Minimality of projective actions of SL(2,R) cocycles non-homotopic to the identity over ergodic translations (Avila–Krikorian, Problem 3.1)

For an ergodic translation $f_\alpha: x \mapsto x + \alpha$ of the torus $\mathbb{R}^d/\mathbb{Z}^d$ and a continuous map $A: \mathbb{R}^d/\mathbb{Z}^d \to SL(2,\mathbb{R})$ that is not homotopic to the identity, the projective action $(f_\alpha, A): (x,w) \mapsto (x+\alpha, \mathring{A}(x)\cdot w)$ on the $(d+1)$-torus $\mathbb{R}^d/\mathbb{Z}^d \times \partial\mathbb{D}$ is known to be transitive, and Avila–Krikorian proved it is minimal whenever $A$ is premonotonic (some iterate is conjugate to one whose projective action rotates all directions monotonically) and of regularity at least $C^{1+\epsilon}$. Their Problem 3.1 asks whether the monotonicity-type hypotheses can be dropped entirely: is the projective action minimal for every continuous $A$ non-homotopic to the identity, over every ergodic translation, in every dimension $d \ge 1$? No proof and no counterexample is known in any regularity class or dimension. The problem remains open; its precise open core is the general continuous (equally, any regularity) non-premonotonic case, in any dimension.

Background

A quasiperiodic $SL(2,\mathbb{R})$ cocycle over a translation is a pair $(f_\alpha, A)$, where $f_\alpha: \mathbb{R}^d/\mathbb{Z}^d \to \mathbb{R}^d/\mathbb{Z}^d$ is the translation $x \mapsto x + \alpha$ on the $d$-dimensional torus and $A: \mathbb{R}^d/\mathbb{Z}^d \to SL(2,\mathbb{R})$ is a matrix-valued function (continuous by default throughout; $SL(2,\mathbb{R})$ denotes the group of real $2 \times 2$ matrices of determinant one). The translation $f_\alpha$ is ergodic (every invariant set has Haar (Lebesgue) measure zero or one) exactly when $1, \alpha_1, \ldots, \alpha_d$ are rationally independent, meaning that no relation $k_0 + k_1\alpha_1 + \cdots + k_d\alpha_d = 0$ with integers $k_j$ not all zero holds. The associated linear skew product $(x, v) \mapsto (x + \alpha, A(x)v)$ on $\mathbb{R}^d/\mathbb{Z}^d \times \mathbb{R}^2$ encodes, for instance, the transfer matrices of one-dimensional quasiperiodic Schrödinger operators, and its basic quantitative invariant is the Lyapunov exponent

$$L(f_\alpha, A) = \lim_{n \to \infty} \frac{1}{n} \int_{\mathbb{R}^d/\mathbb{Z}^d} \ln \|A^{(n)}(x)\| \, dx,$$

the asymptotic exponential growth rate of the cocycle iterates $A^{(n)}(x) = A(x + (n-1)\alpha) \cdots A(x)$.

Since $SL(2,\mathbb{R})$ deformation retracts onto $SO(2,\mathbb{R}) \cong S^1$ (the planar rotation group), every $A: \mathbb{R}^d/\mathbb{Z}^d \to SL(2,\mathbb{R})$ is classified up to homotopy by a vector $l \in \mathbb{Z}^d$: $A$ is homotopic to the unique cocycle of the form $x \mapsto R_{\langle l, x\rangle}$, where $R_\theta$ denotes the planar rotation of angle $2\pi\theta$. "Homotopic to the identity" (equivalently, homotopic to a constant) means $l = 0$. The class $l = 0$ contains the Schrödinger cocycles, of the form $\begin{pmatrix} \lambda v(x) - E & -1 \\ 1 & 0 \end{pmatrix}$, which anchor the spectral theory of quasiperiodic Schrödinger operators such as the almost Mathieu operator, the most studied example of the class, whose sampling function is a pure cosine. The complementary class $l \ne 0$ (the "non-homotopic" cocycles of this problem; the simplest examples are $SO(2,\mathbb{R})$-valued maps of nonzero degree) is the setting of the source paper; M.-R. Herman obtained the first positivity results for Lyapunov exponents there, for cocycles of a very specific form.

The topological dynamics of a cocycle is studied through its projective action (projectivized skew product): the homeomorphism of the compact $(d+1)$-torus $\mathbb{R}^d/\mathbb{Z}^d \times \partial\mathbb{D}$ — with $\partial\mathbb{D}$ the unit circle, identified in the usual way with the projective line of directions in $\mathbb{R}^2$ — given by

