Conjugacy-class closures, analytic Cantor spectrum, and matrix-class projection for Schrödinger cocycles over generalized skew-shifts: the open problems of Avila–Bochi–Damanik (2009)

Avila–Bochi–Damanik (Duke Math. J. 146 (2009)) proved that for Schrödinger operators whose potentials are continuous samplings over generalized skew-shift dynamics, uniformly hyperbolic Schrödinger cocycles are C^0-dense, so the generic continuous potential produces Cantor spectrum. The paper itself states three open directions and one strengthening remark: (i) determine the closure of an arbitrary PSL(2,R)-conjugacy class of SL(2,R) cocycles over such bases, starting from the constant hyperbolic case; (ii) decide whether the generic-Cantor phenomenon survives regularization — can the spectrum of a skew-shift Schrödinger operator with analytic potential ever be a Cantor set; (iii) extend the projection/reduction lemmas beyond the one-parameter Schrödinger slice of SL(2,R); (iv) clarify the role of homotopy and topological obstructions when the closure statements are strengthened from PSL(2,R)-conjugacy to SL(2,R)-conjugacy, beyond the known sufficient hypothesis. No complete resolution of any component is known: Krüger's large-coupling interval theorem resolves a subregime of (ii), Jun's extensions to Jacobi and CMV matrices (two standard analogs of the Schrödinger operator) partially address (iii), and the central regularity question remains essentially wide open.

Background

Discrete one-dimensional Schrödinger operators act on sequences $\psi \in \ell^2(\mathbb{Z})$ by $(H\psi)_n = \psi_{n+1} + \psi_{n-1} + V(n)\psi_n$. A central way of producing ergodic models is to define the potential dynamically: $V(n) = f(T^n x)$, where $T$ is a homeomorphism of a compact metric space $X$, $f : X \to \mathbb{R}$ is a sampling function, and $x \in X$ is a phase. When $T$ is strictly ergodic (that is, minimal — every orbit is dense — and uniquely ergodic — it has a unique invariant probability measure), the spectrum $\Sigma$ of $H_x$ is independent of the phase. The best-studied base dynamics are irrational rotations (quasi-periodic potentials) and shift spaces (random potentials). Between them sits the skew-shift $T(x,y) = (x+\alpha, y+x)$ on the two-torus $\mathbb{T}^2$ with $\alpha$ irrational: like a rotation it is distal (distinct orbits stay a uniformly positive distance apart) and has zero topological entropy, but its orbits carry a quadratic phase, and 'random-like' spectral behavior is expected for it. Avila–Bochi–Damanik work with generalized skew-shifts: strictly ergodic homeomorphisms admitting an equicontinuous factor (a rotation of an infinite compact abelian group), the class generated by iterated skewing over such rotations.

Spectral analysis is transferred to dynamics via the Schrödinger cocycle: for an energy $E$, $A^{(E,V)}(x) = \begin{pmatrix} E - V(x) & -1 \\ 1 & 0 \end{pmatrix} \in \mathrm{SL}(2,\mathbb{R})$, whose iterates carry the solutions of the eigenvalue equation. A cocycle $(T, A)$ is uniformly hyperbolic if there is a decomposition of $\mathbb{R}^2$ into line bundles that is contracted/expanded exponentially along orbits; by Johnson's theorem, $E$ belongs to $\Sigma$ exactly when $(T, A^{(E,V)})$ fails to be uniformly hyperbolic. The (top) Lyapunov exponent $L(E) = \lim_n \frac{1}{n}\int \log\|A^{(n)}(x)\| \, d\mu(x)$ measures the exponential growth of the iterates. A cocycle is reducible if it is conjugate to a constant cocycle, $\tilde{A}(x) = B(Tx)A(x)B(x)^{-1}$, with $B$ valued in $\mathrm{PSL}(2,\mathbb{R}) = \mathrm{SL}(2,\mathbb{R})/\{\pm I\}$; conjugacy by $\mathrm{SL}(2,\mathbb{R})$-valued $B$ is strictly stronger, and the passage from $\mathrm{PSL}(2,\mathbb{R})$ to $\mathrm{SL}(2,\mathbb{R})$ is a lifting problem through the twofold cover $\mathrm{SL}(2,\mathbb{R}) \to \mathrm{PSL}(2,\mathbb{R})$ obstructed by homotopy and homology data of the base. Uniformly hyperbolic cocycles are always reducible up to homotopy (homotopic, as maps $X \to \mathrm{SL}(2,\mathbb{R})$, to a reducible cocycle), and this is the only obstruction in the picture below.

