We study Schrödinger operators on the real line whose potentials are generated by the Fibonacci substitution sequence and a rule that replaces symbols by compactly supported potential pieces. We consider the case in which one of those pieces is identically zero, and study the dimension of the spectrum in the large-coupling regime. Our results include a generalization of theorems regarding explicit examples that were studied previously and a counterexample that shows that the naïve generalization of previously established statements is false. In particular, in the aperiodic case, the local Hausdorff dimension of the spectrum does not necessarily converge to zero uniformly on compact subsets as the coupling constant is sent to infinity.

We consider the spectrum of the Almost Mathieu operator (AMO) and show that the moments of the restriction of the Lebesgue measure to the intersection spectrum Leb|_Σ_α,λ are polynomials in coupling λ with coefficients that are trigonometric polynomials in frequency α. The statement can be considered as a generalization of the Aubry-André formula for the measure of the spectrum of AMO. As a corollary, we obtain that the restriction of the Lebesgue measure to the intersection spectrum that we denote by μ^-_α, λ depends continuously on the parameters (frequency α and coupling λ) in weak-* topology. Moreover, we prove that the dependence is not just continuous but analytic in λ and C^∞ in α in a sense that an integral of an analytic test function φ(x) with respect to μ^-_α, λ has the same kind of dependence. In particular, this implies that the Lebesgue measure of the part of the spectrum Σ_α,λ that lies between two gaps depends analytically on the coupling constant λ and C^∞ on the frequency α in an open domain (away from the critical coupling λ=1) where these gaps do not bifurcate.

We consider discrete Schrödinger operators on ℓ 2 ( Z ) \ell ^2(\mathbb {Z}) with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as well as dynamical localization for this model. An important ingredient of the proof is a non-stationary version of the parametric Furstenberg Theorem on random matrix products, which is also of independent interest.

We consider discrete one-dimensional Schrödinger operators with random potentials obtained via a block code applied to an i.i.d. sequence of random variables. It is shown that, almost surely, these operators exhibit spectral and dynamical localization, the latter away from a finite set of exceptional energies. We make no assumptions beyond non-triviality, neither on the regularity of the underlying random variables, nor on the linearity, the monotonicity, or even the continuity of the block code. Central to our proof is a reduction to the non-stationary Anderson model via Fubini.

We consider one-parameter families of smooth circle cocycles over an ergodic transformation in the base, and show that their rotation numbers must be log-Hölder regular with respect to the parameter. As an immediate application, we get a dynamical proof of the one-dimensional version of the Craig–Simon theorem that establishes that the integrated density of states of an ergodic Schrödinger operator must be log-Hölder.

We provide an explicit formula for an increment of the fibered rotation number of a one-parameter family of circle cocycles over any ergodic transformation in terms of invariant measures. As an application, for a family of random dynamical systems on the circle, this gives a formula for an increment of the rotation number in terms of the stationary measures. In the case of projective Schrödinger cocycles associated with the Anderson Model, that provides a relation between the properties of the stationary measures on the projective space and the integrated density of states (IDS) of the corresponding family of operators. In particular, it gives a dynamical proof of Hölder regularity of the IDS in Anderson Model. Finally, we prove that the IDS for the Anderson Model with an ergodic background must be Hölder continuous.

We consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be Hölder continuous. The result is a vast generalization of the classical statement on Hölder continuity of stationary measures of random walks on linear groups.

We prove Central Limit Theorem for non-stationary random products of SL(2, ℝ) matrices, generalizing the classical results by Le Page and Tutubalin that were obtained in the case of iid random matrix products.

The aim of this note is to show the existence of a large family of Cantorvals arising in the projection description of primitive two-letter substitutions. This provides a common and naturally occurring class of Cantorvals.

We study the space of ergodic measures of the map f : T-2 -> T-2, f (x, y) = (x, x + y)(mod 1),and show that its structure is similar to the graph of Thomae's function.

This file is composed of questions that emerged or were of interest during the workshop "Interactions between Descriptive Set Theory and Smooth Dynamics" that took place in Banff, Canada on 2022.

We prove a version of pointwise Ergodic Theorem for non-stationary random dynamical systems. Also, we discuss two specific examples where the result is applicable: non-stationary iterated function systems and non-stationary random matrix products.

