Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystals

Abstract

The spectrum of one-dimensional discrete Schrödinger operators associated to strictly ergodic dynamical systems is shown to coincide with the set of zeros of the Lyapunov exponent if and only if the Lyapunov exponent exists uniformly. This is used to obtain Cantor spectrum of zero Lebesgue measure for all aperiodic subshifts with uniform positive weights. This covers, in particular, all aperiodic subshifts arising from primitive substitutions including new examples as e.g. the Rudin-Shapiro substitution. Our investigation is not based on trace maps. Instead it relies on an Oseledec type theorem due to A. Furman and a uniform ergodic theorem due to the author.

14 pages

Cited by in corpus (19)

• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Cantor Spectrum for Schrödinger Operators with Potentials arising from Generalized Skew-shifts
• Some Characterizations of Domination
• Learning-based Intelligent Surface Configuration, User Selection, Channel Allocation, and Modulation Adaptation for Jamming-resisting Multiuser OFDMA Systems
• Spectra of Discrete Schrödinger Operators with Primitive Invertible Substitution Potentials
• Nature of Protected Zero Energy States in Penrose Quasicrystals
• Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential
• Quantitative continuity of singular continuous spectral measures and arithmetic criteria for quasiperiodic Schrödinger operators
• On the Hausdorff dimension of the spectrum of Thue-Morse Hamiltonian
• Spectrum of Lebesgue measure zero for Jacobi matrices of quasicrystals
• Parametric Furstenberg Theorem on Random Products of $SL(2,\mathbb{R})$ matrices
• Spectral properties of Schrödinger operators associated to almost minimal substitution systems
• Simple Toeplitz subshifts: combinatorial properties and uniformity of cocycles
• "Mixed spectral nature" of Thue--Morse Hamiltonian
• Orthogonal polynomials on the unit circle: New results
• The spectrum of period-doubling Hamiltonian
• Linearly repetitive Delone sets Delone sets with finite local complexity: Linear repetitivity versus positivity of weights
• Gibbs-like measure for spectrum of a class of one-dimensional Schrödinger operator with Sturm potentials
• Quantum Hamiltonians with Quasi-Ballistic Dynamics and Point Spectrum