Open fronts in the Avila–Bochi–Damanik gap-opening program for strictly ergodic Schrödinger operators: simultaneous gap opening, analytic generic Cantor spectrum, and aperiodic dynamics without finite-dimensional factors

Avila, Bochi and Damanik (J. Eur. Math. Soc. 2012) proved that for Schrödinger operators whose potentials are obtained by continuous sampling over strictly ergodic dynamics admitting a non-periodic finite-dimensional factor, any collapsed spectral gap can be opened by a continuous deformation of the sampling function, and consequently that for a generic continuous sampling function every gap allowed by the Gap Labeling Theorem is open. Three of the four issues that paper explicitly leaves open remain open. First, it is unknown whether a single continuous path of sampling functions starting at a given potential can open all gaps simultaneously. Second, in the analytic category it is unknown whether the generic converse of the Gap Labeling Theorem — or even generic Cantor spectrum — holds for quasi-periodic operators on tori of dimension at least two; even in one frequency, the statement for general analytic potentials and Diophantine frequencies remains open, despite major recent progress on the almost Mathieu operator and related one-frequency models. Third, it is unknown whether the finite-dimensional factor hypothesis on the base dynamics can be removed, for instance for strictly ergodic homeomorphisms of infinite-dimensional compact spaces that admit no non-periodic finite-dimensional factor. The fourth issue, a purely topological question about finite tower covers for aperiodic homeomorphisms of compact (possibly infinite-dimensional) spaces posed in Remark 5.5 of the source as the dynamical ingredient behind the third issue, has since been answered negatively: Tsukamoto–Tsutaya–Yoshinaga (Israel J. Math. 2022) constructed a free homeomorphism of a compact metrizable space without the marker property of Gutman, which is equivalent to the tower property asked for, so this front is discharged by an external counterexample and the open problem consists of the first three fronts.

Background

One-dimensional Schrödinger operators with aperiodic potentials are a central object in the mathematical physics of quasicrystals, and the most unified way to study them is dynamical. Let $X$ be a compact metric space, let $f : X \to X$ be a strictly ergodic homeomorphism — meaning that $f$ is minimal (every orbit is dense in $X$) and uniquely ergodic (there is exactly one $f$-invariant Borel probability measure $\mu$) — and let $v : X \to \mathbb{R}$ be a continuous sampling function. The associated dynamically defined Schrödinger operators act on $\ell^2(\mathbb{Z})$ by

$[H_{f,v,x}\psi](n) = \psi(n+1) + \psi(n-1) + v(f^n x)\,\psi(n).$

Minimality implies that the spectrum $\Sigma_{f,v}$ is a compact perfect subset of $\mathbb{R}$ independent of the base point $x$. The integrated density of states $N_{f,v} : \mathbb{R} \to [0,1]$ is the distribution function of the orbit-averaged spectral measure of $H_{f,v,x}$; it is continuous, non-decreasing and onto, and it is locally constant exactly on the complement of the spectrum. Each bounded connected component of $\mathbb{R} \setminus \Sigma_{f,v}$, called a spectral gap, therefore carries a label $\ell \in (0,1)$: the constant value of $N_{f,v}$ on it. The Gap Labeling Theorem asserts that every label belongs to $\mathcal{L} = G \cap (0,1)$, where $G \subset \mathbb{R}$ is the range of the Schwartzman asymptotic cycle — the subgroup of all integrals $\int_X \phi\, d\mu$ over continuous functions $\phi : X \to \mathbb{R}$ that are coboundaries modulo $\mathbb{Z}$, i.e. $\phi(x) = \psi(f x) - \psi(x) \bmod 1$ for some continuous $\psi : X \to \mathbb{R}/\mathbb{Z}$. Which labels actually occur depends on the sampling function: a label $\ell \in \mathcal{L}$ that does not label an actual gap is said to correspond to a collapsed gap, and actual gaps are called open. When $\mathcal{L}$ is dense in $(0,1)$ — as for almost periodic dynamics, where $X$ is an infinite compact abelian group and $f$ is a minimal translation, including quasi-periodic dynamics on tori $X = \mathbb{T}^d$ and limit-periodic dynamics — having all gaps open is equivalent to $\Sigma_{f,v}$ being a Cantor set, and the statement that all gaps are open for generic $v$ (generic in the sense of Baire: containing a countable intersection of open dense sets) is called the converse of the Gap Labeling Theorem. In the early 1980s Bellissard asked whether mixing dynamics could prevent Cantor spectrum ('whether mixing avoids Cantor spectrum') and whether Cantor spectrum is generic whenever the Gap Labeling Theorem permits it. The most studied special case, the almost Mathieu operator ($X = \mathbb{R}/\mathbb{Z}$, $v(x) = 2\lambda\cos(2\pi x)$), generated the famous Ten Martini Problem (is the spectrum a Cantor set?) and its stronger Dry version (are all allowed gaps open?); the Ten Martini Problem was solved by Avila–Jitomirskaya.