We consider Schrödinger operators in ℓ ^2(ℤ) whose potentials are given by the sum of an ergodic term and a random term of Anderson type. Under the assumption that the ergodic term is generated by a homeomorphism of a connected compact metric space and a continuous sampling function, we show that the almost sure spectrum arises in an explicitly described way from the unperturbed spectrum and the topological support of the single-site distribution. In particular, assuming that the latter is compact and contains at least two points, this explicit description of the almost sure spectrum shows that it will always be given by a finite union of non-degenerate compact intervals. The result can be viewed as a far reaching generalization of the well known formula for the spectrum of the classical Anderson model.

We consider Schrödinger operators in $$\ell ^2({\mathbb Z})$$ whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables (with some small-gap assumption for the support of the single-site distribution). The proof proceeds by extending a result about the presence of ground states for atypical realizations of the classical Anderson model, which we prove here as well and which appears to be new.

The paper considers the equivalence relation of conjugacy-by-homeomorphism on diffeomorphisms of smooth manifolds. In dimension 2 and above it is shown that there is no Borel method of attaching complete numerical invariants. In dimension 5 and above it is shown that the equivalence relation is not Borel, and in fact is complete analytic.

We prove a non-stationary analog of the Furstenberg Theorem on random matrix products (that can be considered as a matrix version of the law of large numbers). Namely, under a suitable genericity conditions the sequence of norms of random products of independent but not necessarily identically distributed $\SL(d, \mathbb{R})$ matrices grow exponentially fast, and there exists a non-random sequence that almost surely describes asymptotical behaviour of that sequence.

Abstract. We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980’s in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schrödinger operators are presented. On the one hand, we show that the Schrödinger operators on l(Z) with limit-periodic potential that have pure point spectrum form a dense subset in the space of all limit-periodic Schrödinger operators on l(Z). More generally, for any bounded potential, one can find an arbitrarily small limit-periodic perturbation so that the resulting operator has pure point spectrum. Our result complements the known denseness of absolutely continuous spectrum and the known genericity of singular continuous spectrum in the space of all limit-periodic Schrödinger operators on l(Z). On the other hand, we show that Schrödinger operators on l(Z) with arbitrarily small one-frequency quasi-periodic potential may have pure point spectrum for some phases. This was previously known only for one-frequency quasiperiodic potentials with ‖ · ‖∞ norm exceeding 2, namely the super-critical almost Mathieu operator with a typical frequency and phase. Moreover, this phenomenon can occur for any frequency, whereas no previous quasi-periodic potential with Liouville frequency was known that may admit eigenvalues for any phase.

We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe–Sommerfeld criterion for sums of Cantor sets which may be of independent interest.

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

In 1977 Philip Anderson shared the Nobel Prize in Physics with his doctoral thesis advisor John van Vleck and his collaborator Nevill Mott. The Nobel Prize was awarded “for their fundamental theoretical investigations of the electronic structure of magnetic and disordered systems”, or, in other words, for the discovery of what is nowadays called Anderson localization. In condensed matter physics, Anderson localization is the absence of diffusion of waves in a random (disordered) medium. A popular, though not quite equivalent, mathematical justification of (spectral) Anderson localization is pure point spectrum of the corresponding Schrödinger operator with random potential, along with exponentially decaying eigenfunctions. Many random models aside from Schrödinger operators with potentials given by iid random variables at each site of a finite-dimensional lattice have been considered, for example sparse random potentials, decaying random potentials, the “trimmed” Anderson model, and the Anderson model on regular trees. But most of the existing methods of proof either require some form of absolute continuity of the randomness (e.g., the Kunz-Souillard method, the fractional moment method, or spectral averaging), or use a highly involved and technically challenging machinery (e.g. multiscale analysis). At the same time, recently it became clear that theory of random matrix products can be used to provide more geometrical and transparent proofs of Anderson Localization, at least in 1D case (see the extended abstracts of talks by Jake Fillman, Victor Kleptsyn, Tom VandenBoom, and Xiaowen Zhu below). New results were obtained in higher dimensional case as well, see the extended abstract of Charles Smart below. The workshop was organized to bring together people who contributed to the recent progress in the field, as well as both experts and graduate students specializing in the areas that are directly related to random matrix products and/or Anderson Localization. The report consists of two sections. In the first one we collected some of the open problems that were presented at the problem sessions that were organized during the workshop. Also, the participants were requested to provide the extended abstracts of the talks, that would contain the formal statements of the main results that were presented, and would be useful as an ”entry point” to the subject. The second one consists of the extended abstracts that were provided by the participants.