$$(x, w) \mapsto (x + \alpha, \mathring{A}(x) \cdot w),$$

where $\mathring{A}(x)$ is the Möbius (fractional-linear) action of $A(x)$ on directions. The action is transitive if it has a dense orbit, and minimal if every orbit is dense. For general quasiperiodically forced circle homeomorphisms (skew products over rotations with arbitrary fiber circle homeomorphisms), transitivity does not imply minimality: Béguin, Crovisier, Jäger, and Le Roux constructed transitive, non-minimal examples, some with minimal sets of Cantor type, within the classification program for quasiperiodically forced circle homeomorphisms of Jäger and Stark; such systems are also the natural habitat of strange nonchaotic attractors — compact invariant attracting sets with complicated geometry but non-chaotic dynamics, investigated numerically since the 1980s and constructed rigorously for forced circle maps by Jäger. For projective actions of one-frequency ($d = 1$) quasiperiodic $SL(2,\mathbb{R})$ cocycles, Béguin et al. proved a strong rigidity of minimal sets: any minimal strict subset of the torus is either a continuous $(p,q)$-invariant graph (the image of a continuous invariant section of a finite covering of the base torus) or meets generic fibers in exactly one or two points (an almost-automorphic configuration: minimal dynamics that is an almost one-to-one extension of the base rotation), so Cantor-type fiber intersections cannot occur. They also noted that, even in this projective setting, transitivity was not known to imply minimality.

Avila and Krikorian's theory for the non-homotopic class is centered on monotonicity. A map $A \in C^1(\mathbb{R}^d/\mathbb{Z}^d, SL(2,\mathbb{R}))$ is monotonic if for some direction $w \in \mathbb{R}^d$, every $x$, and every nonzero $y \in \mathbb{R}^2$, any determination of the argument of $A(x + tw)\cdot y$ has positive derivative in $t$: the projective action rotates every direction monotonically as the base moves in direction $w$. Given a frequency vector, the cocycle $(f_\alpha, A)$ is premonotonic if some iterate is $C^1$-conjugate to a monotonic cocycle; cocycles homotopic to a constant are never premonotonic, while non-homotopic cocycles that are $C^0$-conjugate to cocycles of rotations are premonotonic when the base is ergodic and $A \in C^1$. Their main topological result (Theorem 1.6 of the source; the monotonic case is Theorem 3.9) is that if $f_\alpha$ is ergodic and $(f_\alpha, A)$ is premonotonic with $A \in C^{1+\epsilon}$ (that is, $A$ differentiable with Hölder-continuous derivative), then the projective action is minimal. Transitivity, by contrast, holds unconditionally in the non-homotopic class: for $d = 1$ it is due to Kim, Kim, Hunt, and Ott (in the context of robust strange nonchaotic attractors), and the source paper extends it to all dimensions (Appendix B). For $L(A) > 0$, classical results of R. Johnson, as presented by Herman and recalled in the source, show that the projective action carries exactly two ergodic invariant measures, supported on the unstable and stable directions, and that in the non-homotopic case there is a unique minimal set $K_u = K_s$; minimality is then equivalent to this set filling the entire phase space — which is what monotonicity delivers in their class, and what remains unknown in general.

Problem Statement

Let $d \ge 1$, and let $\alpha \in \mathbb{R}^d$ be such that $1, \alpha_1, \ldots, \alpha_d$ are rationally independent (no relation $k_0 + k_1\alpha_1 + \cdots + k_d\alpha_d = 0$ with integers $k_j$ not all zero), so that the translation $f_\alpha: x \mapsto x + \alpha$ of the torus $\mathbb{R}^d/\mathbb{Z}^d$ is ergodic (every invariant set has Haar (Lebesgue) measure zero or one). Let $A \in C^0(\mathbb{R}^d/\mathbb{Z}^d, SL(2,\mathbb{R}))$ (a continuous matrix-valued map into the group of real $2 \times 2$ matrices of determinant one) be non-homotopic to the identity, that is, of nonzero homotopy class $l \in \mathbb{Z}^d$ ($A$ homotopic to $x \mapsto R_{\langle l, x\rangle}$ with $l \ne 0$, where $R_\theta$ denotes the planar rotation of angle $2\pi\theta$). Let $(f_\alpha, A)$ denote the projective action on the $(d+1)$-torus $\mathbb{R}^d/\mathbb{Z}^d \times \partial\mathbb{D}$ (with $\partial\mathbb{D}$ the unit circle, identified with the projective line of directions in $\mathbb{R}^2$),

$$(x, w) \mapsto (x + \alpha, \ \mathring{A}(x) \cdot w),$$

where $\mathring{A}(x)$ is the Möbius (fractional-linear) action of $A(x)$ on directions.

This is Problem 3.1 of Avila–Krikorian, Monotonic cocycles: is $(f_\alpha, A)$ minimal — that is, is every orbit dense in $\mathbb{R}^d/\mathbb{Z}^d \times \partial\mathbb{D}$ — for every such choice of $d$, $\alpha$, and $A$?

The same question makes sense under any additional smoothness assumption on $A$ ($C^r$, analytic). It is answered affirmatively when $(f_\alpha, A)$ is premonotonic (some iterate is $C^1$-conjugate to a monotonic cocycle, one whose projective action rotates every direction monotonically as the base moves along some fixed direction) and $A \in C^{1+\epsilon}$ (Avila–Krikorian, Theorem 1.6), and no counterexample is known in any regularity class or dimension outside the premonotonic framework, including the one-frequency analytic case. A negative answer is equivalent to exhibiting some ergodic translation, some non-homotopic continuous $A$, and a nonempty proper closed subset of $\mathbb{R}^d/\mathbb{Z}^d \times \partial\mathbb{D}$ invariant under the projective action.

References

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