Global transversality, complete gap labeling, and joint analyticity of the spectral dimension for Sturmian substitution Jacobi Hamiltonians

For every primitive invertible two-letter substitution sequence $v$ (equivalently, a Sturmian sequence with quadratic-irrational rotation number), the Jacobi operator on $\ell^2(\mathbb Z)$ with off-diagonal weights $1,\mathfrak p$ and diagonal weights $0,\mathfrak q$ has a Cantor spectrum $\Sigma_{v,(\mathfrak p,\mathfrak q)}$ of zero Lebesgue measure with purely singular continuous spectral type. The spectrum is characterized dynamically: an energy belongs to $\Sigma_{v,(\mathfrak p,\mathfrak q)}$ if and only if the forward orbit of a corresponding point on the curve of initial conditions is bounded under the trace map $T_s$ associated with the primitive invertible substitution $s$ generating $v$ — a polynomial diffeomorphism preserving the Fricke–Vogt invariant; for the Fibonacci substitution, $T_s(x,y,z)=(2xy-z,x,y)$. The central open problem left by Mei–Yessen (2014) and Yessen (2013) is whether the curve of initial conditions meets the center-stable manifolds of the trace-map dynamics transversally for every parameter pair $(\mathfrak p,\mathfrak q)\neq(1,0)$ — currently proved only near the free operator $(1,0)$ — and whether the two conjectured consequences hold without the weak-coupling restriction: complete gap labeling (all gaps predicted by the gap-labeling theorem are open) for all nontrivial Jacobi parameters, and joint analyticity of the Hausdorff dimension of the spectrum in $(\mathfrak p,\mathfrak q)$ away from $(1,0)$. The Schrödinger instances of these questions were resolved after the source paper appeared (Damanik–Gorodetski–Yessen 2016 for the Fibonacci Hamiltonian; Band–Beckus–Loewy 2024–2026 for gap opening in the entire Sturmian Schrödinger class), so the Jacobi parameter plane is the surviving open core.

Background

One-dimensional discrete Schrödinger and Jacobi operators with aperiodic deterministic coefficients are the standard mathematical models of one-dimensional quasicrystals, discovered in the 1980s. On the Jacobi side, a bounded self-adjoint operator on $\ell^2(\mathbb Z)$ is specified by an off-diagonal (hopping) sequence and a diagonal (potential) sequence; the Schrödinger case is the special case of constant hopping. A two-letter substitution is a map $s:\{0,1\}\to\{0,1\}^*$ (finite words), extended to infinite words by concatenation; it is primitive if some power of $s$ applied to either letter produces a word containing both letters, and invertible if $s$ extends to an automorphism of the free group on two generators. Fixed points of primitive invertible substitutions are Sturmian sequences — the aperiodic sequences of minimal symbolic complexity — and their rotation numbers $\alpha_v$ are quadratic irrationals; the Fibonacci substitution $s(0)=01$, $s(1)=0$ with $\alpha_v=(\sqrt5-1)/2$ is the canonical example. For a substitution sequence $v$ one takes the hull (the orbit closure of $v$ under the shift, a compact minimal uniquely ergodic space) and defines, for real parameters with $\mathfrak p\neq0$,

$$(H_{v,(\mathfrak p,\mathfrak q)}\phi)_n=p(v_{n+1})\phi_{n+1}+p(v_n)\phi_{n-1}+q(v_n)\phi_n,\qquad p(0)=1,\ p(1)=\mathfrak p,\ q(0)=0,\ q(1)=\mathfrak q,$$

so that $(\mathfrak p,\mathfrak q)=(1,0)$ is the free Laplacian, $\mathfrak p=1$ is the Schrödinger case (potential modulated by $v$), and $\mathfrak q=0$ is the off-diagonal case. The spectrum $\Sigma_{v,(\mathfrak p,\mathfrak q)}$ is independent of the hull element, and for every $(\mathfrak p,\mathfrak q)\neq(1,0)$ it is a Cantor set of zero Lebesgue measure with purely singular continuous spectral measures (Yessen 2013; Mei–Yessen 2014).

