (instead of the smooth one, as in the [42]).

We point out that results obtainable by the fibered Anosov-Katok method can be obtained for cocycles over a fixed Liouvillean rotation, but the case of a Diophantine rotation is more difficult, and therefore more interesting.