Anderson localization for general Hölder sampling over hyperbolic transformations, and the large-coupling exceptional set

The Avila–Damanik–Zhang series studies discrete one-dimensional Schrödinger operators whose potentials are obtained by sampling a nonconstant Hölder function along the orbits of a hyperbolic transformation — concretely, a subshift of finite type with a fully supported ergodic measure with local product structure. Part I (Inventiones mathematicae, 2023) proved that the Lyapunov exponent is positive away from a discrete exceptional set $Z_f$ for every nonconstant Hölder sampling function, and that $Z_f$ is finite for locally constant or globally fiber bunched functions; it announced a second paper implementing the road map to Anderson localization (discreteness of $Z_f$, uniform positivity away from $Z_f$, large deviation estimates, elimination of double resonances). That second paper now exists as a preprint (arXiv:2402.00215, 2024) and proves spectral localization, together with exponential dynamical localization away from a finite exceptional set of energies — but only for locally constant sampling functions (arbitrary coupling) and small-supremum-norm, globally fiber bunched ones (small coupling). The problem studied here is the surviving open core of this program: prove Anderson localization for general nonconstant Hölder sampling functions of arbitrary norm — equivalently, for an arbitrary but fixed Hölder sampling function at arbitrary coupling, the doubling map and Arnold's cat map being standard examples of the base dynamics — and determine the fate of the exceptional set $Z_{\lambda f}$ as the coupling $\lambda \to \infty$ for globally bunched functions, for which only discreteness is currently guaranteed.

Background

Discrete one-dimensional Schrödinger operators act on the space $\ell^2(\mathbb{Z})$ of square-summable bilateral sequences by

$[H_\omega u](n) = u(n+1) + u(n-1) + V_\omega(n)\,u(n),$

where the potential $V_\omega$ is drawn from a probability space $(\Omega, \mu)$. In the Anderson model the values $V_\omega(n)$ are independent, identically distributed random variables, and the operator is almost surely localized: it has pure point spectrum with exponentially decaying eigenfunctions (spectral localization), and its unitary time-evolution group has exponential off-diagonal decay relative to the standard basis, uniformly in time (dynamical localization). The standard proof combines positivity of the Lyapunov exponent for independent matrix products (Furstenberg's theorem) with large deviation estimates and an elimination of double resonances (two nearby sites simultaneously producing abnormally small transfer denominators at close energies); a self-contained account for the Bernoulli case is given by Bucaj, Damanik, Fillman, Gerbuz, VandenBoom, Wang, and Zhang (2019).

The Avila–Damanik–Zhang series asks how much of this picture survives when the underlying randomness is structural rather than independent. Writing the potential as $V_\omega(n) = \lambda f(T^n\omega)$ with a sampling function $f$, a coupling constant $\lambda > 0$, and an ergodic transformation $(\Omega, T, \mu)$, the Anderson model corresponds to the Bernoulli shift with $f(\omega) = \omega(0)$. The series treats uniformly hyperbolic base dynamics: $(\Omega, T)$ is a subshift of finite type (a closed, shift-invariant subset of a finite-alphabet two-sided sequence space specified by finitely many forbidden words, with $T$ the left shift), and $\mu$ is a fully supported ergodic measure with a local product structure, meaning that locally the measure decomposes into conditional measures along the stable and unstable directions of the shift. A standard strengthening is the bounded distortion property, a quantitative approximate-independence statement under which the probabilities of cylinder sets depending on far-apart coordinate blocks factor up to a multiplicative constant; it holds for equilibrium states (invariant measures characterized by a variational principle) of Hölder continuous potentials, including Lebesgue measure for expanding circle maps such as the doubling map $x \mapsto 2x \bmod 1$, hyperbolic toral automorphisms (invertible linear maps of the torus whose defining matrix has no eigenvalue of modulus one, such as Arnold's cat map), and Markov measures.

