Cantor Spectrum Research Articles

Density of mode-locking property for quasi-periodically forced Arnold circle maps

AbstractWe show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39(5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J.146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math.348 (2019), 353–377] and Zhang (2020).

Zero measure spectrum for multi-frequency Schrödinger operators

Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.

Cantor spectrum for CMV matrices with almost periodic Verblunsky coefficients

We consider extended CMV matrices with analytic quasi-periodic Verblunsky coefficients with Diophantine frequency vector in the perturbatively small coupling constant regime and prove the analyticity of the tongue boundaries. As a consequence we establish that, generically, all gaps of the spectrum that are allowed by the Gap Labeling Theorem are open and hence the spectrum is a Cantor set. We also prove these results for a related class of almost periodic Verblunsky coefficients and present an application to suitable quantum walk models on the integer lattice.

Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

Honeycomb structures in magnetic fields

We consider the nearest-neighbour tight binding model of the honeycomb lattice in magnetic fields and discover surprizing new analytical results that fully explain fractal spectra and experimentally observed asymmetries in the density of states of molecular graphene. We describe a fractal Cantor spectrum for irrational magnetic flux through a honeycomb, and establish the existence of zero energy Dirac cones for each rational flux with fully explicit estimates on the cone angle. Our results give a substantially more refined description of subtleties in the de Haas–van Alphen and quantum Hall effects, and provide the first quantitative bounds on transport coefficients for the tight-binding model under disorder.

Cantor spectrum for CMV and Jacobi matrices with coefficients arising from generalized skew-shifts

AbstractWe consider continuous cocycles arising from CMV and Jacobi matrices. Assuming that the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$ -dense. This implies that the associated CMV and Jacobi matrices have a Cantor spectrum for a generic continuous sampling map.

Reducibility of Finitely Differentiable Quasi-Periodic Cocycles and Its Spectral Applications

In this paper, we prove the generic version of Cantor spectrum property for quasi-periodic Schrödinger operators with finitely smooth and small potentials, and we also show pure point spectrum for a class of multi-frequency \(C^k\) long-range operators on \(\ell ^2({\mathbb Z}^d)\). These results are based on reducibility properties of finitely differentiable quasi-periodic \(SL(2,{\mathbb R})\) cocycles. More precisely, we prove that if the base frequency is Diophantine, then a \(C^k\) \(SL(2,{\mathbb R})\)-valued cocycle is reducible if it is close to a constant cocycle, sufficiently smooth and the rotation number of it is Diophantine or rational with respect to the frequency.

Singular continuous Cantor spectrum for magnetic quantum walks

In this note, we consider a physical system given by a two-dimensional quantum walk in an external magnetic field. In this setup, we show that both the topological structure as well as its type depend sensitively on the value of the magnetic flux $\Phi$: while for $\Phi/(2{\pi})$ rational the spectrum is known to consist of bands, we show that for $\Phi/(2{\pi})$ irrational the spectrum is a zero-measure Cantor set and the spectral measures have no pure point part.

Construction of QuasiPeriodic Schrödinger Operators with Cantor Spectrum

It is well known that, for fixed Diophantine frequencies and generic small smooth or analytic quasiperiodic potentials, both continuous and discrete Schrodinger operators have Cantor spectrum. Although there have been several examples of Schrodinger operators with Cantor spectrum since Moser’s pioneering work (Bellissard, in: Luck, Moussa, Waldschmidt (eds) Number theory and physics (Les Houches, 1989), Springer, Berlin, 1990; Bellissard et al. in Phys Rev Lett 49:701–704, 1982; Damanik et al. in Ann Henri Poincare 15:1123–1144, 2014, J Spectr Theory 7:1101–1118, 2017; Moser in Comment Math Helv 56:198–224, 1981); however, so far there is no concrete quasiperiodic example in the continuous case, and there is no concrete quasiperiodic example in the discrete case besides the cosine-like potentials (Avila and Jitomirskaya in Ann Math 170:303–342, 2009; Puig in Commun Math Phys 244:297–309, 2004; Sinai in J Stat Phys 46:861–909; 1987; Wang and Zhang in Int Math Res Not 2017:2300–2336, 2017). In this paper, we present a strategy for explicitly constructing quasiperiodic Schrodinger operators with Cantor spectrum.

