Realization of Avila's acceleration for one-frequency SL(2,R) Schrödinger cocycles: degree lower bound and premonotonic characterization, and prevalence of fixed-acceleration energies

Avila's global theory of one-frequency Schrödinger operators attaches to every analytic quasiperiodic SL(2,C) cocycle with irrational frequency an integer invariant, the acceleration, given by the slope of the complexified Lyapunov exponent at the real axis. The source paper (Avila, arXiv:0905.3902, published in Acta Mathematica 2015) proved quantization and stratified analyticity, and posed several follow-up problems. Two of them, the absence of singular continuous spectrum for typical operators (Conjecture 1) and the Almost Reducibility Conjecture (Conjecture 2), have since been resolved in the real-symmetric (Schrödinger) setting by Avila and by independent authors. The open core posed in the same paper is: (i) whether the norm of the topological degree is always a lower bound for the acceleration of SL(2,R) cocycles, and whether, for nonzero degree, this bound is achieved precisely by premonotonic cocycles; and (ii) whether, for a typical perturbation of the potential $2\lambda\cos 2\pi n x$ with $\lambda>1$, energies with any fixed acceleration $1\le k\le n$ form a set of positive measure. No resolution of these two questions is known in the literature to date.

Background

A one-frequency quasiperiodic Schrödinger operator acts on $\ell^2(\mathbb{Z})$ by $(H_{\alpha,v}u)_n=u_{n+1}+u_{n-1}+v(x+n\alpha)u_n$, where the potential $v$ is a real analytic function on the circle $\mathbb{R}/\mathbb{Z}$, the frequency $\alpha$ is irrational, and $x$ is a phase. Such operators are studied through the associated Schrödinger cocycle $(\alpha,A^{(E-v)})$, the skew product over the rotation $x\mapsto x+\alpha$ whose fiber map is the analytic matrix function $A^{(E-v)}(x)=\begin{pmatrix}E-v(x)&-1\\1&0\end{pmatrix}\in\mathrm{SL}(2,\mathbb{R})$ transferring the solution vector $(u_{n+1},u_n)$. The Lyapunov exponent $L(\alpha,A)=\lim_{n\to\infty}\frac{1}{n}\int\ln\|A(x+(n-1)\alpha)\cdots A(x)\|\,dx$ measures the asymptotic growth of solutions and governs the spectral type (by Kotani theory, a general structure theory for zero exponents: vanishing $L$ on a set of positive measure forces absolutely continuous spectrum; positive $L$ is associated with localized regimes). Avila's global theory (arXiv:0905.3902; Acta Math. 215 (2015) 1–54) is built on the convexity of the complexified exponent $\epsilon\mapsto L(\alpha,A_\epsilon)$, where $A_\epsilon(x)=A(x+i\epsilon)$ is the cocycle evaluated on the shifted (complex) circle. A cocycle is regular if this function is affine in $\epsilon$ on a neighborhood of $0$ (on both sides), and its acceleration is the integer-valued quantity $\omega(\alpha,A)=\lim_{\epsilon\to0^+}\frac{L(\alpha,A_\epsilon)-L(\alpha,A)}{2\pi\epsilon}$, which Theorem 5 of the source paper proves to be always an integer for irrational frequencies (quantization). The acceleration stratifies the energy axis: with $\kappa(E)=\omega(\alpha,A^{(E-v)})$, the energies in the spectrum with $\kappa(E)=k\ge1$ and $L(E)>0$ form the $k$-th supercritical stratum, on which the complexified exponent grows at slope $2\pi k$ (part of the later literature indexes these sets as $S_{k+1}^{+}$, so that $S_2^+$ is the first supercritical stratum, of acceleration 1); $\omega=0$ with $L=0$ on a strip is the subcritical regime (conjugate to constants up to small perturbation, by almost reducibility), and critical energies form the boundary. Classical results include Jitomirskaya's resolution of the almost Mathieu operator (AMO, the case $v=2\lambda\cos2\pi x$) transition, Eliasson's and Bourgain–Goldstein's reducibility and nonperturbative localization theorems, and Herman's lower bounds for the exponent. For real cocycles one has two further invariants. The topological degree $\deg(A)\in\mathbb{Z}$ is the winding number of $A:\mathbb{R}/\mathbb{Z}\to\mathrm{SL}(2,\mathbb{R})\simeq S^1$, whose norm bounds from below the complexity of the cocycle within its homotopy class ($\pi_1(\mathrm{SL}(2,\mathbb{R}))=\mathbb{Z}$; cocycles homotopic to a constant have degree $0$). A cocycle is premonotonic if some iterate is $C^1$-conjugate to one whose entries are monotone functions of $x$; Avila–Krikorian ('Monotonic cocycles', Invent. Math. 202 (2015) 271–331) showed this class includes $C^1$ perturbations of nonconstant $\mathrm{SO}(2,\mathbb{R})$-valued cocycles with nonzero degree and proved rigidity theorems for it. The source paper establishes: $|\omega|\le n$ for cocycles whose entries are trigonometric polynomials of degree at most $n$; $\omega=n$ on the whole spectrum for the potential $v=2\lambda\cos2\pi nx$ with $|\lambda|\ge1$; and $\omega=|\deg|$ for $\mathrm{SO}(2,\mathbb{R})$-valued cocycles and, by Avila–Krikorian's 'Monotonic cocycles', for all premonotonic cocycles. In the same paper, two headline conjectures were posed: Conjecture 1 (typical operators have no singular continuous spectrum) and Conjecture 2 (the Almost Reducibility Conjecture, ARC: regularity with $L(\alpha,A_\epsilon)=0$ for $a<\epsilon<b$ implies almost reducibility, i.e. the existence of holomorphic conjugacies $B_n$ on shrinking strips making $B_n(z+\alpha)A(z)B_n(z)^{-1}$ arbitrarily close to the identity). Both are now resolved in the real-symmetric (Schrödinger) setting: Avila's 2023 preprint 'KAM, Lyapunov exponents, and the Spectral Dichotomy' (arXiv:2307.11071, Theorem 1 and Theorem 3) proves that regularity implies almost reducibility without arithmetic assumptions and derives the Spectral Dichotomy, giving Theorem 3: typical one-frequency Schrödinger operators have no singular continuous spectrum; earlier work of Avila (arXiv:1006.0704) covered exponentially Liouville frequencies, and independent proofs of important ARC cases are due to Avila–Jitomirskaya, Hou–You, Ge, and Han–Schlag. What survives untouched is the finer structure of the acceleration stratification itself — which integer values are realized within given classes of cocycles and with what prevalence. As established background, the source's Remark 11 notes the degeneration of the stratified-analyticity argument for cocycles not homotopic to a constant, where Avila–Krikorian nevertheless show that near a nonconstant $\mathrm{SO}(2,\mathbb{R})$-valued cocycle of degree $d$ the zero-exponent locus is an analytic submanifold of codimension $4|d|$, so that, among cocycles non-homotopic to constants with a given irrational frequency, the zero-exponent locus is contained in a countable union of positive-codimension submanifolds.