The spectral analysis is encoded in the Schrödinger cocycle: solutions of the eigenvalue equation propagate by the transfer matrices $A^E(\omega) = \begin{pmatrix} E - \lambda f(\omega) & -1 \\ 1 & 0 \end{pmatrix}$ in $SL(2,\mathbb{R})$. The Lyapunov exponent

$L(E) = \lim_{n\to\infty} \frac{1}{n} \int_\Omega \log \|A^E_n(\omega)\| \, d\mu(\omega),$

whose existence follows from Kingman's subadditive ergodic theorem (a standard convergence result for subadditive sequences of integrals), measures the asymptotic exponential growth of the transfer matrices; its positivity is the primary mechanism of localization, and the object of Part I of the series is the exceptional set $Z_f = \{E \in \mathbb{R} : L(E) = 0\}$.

Part I (Avila–Damanik–Zhang, Inventiones mathematicae 231 (2023), 851–927) proves that for every nonconstant Hölder continuous $f$ (with $\mu$ as above and $T$ having a fixed point), $Z_f$ is discrete, and that $Z_f$ is finite when $f$ is locally constant (depending only on finitely many coordinates of $\omega$) or globally fiber bunched — the latter concretely meaning that $\|f\|_\infty$ is below an explicit threshold depending on the Hölder exponent, so that the cocycle is fiber bunched, that is, its orbit-norm growth is dominated by its oscillation, the regime in which continuous stable and unstable holonomies (canonical identifications of nearby cocycle fibers along the stable and unstable orbits of the base dynamics) exist and the Lyapunov exponent is continuous in the cocycle by the theorem of Backes, Brown, and Butler (2018). Part I moreover shows that $Z_f$ is empty for a residual (containing a countable intersection of open dense sets) collection of Hölder $f$, and that within the restricted classes an open dense set of $f$ gives $L$ uniformly bounded away from zero. Its Remark 1.6 describes the road map from these results to localization: discreteness of $Z_f$, uniform lower bounds for $L(E)$ on compact intervals avoiding $Z_f$ (typically obtained from continuity of $L(E)$), suitable large deviation estimates, and an argument ruling out infinitely many double resonances together imply spectral localization for $\mu$-almost every $\omega$; the authors announced that this implementation would be carried out in the second paper of the series.

The second paper (Avila–Damanik–Zhang, arXiv:2402.00215, 2024) carries out the program, but only in the two classes where $Z_f$ is finite. It proves a uniform large deviation estimate (ULD: exponential decay, uniform in the energy on compact intervals away from a finite exceptional set $F_f \supseteq Z_f$, of the probability that $\frac{1}{n}\log\|A^E_n(\omega)\|$ deviates from $L(E)$) for nonconstant locally constant or globally fiber bunched sampling functions over bounded-distortion measures (Theorem 2.10), and the localization criterion that positivity plus ULD on an interval imply spectral localization — indeed exponential dynamical localization — there (Theorem 2.12). Consequently, full Anderson localization holds for nonconstant locally constant $f$ at arbitrary coupling (Corollaries 2.14 and 2.17, covering Markov chains and the Bernoulli case) and for globally fiber bunched $f$, hence for the doubling map and Arnold's cat map with Lebesgue measure at small coupling $0 < \lambda \le \lambda_0$ (Corollaries 2.15 and 2.16). For the cat map, the previous state of the art was the work of Bourgain and Schlag (2000) on strongly mixing potentials, which had to exclude energy intervals of positive length.