The main tool is the trace map. To every primitive invertible substitution $s$ there corresponds its own polynomial diffeomorphism $T_s$ of $\mathbb R^3$ — its trace map — governing the traces of the transfer matrices along the sequences generated by $s$; for the Fibonacci substitution, $T_s(x,y,z)=(2xy-z,x,y)$. Each $T_s$ preserves the Fricke–Vogt invariant $I(x,y,z)=x^2+y^2+z^2-2xyz-1$ and hence the family of cubic surfaces $S_V=\{I=V^2/4\}$, $V>0$. In the Schrödinger case, Sütö's observation identifies the spectrum with the set of energies whose initial-condition line meets the stable lamination of a compact hyperbolic set on a single surface $S_V$; if the line meets the lamination transversally (an intersection angle bounded away from zero), the spectrum is a dynamically defined Cantor set whose Hausdorff dimension, gap structure, and scaling exponents can be computed through the thermodynamic formalism (topological pressure) of the hyperbolic dynamics. In the Jacobi case the curve of initial conditions $l_{(\mathfrak p,\mathfrak q)}$ no longer lies on a single invariant surface — the invariant along it is a nonconstant function of the energy — so one works with the two-dimensional lamination $L=\bigcup_{V>0}L_V$ inside the non-compact region $\bigcup_{V>0}S_V$ and with the center-stable manifolds (the invariant manifolds of the dynamics of $T_s$ whose linearized flow has no expanding direction).

Transversality is the linchpin: it is known for $\|(\mathfrak p,\mathfrak q)-(1,0)\|$ small (the weak-coupling regime; Yessen 2013, Mei–Yessen 2014), and, in the Fibonacci case, should it fail anywhere, it can fail at no more than finitely many points of the curve. Two consequences are conjectured to hold once the weak-coupling restriction is removed. First, complete gap labeling: the integrated density of states $N_{v,(\mathfrak p,\mathfrak q)}$ (the limiting normalized distribution of eigenvalues of finite-volume restrictions, constant on each gap of the spectrum) takes, on the complement of the spectrum, values in the countable gap-labeling set $\{\{m\alpha_v\}:m\in\mathbb Z\}$ of fractional parts, and the conjecture is that all these predicted gaps are actually open — the analogue for these models of the Dry Ten Martini Problem, the name given in the almost-Mathieu literature to the statement that every gap allowed by the gap-labeling theorem occurs. This is proved for $\|(\mathfrak p,\mathfrak q)-(1,0)\|$ small (Mei–Yessen, Theorem 3.5) but open beyond. Second, regularity of the fractal dimension: $\dim_{\mathrm H}(\Sigma_{v,(\mathfrak p,\mathfrak q)})$ is known to be continuous in $(\mathfrak p,\mathfrak q)$ and analytic along the axes $\mathfrak p=1$ and $\mathfrak q=0$ away from $(1,0)$, and is conjectured to be jointly analytic in $(\mathfrak p,\mathfrak q)$ away from $(1,0)$; a stronger conjecture states that in the Fibonacci Schrödinger case the dimension is monotone in the coupling, from which the analyticity conjecture would follow (Mei–Yessen, Remark 3.8).

The Schrödinger counterparts of all of these questions were open when the source paper was written and have since been settled: Damanik–Gorodetski–Yessen (Invent. Math. 2016) proved that for the Fibonacci Hamiltonian the line of initial conditions meets the stable manifolds transversally for every coupling $\lambda>0$, and deduced that all gaps allowed by the gap-labeling theorem are open for every $\lambda>0$; Band–Beckus–Loewy (preprints 2024–2026) subsequently solved the Dry Ten Martini problem for Schrödinger operators with arbitrary Sturmian potentials at every nonzero coupling, using a tree representation of the spectrum rather than trace-map transversality. What remains open is the extension of this all-parameter theory from the Schrödinger axis to the full Jacobi parameter plane, which is the subject of this problem.