Cantor spectrum of graphene in magnetic fields

We consider a quantum graph as a model of graphene in magnetic fields and give a complete analysis of the spectrum, for all constant fluxes. In particular, we show that if the reduced magnetic flux Phi / 2pi through a honeycomb is irrational, the continuous spectrum is an unbounded Cantor set of Lebesgue measure zero.

Diagonal automorphisms of the 2-adic ring C*-algebra

The $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$ naturally contains a copy of the Cuntz algebra $\mathcal{O}_2$ and, a fortiori, also of its diagonal subalgebra $\mathcal{D}_2$ with Cantor spectrum. This paper is aimed at studying the group ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ of the automorphisms of $\mathcal{Q}_2$ fixing $\mathcal{D}_2$ pointwise. It turns out that any such automorphism leaves $\mathcal{O}_2$ globally invariant. Furthermore, the subgroup ${\rm Aut}_{\mathcal{D}_2}(\mathcal{Q}_2)$ is shown to be maximal abelian in ${\rm Aut}(\mathcal{Q}_2)$. Saying exactly what the group is amounts to understanding when an automorphism of $\mathcal{O}_2$ that fixes $\mathcal{D}_2$ pointwise extends to $\mathcal{Q}_2$. A complete answer is given for all localized automorphisms: these will extend if and only if they are the composition of a localized inner automorphism with a gauge automorphism.

Dry Ten Martini problem for the non-self-dual extended Harper’s model

In this paper we prove the dry version of the Ten Martini problem: Cantor spectrum with all gaps open, for the extended Harper’s model in the non-self-dual region for Diophantine frequencies.

Spectral homogeneity of discrete one-dimensional limit-periodic operators

We prove that a dense subset of limit periodic operators have spectra which are homogeneous Cantor sets in the sense of Carleson. Moreover, by using work of Egorova, our examples have purely absolutely continuous spectrum. The construction is robust enough to extend the results to arbitrary p-adic hulls by using the dynamical formalism proposed by Avila. The approach uses Floquet theory to break up the spectra of periodic approximants in a carefully controlled manner to produce Cantor spectrum and to establish the lower bounds needed to prove homogeneity.

Purely singular continuous spectrum for limit-periodic CMV operators with applications to quantum walks

We show that a generic element of a space of limit-periodic CMV operators has zero-measure Cantor spectrum. We also prove a Craig–Simon type theorem for the density of states measure associated with a stochastic family of CMV matrices and use our construction from the first part to prove that the Craig–Simon result is optimal in general. We discuss applications of these results to a quantum walk model where the coins are arranged according to a limit-periodic sequence. The key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a CMV matrix and its Schur function.

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter's butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila's global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequency-dependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila's Global Theory to the present setting, self-duality via the Fourier transform, and a Johnson-type theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.

A spectral alternative for continuous families of self-adjoint operators

We consider a continuous family of self-adjoint operators A_s in a separable Hilbert space, the parameter s being a point of a complete metric space S . It is well known that isolated simple eigenvalues (assuming that the operators are bounded and the mapping s\mapsto A_s is continuous in the norm sense) behave “well”: under small changes of the parameter they do not disappear and change continuously. Unlike this, the eigenvalues embedded in the essential spectrum can display a very “bad” behavior. It turns out, nevertheless, that if the set of eigenvalues is non-empty for a topologically rich (e.g., open) set of values of the parameter, then the (multi-valued) eigenvalue function has continuous branches. One application is as follows. Suppose a one-dimensional quasi-periodic Schrödinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues.

AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts

We consider continuous SL(2,R)-cocycles over a strictly ergodic home- omorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruc- tion to uniform hyperbolicity, then it can be C 0 -perturbed to become uniformly hyperbolic. For cocycles arising from Schrodinger operators, the obstruction van- ishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrodinger operator is a Cantor set. 1. Statement of the Results

On the spectrum of the one-dimensional Schrödinger operator

The spectral theory of the one-dimensional Schrodinger operator with a quasi-periodic potential can be fruitfully studied considering the corresponding differential system. In fact the presence of an exponential dichotomy for the system is equivalent to the statement that the energy $E$ belongs to the resolvent of the operator. Starting from results already obtained for the spectrum in the continuous case, we show that in the discrete case a generic bounded measurable Schrodinger cocycle has Cantor spectrum.

Cantor singular continuous spectrum for operators along interval exchange transformations

It is shown that Schrödinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.