What remains beyond these methods is the general Hölder case: a fixed nonconstant Hölder sampling function of arbitrary supremum norm, or equivalently the large-coupling regime of a fixed one. Here $Z_f$ is only known to be discrete (possibly infinite); neither continuity of $L(E)$ in $E$ nor uniform large deviations is known, since the available proofs of both require the holonomies supplied by fiber bunching or by local constancy; and the authors state that their techniques would need further development to yield stronger ULD results, planned for a third paper of the series that had not appeared on arXiv as of September 2026. On the positivity side, for large coupling the exceptional set $Z_{\lambda f}$ is known to be discrete for all $\lambda > 0$ and finite only for small $\lambda$ (Part I, Corollary 7.2); large-coupling information exists only in special classes — for $C^1$ monotone doubling-map potentials Zhang (2024) proved $L(E) \ge \log\lambda - C$ for all energies, Chiem (2026) extended this to $C^1$ monotone cat-map potentials, and Bjerklöv and Rajasekar (2025) proved large-coupling asymptotics over strongly expanding circle endomorphisms, and Peng, Wang, and Piao (2026) proved Anderson localization itself for $C^1$ monotone doubling-map potentials at sufficiently large coupling (and at both small and large coupling when the potential has zero mean) — while for the Bernoulli base with locally constant $f$, $Z_{\lambda f}$ is finite for every $\lambda$ (Part I, Corollary 7.8).

Problem Statement

Let $(\Omega, T)$ be a subshift of finite type and let $\mu$ be a fully supported $T$-ergodic probability measure with a local product structure (the known partial results assume the stronger bounded distortion property), with $T$ having a fixed point. Fix $0 < \alpha \le 1$ and a nonconstant sampling function $f \in C^\alpha(\Omega, \mathbb{R})$, and consider the ergodic family of Schrödinger operators

$[H_\omega u](n) = u(n+1) + u(n-1) + \lambda f(T^n\omega)\,u(n), \qquad \omega \in \Omega,$

on $\ell^2(\mathbb{Z})$, with the associated Lyapunov exponent $L(E)$ and exceptional set $Z_{\lambda f} = \{E \in \mathbb{R} : L_{\lambda f}(E) = 0\}$. The problem — the surviving core of the Avila–Damanik–Zhang program after its second paper settled the locally constant and small-norm classes — has two linked components with one common objective: extending the Anderson-localization phenomenon from the bunched and locally constant classes to the full Hölder class.

(1) Localization in the general Hölder (equivalently, arbitrary-coupling) regime. Prove, or disprove, that for $\mu$-almost every $\omega$, the operator $H_\omega$ exhibits Anderson localization — pure point spectrum with exponentially decaying eigenfunctions; the natural strengthening is time-uniform exponential dynamical localization away from the exceptional energies — when $f$ is an arbitrary nonconstant Hölder function, neither locally constant nor subject to any supremum-norm restriction. Equivalently, for an arbitrary but fixed nonconstant Hölder $f$ over any base in the stated class (the doubling map or Arnold's cat map with Lebesgue measure being the standard examples), establish localization for every coupling $\lambda > 0$, removing the smallness restriction of the known corollaries; a proof for one particular $f$ or one particular base would be a special case of this component, not the component itself. The recognized missing inputs are uniform positivity (or continuity) of $L(E)$ on compact energy intervals avoiding the discrete set $Z_{\lambda f}$, a uniform large deviation estimate for the transfer-matrix growth in the non-fiber-bunched regime, and an elimination of double resonances that can handle a possibly infinite discrete exceptional set.

(2) The exceptional set at large coupling. For a globally fiber bunched (small-norm) nonconstant Hölder $f$, determine the fate of $Z_{\lambda f}$ as $\lambda \to \infty$: is it finite — or empty — for all sufficiently large $\lambda$, or does there exist such an $f$ for which $Z_{\lambda f}$ is an infinite discrete set for some large $\lambda$? Currently only discreteness for every $\lambda$ and finiteness for small $\lambda$ are known, with the monotone, strongly expanding, and Bernoulli-locally-constant special cases settled as described in the background.

Component (1) is the spectral conclusion the program aims at; component (2) is its positivity-side twin, decisive because large coupling is precisely what drives a fixed sampling function out of the bunched class. The road map of the series identifies continuity of $L(E)$, a uniform large deviation estimate, and double-resonance elimination as the missing inputs; these are intermediate steps toward the stated goal rather than substitutes for it, and results obtained under re-imposed bunching, local-constancy, or small-coupling hypotheses fall outside the stated generality of the two components.

References

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