Problem Statement

Let $v$ be the fixed point of a primitive invertible substitution on two letters (a Sturmian sequence with quadratic-irrational rotation number $\alpha_v$), and for parameters $(\mathfrak p,\mathfrak q)\in\mathbb R^2$ with $\mathfrak p\neq0$ let $\Sigma_{v,(\mathfrak p,\mathfrak q)}$ denote the spectrum of the Jacobi operator $(H\phi)_n=p(v_{n+1})\phi_{n+1}+p(v_n)\phi_{n-1}+q(v_n)\phi_n$ with $p(0)=1$, $p(1)=\mathfrak p$, $q(0)=0$, $q(1)=\mathfrak q$, so that $(\mathfrak p,\mathfrak q)=(1,0)$ is the free Laplacian. Let $s$ denote the primitive invertible substitution generating $v$, $T_s$ its trace map (the polynomial diffeomorphism of $\mathbb R^3$ governing the traces of the transfer matrices along $v$; for the Fibonacci substitution, $T_s(x,y,z)=(2xy-z,x,y)$), $I=x^2+y^2+z^2-2xyz-1$ the Fricke–Vogt invariant preserved by every $T_s$, $S_V=\{I=V^2/4\}$, $V>0$, the invariant surfaces, and $l_{(\mathfrak p,\mathfrak q)}:\mathbb R\to\mathbb R^3$ the curve of initial conditions of the (normalized) Jacobi transfer matrices, $l_{(\mathfrak p,\mathfrak q)}(E)=\big(\frac{E^2-\mathfrak q E-\mathfrak p^2-1}{2\mathfrak p},\frac{E-\mathfrak q}{2\mathfrak p},\frac{E}{2}\big)$, which depends on $(\mathfrak p,\mathfrak q)$ but not on $v$, so that $E\in\Sigma_{v,(\mathfrak p,\mathfrak q)}$ if and only if the forward $T_s$-orbit of $l_{(\mathfrak p,\mathfrak q)}(E)$ is bounded. The problem is to settle the following three conjectured statements of Mei–Yessen (2014) and Yessen (2013) — items (ii) and (iii) are conjectured in the sources for every primitive invertible substitution (Mei–Yessen, Remarks 3.6 and 3.8), while item (i) is conjectured there for the Fibonacci substitution only (Yessen, Conjecture 4.1; Mei–Yessen, Remark 2.35) and extends that conjecture to every primitive invertible $v$ via its own trace map $T_s$, within the general-$v$ trace-map formalism of Mei–Yessen, Section 4 — each of which may be proved or refuted, for every primitive invertible substitution $v$ and every $(\mathfrak p,\mathfrak q)\neq(1,0)$ with $\mathfrak p\neq0$ (the parameter instances already settled — the Schrödinger axis $\mathfrak p=1$ for the Fibonacci substitution, and gap openness on the Schrödinger axis for the whole Sturmian class — may be assumed):

(i) Global transversality: the curve $l_{(\mathfrak p,\mathfrak q)}$ intersects the center-stable manifolds of the dynamics of $T_s$ on $\bigcup_{V>0}S_V$ transversally at every point of the intersection (equivalently, the conjectured transversality breakdown never occurs away from $(1,0)$; in the Fibonacci case it is known that failures could occur at no more than finitely many points, and transversality holds for $\|(\mathfrak p,\mathfrak q)-(1,0)\|$ sufficiently small — the latter for every primitive invertible $v$).

(ii) Complete gap labeling (the Dry Ten Martini statement for substitution Jacobi operators): with $N_{v,(\mathfrak p,\mathfrak q)}$ the integrated density of states, every gap predicted by the gap-labeling theorem is open, that is,

$$\{N_{v,(\mathfrak p,\mathfrak q)}(E):E\in\mathbb R\setminus\Sigma_{v,(\mathfrak p,\mathfrak q)}\}=\{\{m\alpha_v\}:m\in\mathbb Z\},$$

in the complete-gap-labeling convention of Mei–Yessen, Theorem 3.5, where this is currently known only for $\|(\mathfrak p,\mathfrak q)-(1,0)\|$ small.