Avila–Bochi–Damanik answered both of Bellissard's questions affirmatively within their setting, under one standing hypothesis: $f$ is strictly ergodic and admits a non-periodic finite-dimensional factor — that is, there exist a homeomorphism $\tilde f$ of an infinite compact subset $\tilde X$ of some Euclidean space $\mathbb{R}^d$ and a continuous surjection $\pi : X \to \tilde X$ with $\pi \circ f = \tilde f \circ \pi$ (a factor map). Their gap-opening theorem states that for every $v \in C(X,\mathbb{R})$ and every $\ell \in \mathcal{L}$ there is a continuous path $v_t$, $0 \le t \le 1$, with $v_0 = v$, such that for every $t > 0$ the spectrum $\Sigma_{f,v_t}$ has an open gap with label $\ell$; since having an open gap with a given label is an open condition on $v$, a Baire-category argument yields the generic converse of the Gap Labeling Theorem: for a generic $v \in C(X,\mathbb{R})$, all gaps are open. The proof passes to the $\mathrm{SL}(2,\mathbb{R})$ Schrödinger cocycle $A(x) = \begin{pmatrix} v(x) & -1 \\ 1 & 0 \end{pmatrix}$, whose iterates are the transfer matrices of the eigenvalue equation, and analyzes uniform hyperbolicity (exponential growth of cocycle iterates, equivalent to an invariant splitting of the fiber), the fibered rotation number (a cocycle-level analogue of the integrated density of states measuring rotation on the projective line), and accessibility of non-hyperbolic cocycles by hyperbolic ones. One key dynamical ingredient is a tower construction (Proposition 5.1 of the source): over a minimal homeomorphism of a finite-dimensional compact space one can find, for each $n$, compact sets whose first $n$ iterates are pairwise disjoint and finitely many of whose further iterates cover the whole space. The finite-dimensional factor is used twice: the tower construction descends to the factor, and a dimension-dependent 'mildness' bound on return times comes from finite-dimensional topology. The paper also establishes the limits of the possible: for Furstenberg's minimal but not uniquely ergodic skew products on $\mathbb{T}^2$, Cantor spectrum is not generic (so strict ergodicity is essential), and in the analytic category the authors state that for translations on tori of dimension at least two it is 'far from clear whether one should expect' the generic converse to hold — comparing the two-frequency localization heuristics of Chulaevsky–Sinai — while in one dimension they expect it but could not prove it even for Diophantine translations, that is, translation vectors quantitatively far from rational ones. Prior analytic results covered only restricted regimes: Eliasson's theorem on sufficiently small generic analytic potentials for Diophantine translations in any dimension, Goldstein–Schlag's opening of a dense set of gaps (almost surely over the frequency) in the positive-Lyapunov-exponent regime, and the same authors' earlier theorem for generalized skew-shifts (dynamics fibering over almost periodic systems).

The paper explicitly leaves four issues open, which together mark the boundary of this gap-opening program: sharpening the deformation theorem to a single path (Remark 1.1); transferring the generic converse to the analytic category, especially in several frequencies (Remark 1.2); removing the finite-dimensional factor hypothesis; and the purely topological tower question for aperiodic homeomorphisms (Remark 5.5), which is the natural dynamical input for the third issue. Of these four, the tower question has since been answered negatively: Tsukamoto–Tsutaya–Yoshinaga constructed a free (periodic-point-free) homeomorphism of a compact metrizable space without the marker property introduced into topological dynamics by Downarowicz and Gutman — a property equivalent to the tower property asked for — so the open boundary of the program is now the first three issues.

