Does the set of analytic one-frequency complex cocycles with trivial Oseledets filtration contain a nonempty open set?

For every irrational frequency, Avila–Jitomirskaya–Sadel (JEMS 2014) proved that on a dense open set of analytic one-frequency complex matrix-valued cocycles (arbitrary dimension) the Oseledets filtration is either dominated or trivial, and stated explicitly that they do not know whether the set of cocycles with trivial filtration — those whose $d$ Lyapunov exponents are all equal — contains a nonempty open subset. The problem is open for every irrational frequency and every dimension $d \ge 2$.

Background

A linear cocycle over a circle rotation is a pair $(\alpha, A)$ consisting of a rotation angle $\alpha \in \mathbb{R}$ (here always irrational) and a matrix-valued function $A: \mathbb{R}/\mathbb{Z} \to L(\mathbb{C}^d, \mathbb{C}^d)$; it defines the skew-product $(x, w) \mapsto (x + \alpha, A(x)w)$ on $\mathbb{R}/\mathbb{Z} \times \mathbb{C}^d$, whose iterates involve the products $A_n(x) = A(x+(n-1)\alpha)\cdots A(x)$ of $d \times d$ complex matrices. The cocycle is analytic if $A$ extends holomorphically to a strip $|\mathrm{Im}\,z| < \delta$ around the circle; the space $C^\omega(\mathbb{R}/\mathbb{Z}, L(\mathbb{C}^d, \mathbb{C}^d))$ of such cocycles carries the inductive-limit topology built from the strip norms (the topology used in the source, in which the interior question below is posed). Because an irrational rotation is uniquely ergodic (Lebesgue measure is the only invariant probability measure), Kingman's subadditive ergodic theorem yields the Lyapunov exponents $L_1(\alpha, A) \ge \cdots \ge L_d(\alpha, A) \in [-\infty, \infty)$, the asymptotic exponential growth rates of the singular values of $A_n(x)$ (the singular values of a matrix $B$ are the square roots of the eigenvalues of $B^*B$). The Oseledets theorem refines this into an invariant measurable filtration of $\mathbb{C}^d$; the filtration is trivial when all exponents coincide, i.e. only one exponent value occurs. An invariant continuous decomposition $\mathbb{C}^d = E^1_x \oplus \cdots \oplus E^k_x$ is dominated if for some $n \ge 1$ one has $\|A_n(x) w_j\| > \|A_n(x) w_{j+1}\|$ for every $x$, every unit vector $w_j \in E^j_x$, and every $j$; domination is a uniform, projective form of hyperbolicity and is an open (robust) condition on the cocycle.

A program of Mañé from the 1980s, realized for generic volume-preserving and symplectic maps and for continuous cocycles by Bochi–Viana, predicts that generically the Oseledets splitting is either dominated or all exponents on the relevant bundle vanish; in the low-regularity setting the mechanism is discontinuity of the Lyapunov exponents: a spectral gap without domination forces a discontinuity point of the exponent functionals. Avila–Jitomirskaya–Sadel showed that for analytic complex one-frequency cocycles a stronger phenomenon holds by a different mechanism: all Lyapunov exponents depend continuously on the cocycle, jointly with the frequency, at every irrational frequency, and nevertheless, for every irrational $\alpha$ and every dimension, there is a dense open set of $A \in C^\omega(\mathbb{R}/\mathbb{Z}, L(\mathbb{C}^d, \mathbb{C}^d))$ on which the Oseledets filtration is either dominated or trivial. The proof relates domination to the regularity of the dependence of summed exponents $\sum_{j \le k} L_j$ on imaginary shifts $A(\cdot + it)$ and to a quantization of the acceleration (the integer-valued jump of the slopes of these convex functions, a concept from Avila's global theory of one-frequency Schrödinger operators). A structural consequence of their own proof: within the dense open set where the number of distinct Lyapunov exponents is locally constant (nonempty because the exponents are continuous), the subset with trivial filtration is open; hence the trivial set has nonempty interior if and only if some trivial cocycle is stably trivial, i.e. all of its sufficiently small analytic perturbations still have equal exponents. The authors state that they do not know whether this happens.

The question is specific to the complex category, and the contrast with the real category shows why it is subtle. For $\mathrm{SL}(2,\mathbb{R})$-valued analytic cocycles, such as the Schrödinger cocycles $\begin{pmatrix} E - V(x) & -1 \\ 1 & 0 \end{pmatrix}$ (which encode the quasiperiodic Schrödinger difference equation $u_{n+1} + u_{n-1} + V(x + n\alpha)u_n = Eu_n$ with energy $E$ and analytic potential $V$), Avila's global theory defines the subcritical regime — an open set on which the Lyapunov exponent vanishes identically, containing for instance the almost Mathieu family (the Schrödinger cocycle with cosine potential $V(x) = 2\lambda\cos(2\pi x)$) at coupling $|\lambda| < 1$ — so open sets of trivial-filtration cocycles do exist when a real (symplectic) structure is present. In the full complex space no symmetry survives, and every known source of trivial examples lies on a positive-codimension subset whose perturbations heuristically split the spectrum: constant matrices whose eigenvalues have equal modulus; scalar multiples $e^{g(x)} U(x)$ of unitary-valued analytic cocycles (for which all singular values of every iterate coincide); analytic coboundaries times scalars (a coboundary being a matrix function of the form $B(x+\alpha)B(x)^{-1}$ with $B$ analytic and invertible); compact-group-valued cocycles such as $\mathrm{SO}(3,\mathbb{R})$-valued ones (all exponents are $0$). The only stratum analyzed to date is the most degenerate: Sadel–Xu proved that an analytic one-frequency cocycle with all exponents $-\infty$ is nilpotent, analytically conjugate to a strictly upper-triangular Jordan-type normal form, and admits arbitrarily small analytic perturbations with distinct Lyapunov exponents, so that stratum has empty interior. If the common exponent is finite, then $\det A(x) \ne 0$ for Lebesgue-almost-every $x$ (the sum of the exponents equals the average of $\ln|\det A|$, which is $-\infty$ exactly when the determinant vanishes on a set of positive measure), so the surviving open core concerns a.e.-invertible cocycles, and nothing is known there already for $d = 2$.