(iii) Regularity of the spectral dimension: $\dim_{\mathrm H}(\Sigma_{v,(\mathfrak p,\mathfrak q)})$ is jointly analytic in $(\mathfrak p,\mathfrak q)$ away from $(1,0)$, that is, on the parameter domain $\{(\mathfrak p,\mathfrak q):\mathfrak p\neq0\}\setminus\{(1,0)\}$ — strengthening the known continuity in $(\mathfrak p,\mathfrak q)$ and analyticity along the axes $\mathfrak p=1$ and $\mathfrak q=0$.

Results restricted to a neighborhood of $(1,0)$ or to the already-settled Schrödinger axis instances do not discharge the problem: the conjectures concern precisely the removal of the weak-coupling restriction (Mei–Yessen, Remark 3.6) and all parameters away from $(1,0)$ (Remark 3.8).

References

1. May Mei and William Yessen, 'Tridiagonal Substitution Hamiltonians', Mathematical Modelling of Natural Phenomena 9 (2014), no. 5, 204-238, DOI 10.1051/mmnp/20149514; preprint arXiv:1312.2259, https://arxiv.org/abs/1312.2259.
2. William Yessen, 'Spectral analysis of tridiagonal Fibonacci Hamiltonians', Journal of Spectral Theory 3 (2013), 101-128, DOI 10.4171/jst/39; preprint arXiv:1111.0953, https://arxiv.org/abs/1111.0953. (Origin of the global Jacobi transversality Conjecture 4.1.)
3. David Damanik, Anton Gorodetski, and William Yessen, 'The Fibonacci Hamiltonian', Inventiones mathematicae 206 (2016), 629-692, DOI 10.1007/s00222-016-0660-x; preprint arXiv:1403.7823, https://arxiv.org/abs/1403.7823.
4. Ram Band, Siegfried Beckus, and Raphael Loewy, 'The Dry Ten Martini Problem for Sturmian Hamiltonians', preprint arXiv:2402.16703, https://arxiv.org/abs/2402.16703.
5. Ram Band, Siegfried Beckus, and Raphael Loewy, 'Complete hierarchical structure of the spectral bands in the Kohmoto model', preprint arXiv:2607.06361, https://arxiv.org/abs/2607.06361.
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7. David Damanik and Anton Gorodetski, 'Spectral transitions for the square Fibonacci Hamiltonian', Journal of Spectral Theory 8 (2018), 1487-1507, DOI 10.4171/jst/232, https://doi.org/10.4171/jst/232.
8. William Yessen, 'Hausdorff dimension of the spectrum of the square Fibonacci Hamiltonian', preprint arXiv:1410.3102, https://arxiv.org/abs/1410.3102.
9. David Damanik, Anton Gorodetski, and Boris Solomyak, 'Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian', Duke Mathematical Journal 164 (2015), no. 8, 1603-1640, DOI 10.1215/00127094-3119739; preprint arXiv:1306.4284, https://arxiv.org/abs/1306.4284.
10. Qinghui Liu, Yanhui Qu, and Xiao Yao, 'The Spectrum of Period-Doubling Hamiltonian', Communications in Mathematical Physics 394 (2022), 1039-1100, DOI 10.1007/s00220-022-04417-7; preprint arXiv:2108.13257, https://arxiv.org/abs/2108.13257.
11. Victor Kleptsyn and Alexandro Luna, 'On the Regularity of the Dimension of Cookie-Cutter-Like Sets', preprint arXiv:2511.07557, https://arxiv.org/abs/2511.07557.
12. David Damanik and Long Li, 'Opening gaps in the spectrum of strictly ergodic Jacobi and CMV matrices', Journal of Functional Analysis 289 (2025), 111182, DOI 10.1016/j.jfa.2025.111182; preprint arXiv:2404.03864, https://arxiv.org/abs/2404.03864. (Adjacent genericity result: all gap-labeling-theorem gaps open for dynamically defined Jacobi and CMV matrices with generic continuous sampling functions.)