Problem Statement

The problem is to complete the gap-opening program of Avila–Bochi–Damanik along the fronts the paper itself leaves open. The common scientific objective is to determine the exact extent to which the base dynamics alone can obstruct the opening of spectral gaps: Front 1 sharpens the source's deformation theorem, Front 2 transfers its generic corollary to the analytic category, and Fronts 3 and 4 push the same corollary to the largest class of strictly ergodic — and then merely aperiodic — dynamical systems, with Front 4 the dynamical ingredient behind Front 3.

(F1) Simultaneous opening by a single path (continuous category, hypotheses as in the source: $f$ strictly ergodic with a non-periodic finite-dimensional factor). Given $v \in C(X,\mathbb{R})$, does there exist a continuous path $v_t \in C(X,\mathbb{R})$, $0 \le t \le 1$, with $v_0 = v$, such that for every $t > 0$ all gaps of $\Sigma_{f,v_t}$ are open, i.e. every label $\ell \in \mathcal{L}$ labels an actual gap? The source's Theorem 1 opens one label at a time; the single-path strengthening is not known, and the paper's own method cannot be directly adapted, since its Remark A.4 shows the natural two-parameter strengthening of the key accessibility theorem is false in general.

(F2) Analytic genericity. For quasi-periodic dynamics ($X = \mathbb{T}^d$, $f$ a translation by a fixed $\alpha$) and analytic sampling functions: is a generic analytic $v$ (Baire-generic in the analytic category, e.g. with respect to uniform convergence on a fixed complex strip) such that all gaps are open, or at least such that $\Sigma_{f,v}$ is a Cantor set? For $d \ge 2$, beyond the perturbative small-potential regime of Eliasson, no case is known even for Diophantine $\alpha$, and the source states it is far from clear whether one should expect the statement to hold; for $d = 1$ the expectation is affirmative but the statement for general analytic $v$ — even for Diophantine $\alpha$ — remains open, with known results covering the small-potential, subcritical, and supercritical/large-coupling regimes, specific families such as the almost Mathieu operator, and $C^2$ cosine-type large-coupling models.

(F3) Removing the finite-dimensional factor. Do the gap-opening deformation theorem and the generic converse of the Gap Labeling Theorem for continuous sampling functions remain true for strictly ergodic $f$ without any non-periodic finite-dimensional factor — for instance, for strictly ergodic homeomorphisms of infinite-dimensional compact spaces that admit no such factor? (Strictly ergodic subshifts do not furnish examples of this class: a subshift over a finite alphabet is a zero-dimensional compact metric space, hence embeds in $\mathbb{R}$ and is itself a non-periodic finite-dimensional factor — of itself, minimality on an infinite space excluding periodic points — so the source's Theorem 1, Corollary 2 and Proposition 5.1 already apply to every strictly ergodic subshift, with or without equicontinuous factors.) (Strict ergodicity itself cannot be dropped: the Furstenberg examples refute the generic statement in Appendix B of the source.) Note that the tower-based method of the source cannot extend to arbitrary infinite-dimensional systems: by the refutation of (F4) below, finite tower covers fail even for some aperiodic homeomorphisms of infinite-dimensional compact metric spaces, so a positive answer in infinite dimension would require genuinely different ideas.

(F4) Tower covers for aperiodic homeomorphisms (the topological ingredient behind F3) — resolved negatively by external literature. The source posed (Remark 5.5): if $f$ is a homeomorphism of a compact (possibly infinite-dimensional) space without periodic points, is it true that for every $n \in \mathbb{N}$ there exists a compact set $K$ whose first $n$ iterates $K, fK, \ldots, f^{n-1}K$ are pairwise disjoint and such that finitely many iterates of $K$ cover the whole space? The source proves this for minimal finite-dimensional systems (Proposition 5.1), and the zero-dimensional analogue follows from Downarowicz's lemma; a related generic-recurrence result is due to Bonatti–Crovisier. The general aperiodic case is false: Gutman introduced the marker property — for every $n$ a set (open by definition, and equivalently closed/compact) whose first $n$ iterates are disjoint and whose finitely many iterates cover the space, which is exactly the tower property asked for — proved that every aperiodic homeomorphism of a finite-dimensional compact metric space has it, and asked whether every aperiodic system does; Tsukamoto–Tsutaya–Yoshinaga answered negatively, constructing a free (periodic-point-free) homeomorphism of a compact metrizable space without the marker property. Hence there is an aperiodic homeomorphism of a compact metric space, and an $n$, for which no compact set has its first $n$ iterates pairwise disjoint with finitely many iterates covering the space. This front is discharged by that published counterexample.

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