Problem Statement

Fix an irrational frequency $\alpha \in \mathbb{R} \setminus \mathbb{Q}$ and a dimension $d \ge 2$ (for $d = 1$ the answer is trivially positive, since every $1 \times 1$ cocycle has a single exponent, so $T_{\alpha,1}$ is the entire space), and endow the space $C^\omega(\mathbb{R}/\mathbb{Z}, L(\mathbb{C}^d, \mathbb{C}^d))$ of analytic complex matrix-valued one-frequency cocycles with its usual analytic topology, i.e. the inductive limit of the strip norms $\|A\|_\delta = \sup_{|\mathrm{Im}\,z| < \delta} \|A(z)\|$ (as used in the source). Let

$$T_{\alpha,d} \;=\; \{\, A \in C^\omega(\mathbb{R}/\mathbb{Z}, L(\mathbb{C}^d, \mathbb{C}^d)) \;:\; L_1(\alpha, A) = \cdots = L_d(\alpha, A) \,\}$$

be the set of cocycles whose Oseledets filtration is trivial, that is, whose $d$ Lyapunov exponents are all equal (the common value being finite or $-\infty$). The problem, stated by Avila–Jitomirskaya–Sadel as an explicit unknown attached to their dichotomy theorem: does $T_{\alpha,d}$ have nonempty interior — equivalently, does there exist an analytic complex one-frequency cocycle that is stably trivial, in the sense that all $d$ Lyapunov exponents remain equal for every cocycle obtained by a sufficiently small analytic perturbation in this topology (the frequency $\alpha$ being held fixed)?

The question is open for every irrational $\alpha$ and every $d \ge 2$. By the Sadel–Xu nilpotency theorem, the stratum where the common exponent is $-\infty$ contributes no interior, so the problem is precisely about the finite-common-exponent stratum, which consists of cocycles invertible for Lebesgue-a.e. base point.

References

1. Artur Avila, Svetlana Jitomirskaya, Christian Sadel, "Complex one-frequency cocycles", Journal of the European Mathematical Society 16 (2014), no. 9, 1915–1935. DOI: 10.4171/jems/479; arXiv:1306.1605. https://doi.org/10.4171/jems/479 (the open question is footnote 3 to Theorem 1.1)
2. Christian Sadel, Disheng Xu, "Singular analytic linear cocycles with negative infinite Lyapunov exponents", Ergodic Theory and Dynamical Systems 39 (2019), no. 4, 1082–1098 (published online 2017). DOI: 10.1017/etds.2017.52; arXiv:1601.06118. https://doi.org/10.1017/etds.2017.52
3. Duxiao Wang, Disheng Xu, Qi Zhou, "Hyperbolicity for one-frequency analytic quasi-periodic (Hermitian)-symplectic cocycles", preprint, 2025. arXiv:2503.15281. https://arxiv.org/abs/2503.15281
4. Artur Avila, "Global theory of one-frequency Schrödinger operators", Acta Mathematica 215 (2015), no. 1, 1–54. DOI: 10.1007/s11511-015-0128-7. https://doi.org/10.1007/s11511-015-0128-7 (preprint arXiv:0905.3902, titled "Global theory of one-frequency Schrödinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity", https://arxiv.org/abs/0905.3902)
5. Jairo Bochi, Marcelo Viana, "The Lyapunov exponents of generic volume-preserving and symplectic maps", Annals of Mathematics (2) 161 (2005), no. 3, 1423–1485. DOI: 10.4007/annals.2005.161.1423. https://doi.org/10.4007/annals.2005.161.1423
6. Jairo Bochi, "Cocycles of isometries and denseness of domination", The Quarterly Journal of Mathematics 66 (2015), no. 3, 773–798. DOI: 10.1093/qmath/hav020; arXiv:1408.3859. https://doi.org/10.1093/qmath/hav020
7. Pedro Duarte, Silvius Klein, "Further Related Problems", chapter in: "Lyapunov Exponents of Linear Cocycles" (Atlantis Studies in Dynamical Systems 3), Springer/Atlantis Press, 2016, pp. 247–260. DOI: 10.2991/978-94-6239-124-6_7. https://link.springer.com/chapter/10.2991/978-94-6239-124-6_7
8. Xuanji Hou, Yi Pan, Qi Zhou, "Dynamical classification of analytic one-frequency quasi-periodic $\mathrm{SO}(3,\mathbb{R})$-cocycles", Advances in Mathematics 457 (2024), 109943. DOI: 10.1016/j.aim.2024.109943; arXiv:2311.17537. https://doi.org/10.1016/j.aim.2024.109943
9. Nikolaos Karaliolios, "A note on cocycles in $\mathbb{T}\times SO(3)$", preprint, 2026. arXiv:2602.08514. https://arxiv.org/abs/2602.08514
10. Disheng Xu, "Density of positive Lyapunov exponents for symplectic cocycles", Journal of the European Mathematical Society 21 (2019), no. 10, 3143–3190. DOI: 10.4171/JEMS/899. https://doi.org/10.4171/JEMS/899