Problem Statement

For an irrational frequency $\alpha$ and a real-analytic cocycle $A:\mathbb{R}/\mathbb{Z}\to\mathrm{SL}(2,\mathbb{R})$, let $L(\alpha,A)$ be its Lyapunov exponent, $L(\alpha,A_\epsilon)$ its complexification with $A_\epsilon(x)=A(x+i\epsilon)$, and $\omega(\alpha,A)=\lim_{\epsilon\to0^+}\frac{L(\alpha,A_\epsilon)-L(\alpha,A)}{2\pi\epsilon}\in\mathbb{Z}$ its (quantized) acceleration; let $\deg(A)\in\mathbb{Z}$ be the topological degree of $A$. The two linked questions, posed in Avila's global theory paper (arXiv:0905.3902, section 1.6 items 4–5; published as Acta Mathematica 215 (2015)), are: (1) Is $|\deg(A)|$ always a lower bound for the acceleration, $|\deg(A)|\le|\omega(\alpha,A)|$, for every real-analytic $\mathrm{SL}(2,\mathbb{R})$ cocycle with irrational frequency; and, in the nonzero-degree case, is this lower bound achieved ($\omega=|\deg|$) precisely by the premonotonic cocycles, i.e. those some iterate of which is $C^1$-conjugate to a cocycle with monotone entries? (2) For $\lambda>1$ and $n\ge1$, the potential $v(x)=2\lambda\cos2\pi nx$ has acceleration $n$ on its entire spectrum; for a typical perturbation of $v$ (in the measure-theoretic sense: for almost every perturbation $v+\varphi$ with $\varphi$ in a small neighborhood of $0$ in the real-analytic space $C^\omega(\mathbb{R}/\mathbb{Z},\mathbb{R})$, with respect to a nondegenerate Gaussian measure on the Fourier coefficients), do the energies with any fixed acceleration $1\le k\le n$ form a set of positive Lebesgue measure — in particular, do the intermediate strata $k=2,\ldots,n-1$ occur at all with positive measure? These two questions share one objective: to determine how Avila's acceleration stratification is realized across the cocycle space — which integer accelerations occur within which classes, and with what prevalence under typical perturbation. The source paper's two headline conjectures (no singular continuous spectrum for typical operators, and the Almost Reducibility Conjecture) are excluded from this statement: both are resolved in the real-symmetric setting by later work (refs. 2–4, 8).

References

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