Avila–Bochi–Damanik (ABD) proved, over a generalized skew-shift base: (a) (their Theorems 2–3) uniform hyperbolicity and reducibility are dense, in the $C^0$ topology, within the class of cocycles reducible up to homotopy; more precisely, if a member of that class is not uniformly hyperbolic, then for every nonhyperbolic constant $A_* \in \mathrm{SL}(2,\mathbb{R})$ with $|\mathrm{tr}\,A_*| \le 2$ it lies in the closure of the $\mathrm{PSL}(2,\mathbb{R})$-conjugacy class of $(T, A_*)$; (b) (their Theorems 4–5, via the projection Lemma 9) a retraction phenomenon for the Schrödinger slice $S = \{\begin{pmatrix} t & -1 \\ 1 & 0 \end{pmatrix} : t \in \mathbb{R}\}$: an $S$-valued cocycle over a minimal homeomorphism that can be approximated (in $C^0$, or in $C^s$ for $0 \le s \le r$ over a $C^r$ base) by cocycles with a conjugacy-invariant property can also be approximated by $S$-valued cocycles with that property — perturbations can be projected back into the Schrödinger class. Combining (a) and (b), uniformly hyperbolic Schrödinger cocycles are $C^0$-dense, and since $\mathbb{R} \setminus \Sigma$ is exactly the set of uniformly hyperbolic energies, for generic (in the sense of Baire category) continuous potentials $V$ the complement $\mathbb{R} \setminus \Sigma$ is dense (their Corollary 6): the generic continuous skew-shift potential gives Cantor spectrum, meaning a closed spectrum with empty interior (perfectness being automatic for such spectra).

This generic-Cantor conclusion stands in tension with the regular picture: for the skew-shift with a sufficiently regular non-constant potential, the expectation recorded in Bourgain's monograph and pursued by his program is positive Lyapunov exponents for almost every energy, pure point spectrum with exponentially decaying eigenfunctions for almost every phase, and a spectrum that is an interval, hence not a Cantor set. This tension is the source of the open problems, which the authors state as the explicit delimitation of their method.

Problem Statement

The problem is the set of open directions stated by the authors of Avila–Bochi–Damanik (ABD; Duke Math. J. 146 (2009)), together with the strengthening question raised by their Remark 9, taken together as a delimitation of the density-of-hyperbolicity mechanism behind generic Cantor spectrum over generalized skew-shifts. Throughout, the base $T$ is a generalized skew-shift (a strictly ergodic homeomorphism of a compact space with an equicontinuous factor; the classical skew-shift $T(x,y) = (x+\alpha, y+x)$ on $\mathbb{T}^2$ with $\alpha$ irrational, when explicitly meant) except in component (iii), where, as in the source's projection results (Theorems 4 and 5, Lemma 9), the base is an arbitrary minimal homeomorphism; cocycles are $\mathrm{SL}(2,\mathbb{R})$-valued with the $C^0$ topology unless a higher regularity is stated, and conjugacy means $\tilde{A}(x) = B(Tx)A(x)B(x)^{-1}$.

(i) Closure of conjugacy classes. Determine the closure of an arbitrary $\mathrm{PSL}(2,\mathbb{R})$-conjugacy class of cocycles over the given base dynamics. In particular, determine the closure of the $\mathrm{PSL}(2,\mathbb{R})$-conjugacy class of a constant hyperbolic cocycle $(T, A_*)$ with $|\mathrm{tr}\,A_*| > 2$, which the authors state already escapes their methods (their technique describes only closures of classes of nonhyperbolic constants, $|\mathrm{tr}\,A_*| \le 2$).

(ii) Analytic Cantor spectrum. Can the spectrum of a Schrödinger operator associated to the skew-shift with analytic potential ever be a Cantor set? Throughout this component the sampling function $f : \mathbb{T}^2 \to \mathbb{R}$ is real-analytic, non-constant, and depends genuinely on the skew-shift dynamics: samplings that depend only on the rotation coordinate, of the form $f(x,y) = v(x)$, are excluded, since they reduce the potential $V(n) = \lambda f(T^n x)$ to a one-frequency quasi-periodic potential, whereas samplings that depend only on the skew coordinate, of the form $f(x,y) = w(y)$ (the canonical skew-shift potentials), remain in scope. The already-covered large-coupling Diophantine regime is excluded throughout (a frequency $\alpha$ badly approximable by rationals, for instance satisfying $|\alpha - p/q| > c/q^{\tau}$ for some $c, \tau > 0$ and all rationals $p/q$). On this common class the question is two-sided: either exhibit a sampling function $f$ in this class, an irrational frequency $\alpha$, and a real coupling $\lambda$ (the potential being $V(n) = \lambda f(T^n x)$) for which the spectrum is a closed set with empty interior, or prove that for every such sampling function $f$, every irrational frequency $\alpha$, and every real coupling $\lambda$, the spectrum is not a Cantor set (as expected under the interval-spectrum conjecture). This decides whether the $C^0$-generic Cantor-spectrum phenomenon is an artifact of weak regularity.

(iii) Projection beyond the Schrödinger class. The reduction results (Theorems 4 and 5, Lemma 9 of the source) hold for the one-parameter slice $S = \{\begin{pmatrix} t & -1 \\ 1 & 0 \end{pmatrix} : t \in \mathbb{R}\}$. Investigate their validity for more general classes of matrix families: for which explicit classes $\mathcal{S} \subset \mathrm{SL}(2,\mathbb{R})$, and which regularities $0 \le s \le r$, does it hold that an $\mathcal{S}$-valued cocycle over a minimal base that can be approximated by cocycles with a conjugacy-invariant property can also be approximated by $\mathcal{S}$-valued cocycles with that property?

(iv) Homotopy versus $\mathrm{SL}(2,\mathbb{R})$-conjugacy. Clarify the precise role of homotopy and topological obstructions in strengthening the closure statements from $\mathrm{PSL}(2,\mathbb{R})$-conjugacy to $\mathrm{SL}(2,\mathbb{R})$-conjugacy. The authors prove the strengthening under the hypothesis that $A$ is homotopic to a cocycle conjugate to a constant (their Remark 9), and give an explicit example over the base $T(x,y) = (x+\alpha, y+2x)$ with $A(x,y) = R_{2\pi x}$ (rotation of angle $2\pi x$) showing the stronger conclusion fails without additional hypotheses; the task is to clarify which additional topological hypotheses on the cocycle and the base allow the $\mathrm{SL}(2,\mathbb{R})$-conjugacy version of the closure statement, and how the obstruction behaves beyond the known sufficient condition.

The common objective of the four components is to establish the true scope of the ABD mechanism: which conjugacy-class closures hold (components i and iv), which matrix classes the projection technique covers (component iii), and whether the resulting generic-Cantor phenomenon for Schrödinger operators over skew-shifts is genuine or a low-regularity artifact (component ii). The four components are independent research targets drawn from the same paper. Component (ii) is a two-sided decision. Components (i), (iii) and (iv) are open-ended: for (i), a determination of the closure of an arbitrary $\mathrm{PSL}(2,\mathbb{R})$-conjugacy class; for (iii), a criterion deciding, for an arbitrary family $\mathcal{S} \subset \mathrm{SL}(2,\mathbb{R})$, regularity $0 \le s \le r$, and an arbitrary minimal base, whether the approximation/projection property holds; for (iv), necessary and sufficient topological conditions deciding, for an arbitrary cocycle over the base, whether the $\mathrm{SL}(2,\mathbb{R})$-conjugacy strengthening holds.

References

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